stochastic integration in banach spaces and …veraar/research/papers/thesis.pdfstochastic...

Stochastic Integration in Banach Spacesand

Applications to Parabolic Evolution Equations

Stochastic Integration in Banach Spacesand

Applications to Parabolic Evolution Equations

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus Prof. dr. ir. J.T. Fokkema,voorzitter van het College voor Promoties,

in het openbaar te verdedigenop dinsdag 19 december 2006 om 15.00 uur

door

Mark Christiaan VERAAR

wiskundig ingenieurgeboren te Delft.

Dit proefschrift is goedgekeurd door de promotores:

Prof. dr. Ph.P.J.E. ClementProf. dr. J.M.A.M. van Neerven

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. dr. Ph.P.J.E. Clement, Technische Universiteit Delft, promotorProf. dr. J.M.A.M. van Neerven, Technische Universiteit Delft, promotorProf. dr. Z. Brzezniak, University of YorkProf. dr. B. de Pagter, Technische Universiteit Delft,Prof. dr. S.M. Verduyn Lunel, Universiteit LeidenProf. dr. L.W. Weis, Universitat KarlsruheProf. dr. J. Zabczyk, Polish Academy of Sciences

Het onderzoek beschreven in dit proefschrift is mede gefinancierd door de Neder-landse Organisatie voor Wetenschappelijk Onderzoek (NWO), onder projectnummer639.032.201.Het Stieltjes Instituut heeft bijgedragen in de drukkosten van het proefschrift.

ISBN-10: 90-9021380-5ISBN-13: 978-90-9021380-4

Copyright c© 2006 by M.C. Veraar

Preface

This Ph.D. thesis was written during my period as a Ph.D. student at the DelftUniversity of Technology. In this preface I would like to express my thanks to all whocontributed in some way to the realization of this thesis.

First of all, I would like to thank my advisor Jan van Neerven, for his guidancein the last four years. He introduced me to the subject of this thesis and he taughtme a lot of mathematics. It has been great working together and I hope that thecollaboration will continue in the future. I wish to thank Philippe Clement for manyuseful discussions and his interest in my research. I also would like to thank Lutz Weisfor his kind hospitality during my stays at the Universitat Karlsruhe (Germany) andthe University of South-Carolina (USA). The always pleasant discussions have beenvery important to me, and I hope that the fruitful collaboration will be continued.

I wish to thank Onno van Gaans, Stefan Geiss, Tuomas Hytonen, Roland Schnaubeltand Mario Walther for helpful discussions. I am grateful to Sonja Cox for reading theintroduction and propositions of this thesis and for her comments.

I would like to thank all the colleagues of the Analysis group in Delft for the pleasantworking atmosphere. In particular I am grateful to my colleagues Guido Sweers,Erik Koelink and Ben de Pagter, and to my fellow Ph.D. students Anna Dall’Acqua,Timofey Gerasimov, Wolter Groenevelt, Jan Maas and Yvette van Norden. Manythanks go to the colleagues in Karlsruhe, Markus Duelli, Bernhard Haak, CorneliaKaiser, Peer Kunstmann and Jan Zimmerschied, for all their help during my stays. Ialso would like to thank the colleagues in South-Carolina for their kind hospitality.

I am grateful to the organizers of the Tulka internet seminar in the years 2003-2006 for the stimulating courses. I wish to thank the people of the European ResearchTraining Network “Evolution Equations for Deterministic and Stochastic Systems”(HPRN-CT-2002-00281) for the interesting workshops in Brest, Delft, Jena, Pisa andVienna.

During my research I was financially supported by the Netherlands Organizationfor Scientific Research (NWO), under project number 639.032.201, and by the MarieCurie Fellowship Program in Karlsruhe.

Delft, October 2006 Mark Christiaan Veraar

Contents

1 Introduction 1

1.1 Some history on stochastic integration . . . . . . . . . . . . . . . . . . 1

1.2 γ-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Stochastic integration in UMD spaces . . . . . . . . . . . . . . . . . . . 5

1.4 Ito’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Some history on stochastic evolution equations . . . . . . . . . . . . . . 9

1.6 Stochastic evolution equations in UMD spaces . . . . . . . . . . . . . . 11

1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7.1 Space-time white noise in higher dimensions . . . . . . . . . . . 14

1.7.2 Zakai’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Preliminaries 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Gaussian random variables . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 UMD spaces and decoupling inequalities . . . . . . . . . . . . . . . . . 23

2.5 Type and cotype . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 R-Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

I Stochastic integration 31

3 γ-Spaces 33

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Embeddings for spaces of type p and cotype q . . . . . . . . . . . . . . 38

3.4 Stochastic integration of operator-valued functions . . . . . . . . . . . . 43

3.5 Measurability of γ-valued functions . . . . . . . . . . . . . . . . . . . . 48

3.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

ix

4 Integration w.r.t. cylindrical Brownian motion 534.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.2 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 544.3 Decoupling inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 Characterizations for the Lp-case . . . . . . . . . . . . . . . . . . . . . 574.5 Characterizations for the localized case . . . . . . . . . . . . . . . . . . 654.6 Criteria for stochastic integrability . . . . . . . . . . . . . . . . . . . . 734.7 Brownian filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.8 The Ito formula 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.9 The Ito formula 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.10 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Integration in randomized UMD spaces 955.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Randomized UMD spaces . . . . . . . . . . . . . . . . . . . . . . . . . 955.3 One-sided estimates for stochastic integrals . . . . . . . . . . . . . . . . 985.4 Necessity of the randomized UMD property . . . . . . . . . . . . . . . 995.5 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Integration w.r.t. continuous local martingales 1096.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.2 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 1106.3 Characterizations of integrability . . . . . . . . . . . . . . . . . . . . . 1116.4 Criteria for stochastic integrability . . . . . . . . . . . . . . . . . . . . 1186.5 The Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

II Stochastic equations 121

7 Basic concepts 1237.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2 Evolution families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.3 Deterministic convolutions . . . . . . . . . . . . . . . . . . . . . . . . . 1267.4 Measurability of stochastic convolutions . . . . . . . . . . . . . . . . . 1317.5 Solution concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.6 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

8 Equations in type 2 spaces 1418.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.2 Stochastic convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1428.3 Lipschitz coefficients and integrable initial values . . . . . . . . . . . . 1458.4 Lipschitz coefficients and general initial values . . . . . . . . . . . . . . 1498.5 Locally Lipschitz coefficients . . . . . . . . . . . . . . . . . . . . . . . . 152

x

8.6 Second order equation with colored noise . . . . . . . . . . . . . . . . . 1568.7 Elliptic equations with space-time white noise . . . . . . . . . . . . . . 1598.8 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

9 Equations in UMD spaces 1659.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.2 Convolutions in Besov spaces and γ-spaces . . . . . . . . . . . . . . . . 1669.3 Stochastic convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1729.4 L2

γ-Lipschitz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1789.5 Lipschitz coefficients and integrable initial values . . . . . . . . . . . . 1809.6 Lipschitz coefficients and general initial values . . . . . . . . . . . . . . 1879.7 Locally Lipschitz coefficients . . . . . . . . . . . . . . . . . . . . . . . . 1899.8 Examples with bounded generator . . . . . . . . . . . . . . . . . . . . . 1929.9 Laplacian in Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1939.10 Second order equation with colored noise . . . . . . . . . . . . . . . . . 1959.11 Elliptic equations with space-time white noise . . . . . . . . . . . . . . 1969.12 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

10 Strong solutions 19910.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910.2 Acquistapace-Terreni conditions . . . . . . . . . . . . . . . . . . . . . . 20010.3 The abstract problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.4 Reduction to a deterministic problem . . . . . . . . . . . . . . . . . . . 20210.5 The deterministic problem for time-varying domains . . . . . . . . . . . 20510.6 Second order equation with time-dependent domains . . . . . . . . . . 21010.7 The deterministic problem for constant domains . . . . . . . . . . . . . 21110.8 Zakai’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21310.9 Notes and comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

List of symbols 217

Bibliography 219

Index 229

Summary 233

Samenvatting 237

Curriculum Vitae 241

xi

Chapter 1

Introduction

Stochastic partial differential equations (SPDEs) of evolution type are usually modelledas ordinary stochastic differential equations (SDEs) in an infinite-dimensional statespace. In many examples such as the stochastic heat and wave equation, this viewpointmay lead to existence and uniqueness results and regularity properties. To model theequation in such a way one needs a stochastic integration theory for processes withvalues in an infinite-dimensional space. Since real-valued stochastic integration theoryextends directly to processes with values in Hilbert spaces, this is the class of spacesin which SPDEs are usually modelled. This approach has been considered by manyauthors using semigroup methods from the 70th’s up to now.

There are situations where it is more natural to model the SPDE in a functionspace which is not a Hilbert space but only a Banach space. The main problem forthis is to find a “good” stochastic integration theory for processes with values in aBanach space. In the 70th’s and 80th’s several authors found negative results in thisdirection, and it turned out that the stochastic integration theory for Hilbert spacesdoes not extend to the Banach space setting.

In this thesis we show that if one reformulates the integration theory for Hilbertspaces, then it does generalize to a certain class of Banach spaces. For this class ofspaces we give a complete description of the stochastically integrable processes and weshow that two-sided estimates for the stochastic integral hold. We develop a stochasticcalculus for the stochastic integral. In particular we show that the Ito formula holds.The results lead to new applications in SPDEs.

1.1 Some history on stochastic integration

Let W be a standard Brownian motion on a probability space (Ω,A,P) with filtration

F = (Ft)t≥0. The stochastic integral∫ 1

0φ(t) dW (t) of a function φ ∈ L2(0, 1) was first

introduced by Wiener in [134] (also see [106]) and an isometry holds:

E∣∣∣ ∫ 1

0

φ(t) dW (t)∣∣∣2 = ‖φ‖2

L2(0,1) for φ ∈ L2(0, 1). (1.1.1)

1

2 Chapter 1. Introduction

The case of adapted and joint measurable processes φ : [0, 1] × Ω → R was firstconsidered by Ito in [60]. In this case the Ito isometry holds:

E∣∣∣ ∫ 1

0

φ(t) dW (t)∣∣∣2 = ‖φ‖2

L2((0,1)×Ω) for adapted φ ∈ L2((0, 1)× Ω). (1.1.2)

It is easy to check that (1.1.1) and (1.1.2) can be generalized to the case whereφ takes values in a Hilbert space. This is due to the fact that the norm comes froman inner product. If φ takes values in a general Banach space (E, ‖ · ‖), stochasticintegration is more difficult. In [137] for E = lp with p ∈ [1, 2), Yor constructeda uniformly bounded measurable and adapted process φ : [0, 1] × Ω → E which isnot stochastically integrable with respect to W in any reasonable sense. A possibledefinition of stochastic integrability can be found in Definition 1.3.1. Later on, in[117] Rosinski and Suchanecki showed that there even exists a uniformly boundedmeasurable function φ : [0, 1] → E that is not stochastically integrable with respectto W .

On the other hand, for the class of Banach spaces E that have type 2 (e.g. Lp-spaces for 2 ≤ p <∞), Hoffmann-Jørgensen and Pisier showed in [59] that thereexists a constant C such that

E∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥2

≤ C2‖φ‖2L2(0,1;E) for φ ∈ L2(0, 1;E). (1.1.3)

In other words a one-sided version of (1.1.1) holds for Banach spaces with type 2.In the case that E has martingale type 2 or equivalently up to renorming, E is 2-uniformly smooth (e.g. Lp-spaces for 2 ≤ p < ∞), one can even show that there is aconstant C such that

E∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥2

≤ C2|φ‖2L2((0,1)×Ω;E) for adapted φ ∈ L2((0, 1)× Ω;E). (1.1.4)

This has been considered by several authors (cf. [13, 19, 42, 102] and the referencestherein).

The results (1.1.3) and (1.1.4) are very useful but they do not give two-sidedestimates or characterizations of integrability. In [117] Rosinski and Suchaneckicharacterized the functions φ : [0, 1] → E that are stochastically integrable withrespect to W in terms of Gaussian characteristic functions. This result has beenextended to more general integrators by Rosinski in [116].

In [50] Garling showed that for UMD spaces E and all p ∈ (1,∞) there existconstants c, C such that for all adapted step processes φ : [0, 1]× Ω → E, one has

cEE∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p ≤ E

∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p ≤ C EE

∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p. (1.1.5)

Here W is a Brownian motion on a different probability space (Ω, A, P). The expec-tations E and E stand for integration over Ω and Ω respectively. The inequalities of

1.2. γ-Spaces 3

the form (1.1.5) are referred to as decoupling inequalities. Garling also showed thatthe validity of (1.1.5) for all adapted step processes φ implies the UMD property.

The UMD property is defined in terms of unconditionality of martingales differencesequences. Examples of UMD Banach spaces are the Lp-spaces for p ∈ (1,∞). UMDspaces are important for vector-valued harmonic analysis. This is due to the fact thatthe Hilbert transform is bounded on Lp(R;E) for all p ∈ (1,∞) if E is a UMD space.This result has been proved by Burkholder in [27]. In [50] Garling has given ashort proof based on (1.1.5). The converse also holds, i.e. if for some p ∈ (1,∞) theHilbert transform is bounded on Lp(R;E), then E is a UMD space. This has beenshowed by Bourgain in [17].

UMD spaces also are the “right” spaces for studying maximal Lp-regularity forevolution equations. For this class of spaces Weis [133] has characterized maximal Lp-regularity in terms of R-boundedness of certain resolvents. For the proof he extendedthe Mikhlin multiplier theorem to a UMD space setting. The randomized notionR-boundedness will play an important role in this thesis as well.

In [88] McConnell has generalized (1.1.5) to so-called tangent martingale dif-ference sequences. This result has been proved independently in [57] by Hitczenko.Their proofs are based on the existence of certain biconcave functions related to theUMD property and constructed by Burkholder in [28]. Using the inequalities fortangent martingale difference sequences, McConnell has given sufficient conditions formeasurable and adapted processes φ : [0, 1] × Ω → E to be stochastically integrablewith respect to W . He showed that if almost all paths of φ are stochastically integrablewith respect to an independent Brownian motion, then φ is stochastically integrablewith respect to W .

In the above results no two-sided estimates for the stochastic integral are given. In[23] by Brzezniak and van Neerven and in [100] by van Neerven and Weis, theauthors have studied stochastic integrability of operator-valued functions Φ : [0, 1] →B(H,E) with respect to WH . Here H is a separable Hilbert space and WH is acylindrical Brownian motion. They have extended the results of [117] to the operator-valued situation and obtained an isometry in terms of γ-spaces. Before we explain thisin more detail we will discuss γ-spaces.

1.2 γ-Spaces

Let E be a Banach space and let H be a separable Hilbert space. Let (γn)n≥1 be aGaussian sequence, i.e. a sequence of independent standard Gaussian random variableson a probability space (Ω,A,P).

A bounded operator R ∈ B(H,E) is said to be γ-radonifying if there exists anorthonormal basis (hn)n≥1 of H such that

∑n≥1 γnRhn converges in L2(Ω;E). For

such an operator we define

‖R‖γ(H,E) :=(E

∥∥∥∑n≥1

γnRhn

∥∥∥2) 12.


This quantity does not depend on the sequence (γn)n≥1 and the basis (hn)n≥1, anddefines a norm on the space γ(H,E) of all γ-radonifying operators from H into E.Endowed with this norm, γ(H,E) is a Banach space and it defines an operator ideal.The space γ(H,E) is a closed linear subspace of the space of almost summing operatorsfrom H into E which was first introduced by Linde and Pietsch in [79].

Let (S,Σ, µ) be a separable measure space. We say that φ : S → E belongs toL2(S) scalarly if for all x∗ ∈ E∗, 〈φ, x∗〉 ∈ L2(S). For a strongly measurable functionφ : S → E which is scalarly in L2(S), consider the integral operator Iφ : L2(S) → Egiven by

Iφf =

∫S

φ(s)f(s) dµ(s), f ∈ L2(S).

This integral is well-defined as a Pettis integral. We define the γ-norm of φ as

‖φ‖γ(S;E) = ‖Iφ‖γ(L2(S),E).

This randomized norm or γ-norm has been introduced by Kalton and Weis in [67].For Hilbert spaces E one has ‖φ‖γ(S,E) = ‖φ‖L2(S;E) and for certain spaces E (forexample Lp-spaces) it is equivalent with a square-function norm. For this reason theγ-norm is sometimes referred to as a generalized square-function. The space γ(S;E)is not complete in general. However, it is a subspace of the Banach space γ(L2(S), E)in a canonical way.

Many classical results from Hilbert space theory do not generalize to Banach spacesin a direct way. In some cases however, a reformulation with γ-norms allows such ageneralization. An illustrative example of this is the Fourier-Plancherel formula. Fora function φ ∈ L1(R;E)∩L2(R;E) it is not true in general that the Fourier transform

φ satisfies the isometry ‖φ‖L2(R;E) = ‖φ‖L2(R;E). More precisely, if this holds for allfunctions φ, then E is isomorphic to a Hilbert space. However, for general Banachspaces E one still has

‖φ‖γ(R;E) = ‖φ‖γ(R;E).

We refer to Chapter 3 for more details.The γ-norms or generalized square-functions have turned out to be useful in various

fields of analysis. In [49, 67, 66] γ-norms have been used to study the H∞-calculus.In [55] applications to control have been obtained and in [62] the γ-spaces have beenapplied for wavelet decompositions. In [48] the randomized norms have been used inthe study of local Banach space theory.

At the end of the previous section we have already mentioned that γ-norms havebeen used in [23] and [100] to study stochastic integrability. We can now formulatethis result more precisely. For notational convenience we do not consider the operator-valued situation. The authors show that a function φ : [0, 1] → E is stochasticallyintegrable if and only if φ ∈ γ(0, 1;E). In this case the following isometry holds:

E∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥2

= ‖φ‖2γ(0,1;E). (1.2.1)

1.3. Stochastic integration in UMD spaces 5

Since (1.2.1) is an isometry, this seems to be the right generalization of (1.1.1) to theBanach space setting. The good structure of γ(0, 1;E) makes it possible to study thestochastic integral in great detail.

It is not difficult to see that one has a continuous embedding

L2(0, 1;E) → γ(0, 1;E) (1.2.2)

if and only if E has type 2. As a consequence of this and (1.2.1) one also obtains(1.1.3). We will extend this embedding result to spaces with type p where p ∈ [1, 2].Recall that for p < ∞, Lp-spaces have type minp, 2 and every Banach space hastype 1.

For s ∈ R, 1 ≤ p, q ≤ ∞ let Bsp,q(Rd;E) denote the vector-valued Besov space.

Theorem 1.2.1. Let E be a Banach space and let p ∈ [1, 2]. Then the space E has

type p if and only if Bdp− d

2p,p (Rd;E) → γ(Rd;E).

Theorem 1.2.1 will be proved in Section 3.3. A similar result holds for domainsS ⊂ Rd. As a consequence we obtain sufficient conditions for stochastic integrability.In particular if E has type p, then for all α > 1

p− 1

2, every function φ ∈ Cα([0, 1];E) is

stochastically integrable. Here Cα([0, 1];E) is the space of E-valued α-Holder contin-uous functions. Conversely, one can show that if α ∈ (0, 1

2) and every φ ∈ Cα([0, 1];E)

is stochastically integrable, then E has type p for all p ∈ (1, 2) with α < 1p− 1

2. In

particular for E = lp with p ∈ [1, 2), for all α < 1p− 1

2, there exists an α-Holder

continuous function φ : [0, 1] → E which is not stochastically integrable with respectto W . This extends examples in [117, 137].

1.3 Stochastic integration in UMD spaces

One of the aims of this thesis is to extend the results in [23] and [100] to operator-valued processes. Again for notational convenience we will explain the results in thecase of E-valued processes. Before we state the result let us give a precise definition ofthe stochastic integral. It is obvious how to define the stochastic integral for adaptedstep processes, and we will extend this to more general processes below.

Recall that for a Banach space E, L0(Ω;E) stands for the space of strongly mea-surable functions from Ω into E, with the topology induced by convergence in proba-bility. For a strongly measurable process φ : [0, 1]× Ω → E we say that φ belongs toL0(Ω;L2(0, 1)) scalarly if for all x∗ ∈ E∗, 〈φ, x∗〉 ∈ L0(Ω;L2(0, 1)).

Definition 1.3.1. For a strongly measurable and adapted process φ : [0, 1] × Ω → Ethat is scalarly in L0(Ω;L2(0, 1)) we say that φ is stochastically integrable with respectto W if there exists a sequence (φn)n≥1 of adapted step processes such that:

(i) for all x∗ ∈ E∗ we have limn→∞

〈φn, x∗〉 = 〈φ, x∗〉 in L0(Ω;L2(0, 1)),


(ii) there exists a process ζ ∈ L0(Ω;C([0, 1];E)) such that

ζ = limn→∞

∫ ·

0

φn(t) dW (t) in L0(Ω;C([0, 1];E)).

The process ζ is called the stochastic integral of φ with respect to W , notation

ζ =

∫ ·

0

φ(t) dW (t).

In this way∫ ·

0φ(t) dW (t) is uniquely defined up to indistinguishability. Moreover, it

is a continuous local martingale starting at zero.We will give a very short proof of the decoupling inequality (1.1.5) and using this

and (1.2.1) we will prove the following result.

Theorem 1.3.2. Let E be a UMD space. For a strongly measurable and adaptedprocess φ : [0, 1]× Ω → E which is scalarly in L0(Ω;L2(0, 1)) the following assertionsare equivalent:

(1) φ is stochastically integrable.

(2) There exists a process ζ ∈ L0(Ω;C([0, 1];E)) such that for all x∗ ∈ E∗, we have

〈ζ, x∗〉 =

∫ ·

0

〈φ(t), x∗〉 dW (t) in L0(Ω;C([0, 1])).

(3) φ ∈ γ(0, 1;E) almost surely.

(4) For almost all ω ∈ Ω, φ(·, ω) is stochastically integrable with respect to W .

In this case, for all p ∈ (1,∞),

E supt∈[0,1]

∥∥∥∫ t

0

φ(s) dW (s)∥∥∥p hp,E E‖φ‖pγ(0,1;E). (1.3.1)

The implication (4) ⇒ (1) is the mentioned result of McConnell from [88] whichhe proved with different methods.

The theorem does not contain the isometry (1.2.1) for processes. However, (1.3.1)is a good replacement since it is a two-sided estimate. The estimate (1.3.1) can beseen as a stochastic integral version of the Burkholder-Davis-Gundy inequalities. Itwill allow us to prove an E-valued version of the Brownian martingale representationtheorem. The equivalent condition in (2) is very useful and rather surprising. It showsessentially that a “Bochner type” integrability property is equivalent to a “Pettis type”integrability property.

As a consequence of (1.2.2) and (1.3.1) we recover (1.1.4) for UMD spaces E withtype 2. Recall that every UMD space with type 2 is a martingale type 2 space.However, there are martingale type 2 spaces without the UMD property.

1.3. Stochastic integration in UMD spaces 7

We will consider several other criteria for stochastic integrability. In Banach func-tion spaces, Lp-moments of the stochastic integral are estimated in terms of Lp-normsof square-functions. We also show that local Lp-martingales are stochastically in-tegrable. In particular the stochastic integral process

∫ ·0φ(t) dW (t) is stochastically

integrable with respect to W .

Theorem 1.3.2 will also be proved in the case that the integrator W is replacedby an arbitrary continuous local martingale M . It seems that the equivalence (4)does not extend to this situation. At some level it is surprising that integration ofa process with respect to a general continuous local martingale can be checked usingconvergence of Gaussian sums in (3). However, seeing the γ-norm as a generalizedsquare-function, this result is very natural. The proof of the result for arbitrarycontinuous local martingales is based on Theorem 1.3.2 and the result of Dambis,Dubins and Schwartz which says that every continuous local martingale is a timetransformed Brownian motion

Although we have been able to give a short proof of (1.1.5), the proof in [50] isstill very interesting. This comes from the fact that Garling proves the one-sidedestimates

E∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p ≤ C EE

∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p (1.3.2)

and

cEE∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p ≤ E

∥∥∥∫ 1

0

φ(t) dW (t)∥∥∥p (1.3.3)

from randomized versions of the UMD property. These randomized properties wereintroduced in [51] by Garling under the abbreviations LERMT (Lower Estimates forRandom Martingale Transforms) and UERMT (Upper Estimates for Random Mar-tingale Transforms). We will prefer to use the notation UMD− and UMD+, becauseit emphasizes the relation with UMD. Here the superscript − stands for Lower andthe superscript + stands for Upper. In [50], (1.3.2) has been proved for UMD− spacesand (1.3.3) has been proved for UMD+ spaces. Therefore, the characterizations inTheorem 1.3.2 have one-sided versions for UMD− and UMD+ space. For example theimplication (3) ⇒ (1) remains valid for UMD− spaces. This is interesting since L1 isa UMD− space.

It is natural to ask for which spaces E, (1.3.2) and (1.3.3) hold. We already notedthat in [50] it has been shown that if both inequalities hold, then E is a UMD space.We will extend this argument. However, we will only show that (1.3.2) implies theUMD− property for the special class of Paley-Walsh martingales. A similar resultholds for UMD+. It is unknown whether this implies the properties for arbitrarymartingales. For UMD spaces this is indeed the case.


1.4 Ito’s formula

One of the main tools of stochastic calculus is the Ito formula. We will also obtain anIto formula for our stochastic integral. The result is formulated below for stochasticintegrals of the form

∫ ·0,Φ(t) dWH(t), where H is a separable Hilbert space, Φ : [0, 1]×

Ω → B(H,E), and WH is a cylindrical Brownian motion.

Although, we have only defined the stochastic integral for E-valued processes in thelast section, we will briefly comment on the case of operator-valued processes. Firstof all, the process Φ is assumed to be H-strongly measurable and adapted, i.e. for allh ∈ H, Φh is strongly measurable and adapted. The cylindrical Brownian motion canbe interpreted as WH(t) =

∑n≥1 hn ⊗Wn, where (hn)n≥1 is an orthonormal basis for

H and (Wn)n≥1 is a sequence of independent standard Brownian motions. Now, onecan think of the stochastic integral of the process Φ as∫ ·

0

Φ(t) dWH(t) =∑n≥1

∫ ·

0

Φ(t)hn dWn(t),

where the series converses in L0(Ω;C([0, 1];E)). The integrable processes Φ canbe characterized as in Theorem 1.3.2, where one has to replace γ(L2(0, 1), E) byγ(L2(0, 1;H), E) and γ(0, 1;E) by γ(0, 1;H,E).

After this brief explanation on the stochastic integral for operator-valued processes,we return to the Ito formula. Let E and F be Banach spaces and let R ∈ γ(H,E)and T ∈ B(E,B(E,F )) be given. Let (hn)n≥1 be an arbitrary orthonormal basis forH and define the trace of T with respect to R as

TrRT :=∑n≥1

(TRhn)(Rhn).

This sum converges unconditionally and it does not depend on the choice of the or-thonormal basis.

We can now state the Ito formula which will be proved in Section 4.8.

Theorem 1.4.1. Let E and F be UMD spaces. Assume that f : R+ × E → F is ofclass C1,2. Let Φ : [0, 1] × Ω → B(H,E) be an H-strongly measurable and adaptedprocess which is stochastically integrable with respect to WH and assume that the pathsof Φ belong to L2(0, 1; γ(H,E)) a.s. Let ψ : [0, 1] × Ω → E be strongly measurableand adapted with paths in L1(0, 1;E) a.s. Let ξ : Ω → E be strongly F0-measurable.Define ζ : [0, 1]× Ω → E as

ζ = ξ +

∫ ·

0

ψ(s) ds+

∫ ·

0

Φ(s) dWH(s).

Then s 7→ D2f(s, ζ(s))Φ(s) is stochastically integrable and almost surely we have for

1.5. Some history on stochastic evolution equations 9

all t ∈ [0, 1],

f(t, ζ(t))− f(0, ξ) =

∫ t

0

D1f(s, ζ(s)) ds+

∫ t

0

D2f(s, ζ(s))ψ(s) ds

+

∫ t

0

D2f(s, ζ(s))Φ(s) dWH(s)

+1

2

∫ t

0

TrΦ(s)

(D2

2f(s, ζ(s)))ds.

(1.4.1)

Here the deterministic integrals are defined as Bochner integrals. The stochasticintegral is defined as in Definition 1.3.1. There is also a version of the Ito formula forUMD− spaces.

The proof of (1.4.1) is based on Theorem 1.3.2 (2) and standard approximationarguments. For martingale type 2 spaces E and F the version of the above result canbe found in [102] (also see [21]). Notice that Theorem 1.4.1 can also be applied tothe situation where f : E × E∗ → R is given by f(x, x∗) = 〈x, x∗〉. We do not knowhow such a result could be obtained with the Ito formula from [21, 102], unless E isisomorphic to a Hilbert space.

The careful reader has noticed the assumption that Φ belong to L2(0, 1; γ(H,E))a.s. in Theorem 1.4.1. This L2-assumption is unnatural in the sense that for theexistence of the stochastic integral in the definition of ζ this is not needed. Theassumption is needed in the approximation argument in the proof of the Ito formula.Although the condition is not very strong, we will show that at least in the specialcase where f : E × E∗ → R is as above, the L2-assumption on Φ can be omitted.

1.5 Some history on stochastic evolution equations

In the theory of deterministic partial differential equations of evolution type it iscommon to formulate partial differential equations as ordinary differential equations ina function space. This approach is also used for stochastic partial differential equations.Therefore, it is useful to study stochastic equations in an abstract formulation. Theabstract equation we will be considering is

dU(t) = A(t, U(t)) dt+B(t, U(t)) dWH(t), t ∈ [0, T ],

U(0) = u0.(1.5.1)

Here (U(t))t∈[0,T ] is the unknown and A and B are certain unbounded and non-linearoperators. The noise term is modelled with the cylindrical Brownian motion WH andu0 is the initial value which could be random.

We can distinguish two important methods for studying equations of type (1.5.1).One method is based on monotonicity of the operators under consideration (see [81]).This is a Hilbert space method and therefore the function space is usually a Hilbert


space. It has been applied in the stochastic case by Pardoux in [107] and Krylovand Rozovskiı in [71].

Semigroup methods are another way to study (1.5.1). Usually this method isapplied in Hilbert spaces, because real-valued stochastic integration theory generalizesto such spaces. Many authors have studied (1.5.1) with semigroup methods. Thisstarted with Curtain and Falb [33] and Dawson [38] and has been continued by DaPrato and collaborators (cf. [37] and the references therein). One of the importanttools that has been developed, is the factorization method from [35] by Da Prato,Kwapien and Zabczyk. This method allows one to prove space-time regularityresults for stochastic convolutions. For a detailed discussion we refer to the monographof Da Prato and Zabczyk [37] and the references therein.

In the papers [19] and [20] Brzezniak has studied particular cases of (1.5.1) in aBanach space E. In [19] maximal regularity results have been considered and in [20]semi-linear equations have been studied using fixed point methods. The stochasticintegral he uses is the one for martingale type 2 spaces E (see (1.1.4)). We will returnto this topic in Section 1.6.

In the linear case with additive noise, i.e.dU(t) = A(t)U(t) dt+B(t) dWH(t), t ∈ [0, T ],

U(0) = u0,(1.5.2)

it is possible to study (1.5.2) in an arbitrary Banach space. This is because (1.2.1)holds in every Banach space. The case where A is the generator of a strongly con-tinuous semigroup and B is constant, has been studied in [23] by Brzezniak andvan Neerven and in [100] by van Neerven and Weis. For analytic semigroupgenerators A, regularity of the solution of (1.5.2) has been studied by Dettweiler,van Neerven and Weis [43]. The situation where A depends on time and generatesa strongly continuous evolution family has been considered in [131]. There the factor-ization method of [35] has been extended to the case where (A(t))t∈[0,T ] satisfies theAcquistapace-Terreni conditions which have been introduced by Acquistapace andTerreni in [4] (see Section 10.2).

In the definition of a solution of a stochastic equation one can either fix the prob-ability space on which the processes live in advance or make this variable as well.The former is usually called a strong solution, whereas the latter is a martingale orweak solution. Martingale solutions are important since it allows one to get existenceresults for equations with less regular coefficients. However, in accordance with mostreferences so far, we will only consider strong solutions.

For the convenience of the reader we give some references to the theory on martin-gale solutions. In the Hilbert space setting martingale solutions have been studied byViot in [132] and continued by Metivier in [90] (also see [37, Chapter 8]). In theBanach space setting martingale solutions have been considered by Dettweiler in[42]. This is done for bounded operators and in martingale type 2 spaces. It has beenextended to the setting of unbounded operators in [22] by Brzezniak and Gatarek.

1.6. Stochastic evolution equations in UMD spaces 11

This has been generalized to the setting of UMD− spaces by Zimmerschied in [140].His methods are based on the integration theory developed in this thesis (see Section1.3) and a representation result of Ondrejat [105].

Another possible approach to study (1.5.1) is Lyons’s rough path analysis. Thishas been taken up in the Banach space setting for bounded operators in [77] byLedoux, Lyons and Qian.

1.6 Stochastic evolution equations in UMD spaces

We will consider the following class of semi-linear equations on a UMD Banach spaceE.

dU(t) = (A(t)U(t) + F (t, U(t))) dt+B(t, U(t)) dWH(t), t ∈ [0, T ],

U(0) = u0.(1.6.1)

Here (A(t))t∈[0,T ] is a family of densely defined closed operators on E that generates anevolution family (P (t, s))0≤s≤t≤T . The processes F and B are defined on certain spacesin between E and the domains of (A(t))t∈[0,T ]. We will define strong, variational, mildand weak solutions for (1.6.1). Here, strong and weak solution are not the probabilisticnotions we discussed in Section 1.5, but analytic solution concepts. The formula of amild solution is given by

U(t) = P (t, 0)u0 +

∫ t

0

P (t, s)F (s, U(s)) ds+

∫ t

0

P (t, s)B(s, U(s)) dWH(s).

We prove several equivalences between the solution types. Most of these equivalencesare part of mathematical folklore. In case of additive noise some of the conceptswere already considered by Chojnowska-Michalik in [30] (also see [37] and thereferences therein).

For Hilbert spaces E in the case that A generates a strong continuous semigroup,(1.6.1) has been considered by Da Prato and Zabczyk in [37, Chapter 7]. Existenceand uniqueness results have been obtained using fixed point arguments. The casewhere A depends on time has been considered by Seidler in [122]. For this purpose,he extended the factorization method from [35] to the non-autonomous setting underrestrictive assumptions on the fractional domain spaces. For UMD spaces E withtype 2 in the case that A generates an analytic semigroup, (1.6.1) has been studiedby Brzezniak. Again, this is based on fixed point arguments and the factorizationmethod.

As a first step we consider (1.6.1) in a UMD space E with type 2. We assume that(A(t))t∈[0,T ] satisfies the Tanabe conditions in [124, Section 5.2]. These condition are:A(t) generates an analytic semigroup in a uniform way in t ∈ [0, T ], each A(t) has thesame domain D and A : [0, T ] → B(D,E) is Holder continuous (see Section 7.2). Wewill extend the factorization method to our setting. We do not need any assumptions


on the fractional domain spaces and we prove the following result in Section 8.5. Fora ∈ [0, 1) let V ∞,loc

a (0, T ;E) be the space of processes φ : [0, T ]×Ω → (E,D)a,1 whichare adapted and have continuous paths. The space (E,D)a,1 is the real interpolationspace between E and D. If a = 0, we let (E,D)a,1 = E.

Theorem 1.6.1. Let E be a UMD space with type 2. Assume that (A(t))t∈[0,T ] satisfiesthe Tanabe conditions and that F and B are locally Lipschitz and of linear growth asin Section 8.5 (A2)′ and (A3)′ and that u0 ∈ (E,D) 1

2,1. Then there exists a unique

mild solution U of (1.6.1) in V ∞,loca (0, T ;E) and it has a version such that for almost

all ω ∈ Ω,t 7→ U(t, ω) ∈ Cλ([0, T ]; (E,D)a+δ,1),

where λ > 0 and δ ≥ 0 satisfy λ+ δ < min1− (a+ θF ), 12− (a+ θB).

The fixed point space we used is of the form

Lr(Ω;Lp(0, T ; (E,D)a,1)). (1.6.2)

Theorem 1.6.1 is applied to a second order equation with colored noise similar tothe equation considered in [119] by Sanz-Sole and Vuillermot. The equation ismodelled in Lp with p ∈ [2,∞). We also prove a maximal regularity result for thisequation. Secondly, we apply Theorem 1.6.1 to a general elliptic equation on boundeddomains with space-time white noise. This will be explained in more detail in Section1.7.1.

We will consider (1.6.1) on a general UMD− space as well. This will be difficultenough already for the case that A generates an analytic semigroup. We will extendthe factorization method to UMD− spaces using the γ-norm. The problem (1.6.1)will be studied using fixed point methods. Since we have to consider deterministic andstochastic convolutions we need Bochner and γ-norms. Therefore, our fixed point spacewill be the intersection of a Bochner and a γ-space. The space has to be small enoughin order to obtain existence and regularity properties for the stochastic convolution.

To be able to apply a fixed point argument we need to estimate both the determinis-tic and stochastic convolution in the Bochner and weighted γ norm. The deterministicconvolution can be estimated in the γ-norm thanks to a combination of Theorem 1.2.1and results for convolutions in Besov spaces. It will however give restrictions on Frelated to the type of the space E. The γ-norm of the stochastic convolution canbe estimated likewise or with Pisier’s property (α). The Lp-spaces with p ∈ [1,∞)satisfy this property. The notion of a Lipschitz function also has to be generalized toγ-norms. This leads to a randomized Lipschitz condition, which we call L2

γ-Lipschitz.The Lipschitz functions that arise in stochastic partial differential equations usuallysatisfy this condition. For p ∈ [1,∞), α ∈ [0, 1

2) and a ∈ R let V p,loc

α,a (0, T ;E) be thespace of processes as in Section 9.6. In Section 9.7 we will prove the following result.

Theorem 1.6.2. Let E be a UMD− space with property (α) and type q ∈ [1, 2].Assume that A generates an analytic and strongly continuous semigroup, F is locally

1.6. Stochastic evolution equations in UMD spaces 13

Lipschitz as in Section 9.7 (A2)′ and of linear growth as in (9.5.1), B is locally L2γ-

Lipschitz as in Section 9.7 (A3)′ and of linear growth as in (9.5.2) and u0 ∈ (E,D) 12,1.

Then for every α ∈ (0, 12) and p ∈ (2,∞) such that α ∈ (a + θB,

12− 1

p) and a + θF <

32− 1

q− 1

p, there exists a unique mild solution U of (1.6.1) in V p,loc

α,a (0, T ;E) and it hasa version such that for almost all ω ∈ Ω,

t 7→ U(t, ω) ∈ Cλ([0, T ]; (E,D)a+δ,1),

where λ > 0 and δ ≥ 0 satisfy λ+ δ < min12− (a+ θB), 1− (a+ θF ).

There is also a version of this result for spaces without property (α). This will giverestrictions on the function B related to the type of E.

As an application of Theorem 1.6.2 we will consider a perturbed heat equation inLp-spaces with p ∈ [1,∞). We will also extend the above examples for Lp-spaces withp ∈ [2,∞) to all p ∈ (1,∞).

The last class of equations we will consider is of the form:

dU(t) = A(t)U(t)dt+N∑n=1

BnU(t) dWn(t), t ∈ [0, T ],

U(0, x) = u0.

(1.6.3)

Here the linear operators (A(t))t∈[0,T ] are closed and densely defined on a Banachspace E, the operators (Bn)

Nn=1 are generators of strongly continuous groups on E,

and (Wn)Nn=1 are real-valued independent F -adapted Brownian motions. The main

problem here is that the (Bn)Nn=1 are unbounded with D(A(t)) ⊂

⋂Nn=1D(B2

n) andthat we want the solution to be a strong solution. Typically (A(t))t∈[0,T ] is a familyof second order differential operators and (Bn)n≥1 are first order operators. For theprecise definition of a strong solution we refer to Section 7.5.

The problem (1.6.3) has been considered by Da Prato, Iannelli and Tubaro[34] and Da Prato and Zabczyk [37, Section 6.5] in the autonomous case andin Hilbert spaces. They showed that (1.6.3) is equivalent to a deterministic problem,which they solve pathwise. We will extend their results to the non-autonomous settingand to UMD Banach spaces. This will be done with the Ito formula from Section 1.4. Inparticular we need this formula for f : E×E∗ → R given by f(x, x∗) = 〈x, x∗〉. To solvethe deterministic problem we will assume that (A(t))t∈[0,T ] satisfies the Acquistapace-Terreni conditions (AT1) and (AT2) introduced by Acquistapace and Terreni in [4](see Section 10.2). In Section 10.5 we will obtain the following result. The hypotheses(H1), (H2), (H3) and (H4) can be found in Section 10.3 and condition (K) is explainedin Section 10.5.

Theorem 1.6.3. Let E be a UMD Banach space. Assume that Hypotheses (H1),(H2), (H3) and (H4) are fulfilled and that (AT1), (AT2) and (K) are satisfied forA(t)− 1

2

∑Nn=1B

2n−µ for all µ ∈ R large enough. If u0 ∈ D(A(0)) almost surely, then

the problem (1.6.3) admits a unique strong solution U for which AU ∈ C([0, T ];E)almost surely.


We apply Theorem 1.6.3 to a second order equation with time varying boundaryconditions and to Zakai’s equation. Both equations are modelled in Lp-spaces withp ∈ (1,∞). The Zakai equation will be explained in more detail in Section 1.7.2.

Before Acquistapace and Terreni wrote their paper [4], they also studied (1.6.3)in Hilbert spaces in [3]. They have assumed that (A(t))t∈[0,T ] satisfies the Kato-Tanabeconditions [124, Section 5.3]. This gives technical problems for the deterministic prob-lem.

In [19], in the setting of martingale type 2 spaces E, Brzezniak has studied a classof equations which is more general than (10.1.1). For E = Lp(Rd) with 2 ≤ p < ∞,the existence of solutions with paths in L2(0, T ;W 2,p(Rd)) has been obtained. We donot know whether the techniques of [34] can be extended to the setting of martingaletype 2 spaces E, since this would require an extension of the Ito formula for the dualitymapping.

Another approach was taken by Krylov [70], who developed an Lp-theory for ageneral class of time-dependent parabolic stochastic partial differential equations onRd by analytic methods. The equations are non-linear and many coefficients are onlyassumed to be measurable. In this high generality he has been able to obtain a solutionwith paths in Lp(0, T ;W 2,p(Rd)) with p ∈ [2,∞).

1.7 Applications

In this section we will give two applications of the theory developed so far. The firstapplication will be concerned with a partial differential equation perturbed with aspace-time white noise in higher dimensions. We will obtain existence and uniquenessconditions and discuss several regularity results. In the second application we considersimilar questions for a special case of the Zakai equation.

1.7.1 Space-time white noise in higher dimensions

Let S ⊂ Rd be a bounded C∞-domain and consider

∂u

∂t(t, s) = A(t, s,D)u(t, s) + f(t, s, u(t, s))

+g(t, s, u(t, s))∂w

∂t(t, s), s ∈ S, t ∈ (0, T ],

Bj(s,D)u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], j = 1, . . . ,m (1.7.1)

u(0, s) = u0(s), s ∈ S.Here A is of the form

A(t, s,D) =∑

|α|≤2m

aα(t, s)Dα,

where D = (∂1, . . . , ∂d), and for j = 1, . . . ,m,

Bj(s,D) =∑|β|≤mj

bjβ(s)Dβ,

1.7. Applications 15

where 1 ≤ mj < 2m is an integer. For the principal part of A, Aπ(t, s,D) =∑|α|=2m aα(t, s)D

α of A we assume that there is a κ > 0 such that

(−1)m+1∑

|α|=2m

aα(t, s)ξα ≥ κ|ξ|2m, t ∈ [0, T ], s ∈ S, ξ ∈ Rd.

For |α| ≤ 2m the coefficients aα are in Cµ([0, T ];C(S)). For the coefficients of theboundary value operator assume that for j = 1, . . . ,m and |β| ≤ mj, bjβ ∈ C2m−mj(S).and (Bj)

mj=1 is a normal system of Dirichlet type, i.e. 0 ≤ mj < m (cf. [124, Section

3.7]).The functions f, g : [0, T ]×Ω×S×R → R are jointly measurable, and adapted in

the sense that for each t ∈ [0, T ], f(t, ·), g(t, ·) is Ft ⊗BS ⊗BR-measurable. Finally, wis a spatio-temporal white noise and u0 : S × Ω → R is a BS ⊗ F0-measurable initialvalue condition.

Consider the following condition:

(C) The functions f and g are locally Lipschitz in the fourth variable uniformly in[0, T ]× Ω× S, i.e. for each R > 0 the exists constants LRf and LRg such that

|f(t, ω, s, x)−f(t, ω, s, y)| ≤ LRf |x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R,

|g(t, ω, s, x)−g(t, ω, s, y)| ≤ LRg |x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R.

The functions f and g are of linear growth in the fourth variable uniformly in[0, T ]× Ω× S, i.e. the exists constants Cf and Cg such that

|f(t, ω, s, x)| ≤ Cf (1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R,

|g(t, ω, s, x)| ≤ Cg(1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R.

The main theorem of this section will be formulated in the terms of the spacesHs,pBj(S), Bs

p,1,Bj(S) and CsBj(S). The definitions of these spaces can be found in

Section 8.7.The problem (1.7.1) will be modelled as a stochastic evolution equation of the form

(1.6.1) in the space E = Lp(S) with p ∈ [2,∞). We say that (1.7.1) has a mild solutionif the corresponding functional analytic model (1.6.1) has a mild solution. The nexttheorem will be proved in Section 8.7 and is a consequence of Theorem 1.6.1.

Theorem 1.7.1. Assume dm< 2. Assume (C) and let p ∈ [2,∞) be such that d

2mp<

12− d

4m. If a ∈ [ d

2mp, 1

2− d

4m) is such that (8.7.2) holds for (a, p), and if for almost

all ω ∈ Ω we have u0(·, ω) ∈ B2map,1,Bj(S), then there exists a unique mild solution

u : [0, T ] × Ω → R of (1.7.1) such that almost surely t 7→ u(t, ·) is continuous as anB2map,1,Bj(S)-valued process.

Moreover, if for almost all ω ∈ Ω, u0(·, ω) ∈ Bm− d

2

p,1,Bj(S), then u has paths in

Cλ([0, T ];B2mδp,1,Bj(S)) for all δ > d

2mpand λ > 0 that satisfy δ + λ < 1

2− d

4mand

(8.7.2) for (δ, p).


By Sobolev embedding one obtains Holder continuous solutions in time and space.

For instance assume that u0 ∈ Cm− d

2

Bj (S). Then it follows in combination with [126,

Theorem 4.6.1] that the solution u has paths in Cλ([0, T ];C2mδBj(S)) for all δ, λ > 0

that satisfy δ + λ < 12− d

4m.

1.7.2 Zakai’s equation

Consider the following instance of the Zakai equation.

∂u

∂t(t, x) = A(t, x,D)u(t, x) +B(x,D)u(t, x)

dW (t)

dt, t ∈ [0, T ], x ∈ Rd

u(0, x) = u0(x), x ∈ Rd.(1.7.2)

Here

A(t, x,D) =d∑

i,j=1

aij(t, x)DiDj +d∑i=1

qi(t, x)Di + r(t, x),

B(x,D) =d∑i=1

bi(x)Di + c(x).

All coefficients are real-valued and we take aij, qi, r ∈ B([0, T ];C1b (R

d)), where Bstands for bounded. The coefficients aij, qi and r are µ-Holder continuous in time forsome µ ∈ (0, 1], uniformly in Rd. Furthermore we assume that the matrices (aij(t, x))i,jare symmetric, and there exists a constant ν > 0 such that for all t ∈ [0, T ], we have

d∑i,j=1

(aij(t, x)−

1

2bi(x)bj(x)

)λiλj ≥ ν

d∑i=1

λ2i , x ∈ Rd, λ ∈ Rd. (1.7.3)

For the coefficients of B we assume bi, c ∈ C2b (Rd). Finally, W is a standard Brownian

motion.The problem (1.7.2) will be modelled as a stochastic evolution equation of the form

(1.6.3) in the space E = Lp(S) with p ∈ (1,∞). We will say that (1.7.2) has a strongsolution if the corresponding functional analytic model (1.6.3) has a strong solution.The following theorem will be proved in Section 10.8 and is a consequence of Theorem1.6.3.

Theorem 1.7.2. Let p ∈ (1,∞) and assume the above conditions. If u0 ∈ W 2,p(Rd)almost surely, then there exists a unique strong solution u of (1.7.2) on [0, T ] withalmost all paths in Cα([0, T ];E) ∩ C([0, T ];W 2,p(Rd)) for all α ∈ (0, 1

2).

The case where A is time independent and p = 2 has been considered in [37,Example 6.31]. If one takes p large in Theorem 1.7.2 one can apply Sobolev embeddingto obtain Holder regularity of the solution.

1.8. Overview 17

1.8 Overview

This thesis consists of two parts. Part I consists of Chapters 3-6 and is devoted tostochastic integration theory. Chapters 7-10 represent Part II which is concerned withstochastic evolution equations.

In Chapter 2 we will explain some preliminary results on measurability, Gaussianrandom variables, UMD spaces, (co)type, R-boundedness and Besov spaces. In Chap-ter 3 we will consider γ-spaces and prove Theorem 1.2.1 which is the main result in[65]. We also give a construction of the stochastic integral for operator-valued func-tions and prove several preliminary results on γ-spaces. The stochastic integrationtheory for operator-valued processes will be presented in Chapter 4. This is based on[96, 95]. We also prove the Ito formula which is taken from [97]. In Chapter 5 weconsider stochastic integration and randomized UMD spaces. This chapter is basedon [129]. Integration with respect to arbitrary continuous local martingales will bestudied in Chapter 6 and is based on [130].

Some basic concepts for convolutions and stochastic evolution equations will bediscussed in Chapter 7. In Chapter 8 we study (1.6.1) for UMD spaces with type 2.The problem (1.6.1) for UMD− spaces will be considered in Chapter 9 and is basedon [98]. In Chapter 10 we study the problem (1.6.3), which comes from [97].

Chapter 2

Preliminaries

2.1 Introduction

In this preliminary chapter we will recall some definitions and results that we willfrequently use. In Section 2.2 we discuss several measurability concepts of functionstaking values in a metric space. Section 2.3 is concerned with Gaussian random vari-ables taking values in a Banach space. Many of the results in this thesis are formulatedfor Banach spaces with the UMD property. This is the subject of Section 2.4. We alsoprove a decoupling inequality for random sums taking values in a UMD space. Thisinequality is fundamental in Chapter 4. In Section 2.5 we recall the Banach spaceproperties (martingale) type and cotype. Moreover, two inequalities for vector-valuedrandom sums are discussed. In Section 2.6 we explain the notion of R-boundedness.This randomized boundedness condition will be useful at several places. In the finalSection 2.7 we define the vector-valued Besov spaces.

For some unexplained notations we refer to the List of symbols at the end of thethesis. Here the reader may also find the Index.

The scalar field of the vector spaces in this thesis will be the real numbers R, unlessstated otherwise.

2.2 Measurability

Let (S,Σ) be a measurable space and let (E, d) be a metric space.A function φ : S → E is called (Borel) measurable if for all B ∈ BE, we have

φ−1(B) ∈ Σ.A function φ : S → E is called strongly measurable if it is the pointwise limit of a

sequence of simple functions. We will often use the following measurability theorem.

Proposition 2.2.1 (Pettis’s measurability theorem). Let E be a complete metricspace. Let Γ be a set of real-valued continuous functions defined on E that separatespoints of E. For a function φ : S → E, the following assertions are equivalent:

(1) φ is strongly measurable.

19

20 Chapter 2. Preliminaries

(2) φ is separably valued and measurable.

(3) φ is separably valued and for all f ∈ Γ, f φ is measurable.

(4) There exists a sequence of countably valued measurable functions (φn)n≥1 suchthat φ = limn→∞ φn uniformly in S.

As a consequence the pointwise limit of a sequence of strongly measurable functionsis again strongly measurable.

Now assume that E is a Banach space. The Pettis measurability theorem is oftenapplied with Γ = E∗. If φ : S → E is such that 〈φ, x∗〉 is measurable for all x∗ ∈ E∗,we say that φ is weakly measurable. In this way a function φ : S → E is stronglymeasurable if and only if φ is separably valued and weakly measurable.

Let E1 and E2 be Banach spaces. An operator-valued function Φ : S → B(E1, E2)will be called E1-strongly measurable if for all x ∈ E1, the E2-valued function Φx isstrongly measurable. Observe that if Φ : S → B(E1, E2) is E1-strongly measurableand f : S → E1 is strongly measurable, then Φf is strongly measurable.

If (S,Σ, µ) is a measure space and φ : S → E is defined as an equivalence class offunctions, then we say that φ is strongly measurable if there is a version of φ which isstrongly measurable.

If (S,Σ, µ) is a finite measure space, we introduce the space L0(S;E) of all stronglymeasurable functions φ : S → E, identifying functions which are almost everywhereequal. This space is a complete metric space under the metric

dS(φ1, φ2) =

∫S

d(φ1(s), φ2(s)) ∧ 1 dµ(s).

For functions φ, φ1, φ2, . . . ∈ L0(S;E) we have φ = limn→∞ φn if and only if φ =limn→∞ φn in measure.

For a detailed study on vector-valued functions we refer to [46, Chapter II]. In par-ticular, we will use the Bochner and Pettis integral. For the definition and propertiesof the conditional expectations and martingale theory we refer to [46, Chapter V]. Weonly comment on the following simple extension of the Doob regularization theoremfor martingales to the vector-valued situation. In particular it gives a condition underwhich every martingale has a jointly measurable version.

Proposition 2.2.2. Let E be a Banach space and let (Ω,A,P) be a complete probabilityspace with a filtration F = (Ft)t∈R+ that satisfies the usual conditions. If M : R+×Ω →E is an F-martingale, then M has a cadlag version.

Proof. It suffices to consider the case that M is a martingale which is constant aftersome time T .

We can find a sequence of FT -simple functions (fn)n≥1 such that MT = limn→∞ fnin L1(Ω;E). Let (Mn)n≥1 be the sequence of martingales defined by Mn

t = E(fn|Ft).It follows from the real-valued case of Doob’s regularization theorem (cf. [63, Theorem

2.3. Gaussian random variables 21

7.27]) that each Mn has a cadlag version Mn. Furthermore, by [63, Proposition 7.15]and the contractiveness of the conditional expectation, for arbitrary δ > 0, we have

δ P(

supt∈[0,T ]

‖Mnt − Mm

t ‖ > δ)≤ E‖Mn

T −MmT ‖ → 0 if n,m→∞.

Hence (Mn)n≥1 is a Cauchy sequence in probability in the space (CL([0, T ];E), ‖ · ‖∞)of cadlag function on [0, T ]. Since CL([0, T ];E) is complete, Mn is convergent to someM in CL([0, T ];E) in probability. For all t ∈ [0, T ] and all δ > 0, we have

δP(‖Mt −Mt‖ > δ) ≤ δP(‖Mt − Mn

t ‖ >δ

2

)+ δP

(‖Mn

t −Mt‖ >δ

2

)≤ δP

(‖Mt − Mn

t ‖ >δ

2

)+ 2E‖Mn

t −Mt‖

≤ δP(‖Mt − Mn

t ‖ >δ

2

)+ 2E‖fn −MT‖.

Since the latter converges to 0 as n tends to infinity and δ > 0 was arbitrary, it followsthat Mt = Mt almost surely. This proves that M is the required version of M .

2.3 Gaussian random variables

Let E be a Banach space and let (Ω,A,P) be a probability space.A mapping η : Ω → E is called a random variable if it is strongly measurable. A

random variable is called Gaussian if for all x∗ ∈ E∗, 〈η, x∗〉 is a real-valued Gaussianrandom variable.

If η is an E-valued Gaussian random variable, then a theorem of Fernique saysthat there exists an ε > 0 such that Eeε‖η‖2 < ∞ (cf. [76, Section 2.7]). In particularall p-th moments of ξ are finite.

For an E-valued Gaussian random variable ξ we define its mean m ∈ E andcovariance operator Q ∈ B(E∗, E) as

m = Eξ, Qx∗ = E〈ξ −m,x∗〉(ξ −m).

Clearly, Q is a positive and symmetric operator. In this case we say that ξ hasdistribution N (m,Q). Conversely, an operator Q ∈ B(E∗, E) is called a Gaussiancovariance operator if there exists an E-valued Gaussian random variable ξ such thatQ is its covariance operator.

Let (γn)n≥1 be a Gaussian sequence, i.e. an i.i.d. sequence of standard real-valuedGaussian random variables. For a sequence (xn)n≥1 in E, the a.s. convergence, con-vergence in probability and Lp-convergence for some (all) p ∈ [1,∞) of∑

n≥1

γnxn


are equivalent and the limit is again an E-valued Gaussian random variable (cf. [76,Theorems 2.1.1 and 2.2.1]).

Let (rn)n≥1 be a Rademacher sequence, i.e. an i.i.d. sequence with

P(r1 = 1) = P(r1 = −1) =1

2.

For N ≥ 1 and x1, . . . , xN ∈ E, recall the Kahane-Khinchine inequalities (cf. [45,Section 11.1] or [76, Proposition 3.4.1]): for all p, q ∈ [1,∞), we have

(E∥∥∥ N∑n=1

rnxn

∥∥∥p) 1p hp,q (E

∥∥∥ N∑n=1

rnxn

∥∥∥q) 1q .

Also for a centered E-valued Gaussian random variable ξ the Kahane-Khinchine in-equalities (cf. [76, Corollary 3.4.1]) hold: for all p, q ∈ [1,∞), we have

(E‖ξ‖p)1p hp,q (E‖ξ‖q)

1q .

As a consequence one obtains the following convergence result for E-valued Gaussianrandom variables (see [117]). Let (ξn)n≥1 be a sequence of E-valued Gaussian randomvariables and ξ : Ω → E is a random variable. If limn→∞ ξn = ξ in probability, then ξis a Gaussian random variable and for all p ∈ [1,∞), limn→∞ ξn = ξ in Lp(Ω;E).

Let I be an index set. A mapping ζ : I × Ω → E is called a Gaussian process iffor all n ≥ 1 and all i1, i2, . . . , in ∈ I,

(ζi1 , ζi2 , . . . , ζin)

is an En-valued Gaussian random variable. Clearly, ζ is a Gaussian process if andonly if for all x∗ ∈ E∗, 〈ζ, x∗〉 is a Gaussian process.

A strongly measurable process W : R+ × Ω → E is called a Brownian motion iffor all x∗ ∈ E∗, 〈W,x∗〉 is a Brownian motion starting at 0. Let Q be the covarianceof W (1). For the process W we have

(1) W (0) = 0,

(2) W has a version with continuous paths,

(3) W has independent increments,

(4) For all 0 ≤ s < t <∞, W (t)−W (s) has distribution N (0, (t− s)Q).

In this situation we say that W is a Brownian motion with covariance Q. Notice thatevery process W that satisfies (3) and (4) has a path-wise continuous version (cf. [63,Theorem 3.23]).

Conversely, for a separable Banach space E and a Gaussian covariance operatorQ ∈ B(E∗, E), there exists a Brownian motion W with covariance Q. To see this wewill use some terminology from Chapter 3. Indeed, as in [16, Remark 3.3.8] let HQ be

2.4. UMD spaces and decoupling inequalities 23

the reproducing kernel Hilbert space or Cameron-Martin space (cf. [16, 37]) associatedwith Q and let i : HQ → E be the inclusion operator. Then ii∗ = Q and i ∈ γ(H,E).Let (hn)n≥1 be an orthonormal basis for HQ and let (Wn)n≥1 be independent real-valued Brownian motions. Then W (t) =

∑n≥1Wn(t)ihn converges in L2(Ω;E) and

satisfies (1), (3) and (4) and there is a version which satisfies (1)-(4).

2.4 UMD spaces and decoupling inequalities

Let (Ω,A,P) be a probability space. Let (rn)n≥1 be a Rademacher sequence on Ω. LetG0 = ∅,Ω and Gn = σ(rk, k = 1, . . . , n). Recall that a martingale difference sequence(dn)

Nn=1 is a Paley-Walsh martingale difference sequence if it is a martingale difference

sequence with respect to the filtration (Gn)Nn=0.A Banach space E is a UMD(p) space for p ∈ (1,∞) if for every N ≥ 1, every

martingale difference sequence (dn)Nn=1 in Lp(Ω, E) and every −1, 1-valued sequence

(εn)Nn=1, we have (

E∥∥∥ N∑n=1

εndn

∥∥∥p) 1p

.E,p

(E

∥∥∥ N∑n=1

dn

∥∥∥p) 1p. (2.4.1)

The smallest constant such that (2.4.1) holds is called the UMD(p) constant of Eand is denoted by βp(E). Similarly, we say that E is a UMDPW(p)-space if one onlyconsiders Paley-Walsh martingale difference sequences in the definition of UMD(p).The corresponding constant is denoted by βPW

p (E). In [85] it has been shown thatUMDPW(p) already implies UMD(p) and βp(E) = βPW

p (E). It was shown in [26]that if E is UMD(p) space for some p ∈ (1,∞), then E is a UMD(p) space for allp ∈ (1,∞). Spaces with this property will be referred to as UMD spaces. UMD standsfor unconditional martingale differences. Some properties of UMD spaces are:

• If E is UMD, then every closed subspace of F ⊂ E is UMD with βp(F ) ≤ βp(E)for all p ∈ (1,∞).

• E is UMD if and only if E∗ is UMD with βq(E∗) ≤ βp(E) for all p, q ∈ (1,∞)

with 1p

+ 1q

= 1.

• If E is UMD and F ⊂ E is a closed subspace, then the quotient space E/F isUMD with βp(E/F ) ≤ βp(E) for all p ∈ (1,∞).

• UMD spaces are super-reflexive.

• UMD spaces have non-trivial type (see Section 2.5).

Most “classical” reflexive spaces are examples of UMD space. For instance thereflexive ranges of the Lp-spaces, the Hardy spaces Hp and the Schatten classes Cp areUMD spaces.

Let (Ω, A, P) be a copy of (Ω,A,P). Below we will consider random variables onthe probability space (Ω × Ω,A ⊗ A,P ⊗ P). If ξ is a random variable on Ω, thenwe sometimes use an extension of ξ to Ω × Ω in the obvious way, (ω, ω) 7→ ξ(ω).


This random variable will again be denoted by ξ. A similar extension will be used forrandom variables on Ω. An independent copy of a random variable (or process) ξ onΩ is defined as ξ(ω, ω) = ξ(ω). Sometimes ξ will be used as a random variable on Ω.Expectations with respect to Ω will be denoted by E and expectation with respect toΩ will be denoted by E. Expectations with respect to Ω× Ω will be denoted by EE.

We prove a general decoupling inequality. Let F = (Fn)n≥0 be a filtration on Ω.Let (ηn)n≥1 be a real-valued sequence of centered F -adapted random variables suchthat for each n ≥ 1, ηn is independent of Fn−1. Let (ηn)n≥1 be an independent copy of(ηn)n≥1 and let F on Ω be a copy of F . Let (vn)n≥1 be an E-valued F ⊗F -predictablesequence, i.e. for each n ≥ 1, vn is Fn−1 ⊗ Fn−1-measurable.

Theorem 2.4.1. Let E be a UMD space and let p ∈ (1,∞). Then for all N ≥ 1,

EE∥∥∥ N∑n=1

ηnvn

∥∥∥p hp,E EE∥∥∥ N∑n=1

ηnvn

∥∥∥p. (2.4.2)

Proof. For n = 1, . . . , N define

d2n−1 := 12(ηnvn + ηnvn) and d2n := 1

2(ηnvn − ηnvn).

Then (dj)2Nj=1 is a martingale difference sequence with respect to the filtration (Gj)2N

j=1,where for n ≥ 1,

G2n = σ(Fn, Fn),G2n−1 = σ(Fn−1, Fn−1, ηn + ηn).

Indeed, (dn)2Nn=1 is (Gn)2N

n=1-adapted. For n = 1, . . . , N ,

EE(d2n+1|G2n) = 12vn+1(Eηn+1 + Eηn+1) = 0,

since ηn+1 and ηn+1 are independent of G2n and centered. For n = 1, . . . , N ,

EE(d2n|G2n−1) =1

2vnEE(ηn − ηn|G2n−1)

(i)=

1

2vnEE(ηn − ηn|ηn + ηn)

(ii)= 0.

Here (i) follows from the independence of σ(ηn, ηn) and σ(Fn−1, Fn−1) (cf. [115, SectionII.41]) and in (ii) we used that ηn and ηn are i.i.d. Since

N∑n=1

ηnvn =2N∑j=1

dj andN∑n=1

ηnvn =2N∑j=1

(−1)j+1dj,

the result follows from the UMD property applied to (dj)2Nj=1 and ((−1)j+1dj)

2Nj=1.

Remark 2.4.2. In the above setting one can also take (ηn)n≥1 and (ηn)n≥1 to be E-valued and (vn)n≥1 to be B(E,F )-valued, where E is an arbitrary Banach space and Fis a UMD space. If one assumes that each vn is an E-strongly Fn−1⊗Fn−1-measurable,then (2.4.2) still holds with a similar proof.

2.5. Type and cotype 25

2.5 Type and cotype

Let p ∈ [1, 2] and q ∈ [2,∞]. A Banach space E is said to have type p if there exists aconstant C ≥ 0 such that for all finite subsets x1, . . . , xN of E we have(

E∥∥∥ N∑n=1

rnxn

∥∥∥2) 12 ≤ C

( N∑n=1

‖xn‖p) 1

p.

The least possible constant C is called the type p constant of E and is denoted byTp(E). A Banach space E is said to have cotype q if there exists a constant C ≥ 0such that for all finite subsets x1, . . . , xN of E we have( N∑

n=1

‖xn‖q) 1

q ≤ C(E

∥∥∥ N∑n=1

rnxn

∥∥∥2) 12,

with the obvious modification in the case q = ∞. The least possible constant C iscalled the cotype q constant of E and is denoted by Cp(E).

Every Banach space has type 1 and cotype ∞ with constant 1. Therefore, we saythat E has non-trivial type (non-trivial cotype) if E has type p for some p ∈ (1, 2](cotype q for some 2 ≤ q < ∞). If a Banach space E has non-trivial type, it hasnon-trivial cotype. Hilbert spaces have type 2 and cotype 2 with constants 1. Forp ∈ [1,∞) the Lp-spaces have type minp, 2 and cotype maxp, 2.

The following two results on random sums will be very useful.

Lemma 2.5.1. Let E be a Banach space. Let (rk)Nn=1 be a Rademacher sequence and

let (ξn)Nn=1 be a sequence of independent symmetric real-valued random variables with

α := minn≤N

E|ξn| ∈ (0,∞). Then the following estimate holds for all p ∈ [1,∞) and all

choices of (xn)Nn=1 in E:(

E∥∥∥ N∑n=1

rnxn

∥∥∥p)1/p

≤ 1

α

(E

∥∥∥ N∑n=1

ξnxn

∥∥∥p)1/p

.

For a real-valued random variable ξ and q ∈ [1,∞) we define

‖ξ‖q,1 =

∫R+

P(|ξ| > t)1/q dt.

Note that if s > q, ‖ξ‖q,1 ≤ qs−q (E|ξ|

s)1/s.

Lemma 2.5.2. Let E be a Banach space with finite cotype q0, let p ∈ [1,∞) and letq = maxq0, p. Let (rn)

Nn=1 be a Rademacher sequence and let (ξn)

Nn=1 be an i.i.d.

sequence of symmetric random variables. Then there exists a constant C > 0 suchthat for all choices of (xn)

Nn=1 in E:(

E∥∥∥ N∑n=1

ξnxn

∥∥∥p)1/p

≤ ‖ξ1‖q,1C(E

∥∥∥ N∑n=1

rnxn

∥∥∥p)1/p

.


In the definitions of type and cotype the role of the Rademacher variables may bereplaced by Gaussian variables without altering the class of spaces under consideration.For type this follows from directly from Lemma 2.5.1, but for cotype this is a delicateproblem (see [87]).

The least constants arising from these equivalent definitions are called the Gaussiantype p constant and the Gaussian cotype q constant of E respectively, notation T γp (E)and Cγ

q (E).Every Hilbert space has both type 2 and cotype 2, and a famous result of Kwapien

asserts that up to isomorphism this property characterizes the class of Hilbert spaces(see [74]).

Let (S,Σ, µ) be a measure space that is non-trivial in the sense that it containsat least one set of finite positive measure. If 1 < p ≤ 2 and p ≤ r < ∞, thena Banach space E has type p if and only if Lr(S;E) has type p (cf. [45, Theorem11.12]). Similarly, if 2 ≤ q < ∞ and 1 ≤ r ≤ q, then E has cotype q if and only ifLr(S;E) has cotype q. Moreover,

Tp(L2(S;E)) = Tp(E), Cq(L

2(S;E)) = Cq(E). (2.5.1)

Let p ∈ [1, 2] and q ∈ [2,∞). A Banach space E is said to have martingale typep if there exists a constant C ≥ 0 such that for every N ≥ 1 and every martingaledifference sequence (dn)n≥1 in Lp(Ω;E) we have

E∥∥∥ N∑n=1

dn

∥∥∥p ≤ Cp

N∑n=1

E‖dn‖p.

The least possible constant C is called the martingale type p constant of E and isdenoted by MTp(E). A Banach space E is said to have martingale cotype q if thereexists a constant C ≥ 0 such that for every N ≥ 1 and every martingale differencesequence (dn)n≥1 in Lp(Ω;E) we have

N∑n=1

E‖dn‖q ≤ CqE∥∥∥ N∑n=1

dn

∥∥∥q.The least possible constant C is called the martingale cotype q constant of E and isdenoted by MCq(E). Usually, the notions martingale type and cotype are introducedfor martingales not necessarily staring at 0. By an easy randomization argument asin [26, Remark 1.1] one can see this gives the same definition.

Every Banach space has martingale type 1. By the Kahane-Khinchine inequalitiesit is clear that every martingale type p space has type p. Conversely, if E is a UMDspace, then every type p space has martingale type p. Similar statements hold formartingale cotype q. In particular, every UMD space has martingale type p andmartingale cotype q for some 1 < p ≤ 2 ≤ q < ∞. The converse is not true. Acounterexample is constructed in [112]. For a counterexample in the Banach latticesituation we refer to [17].

2.6. R-Boundedness 27

In [111] it has been proved that a space E has martingale type p (martingalecotype q) if and only if there is an equivalent norm on E which is p-uniformly smooth(q-uniformly convex).

2.6 R-Boundedness

Let (rn)n≥1 be a Rademacher sequence on a probability space (Ω,A,P) and let E1 andE2 be Banach spaces. A collection T ⊂ B(E1, E2) is said to be R-bounded if thereexists a constant M ≥ 0 such that(

E∥∥∥ N∑n=1

rnTnxn

∥∥∥2

E2

) 12 ≤M

(E

∥∥∥ N∑n=1

rnxn

∥∥∥2

E1

) 12,

for all N ≥ 1 and all sequences (Tn)Nn=1 in T and (xn)

Nn=1 in E1. The least constant

M for which this estimate holds is called the R-bound of T , notation R(T ). By theKahane-Khinchine inequalities, the role of the exponent 2 may be replaced by anyexponent 1 ≤ p < ∞ (at the expense of a possibly different constant). Replacingthe role of the Rademacher sequence by a Gaussian sequence, we obtain the relatednotion of γ-boundedness. Again by the Kahane-Khinchine inequalities, the role of theexponent 2 may be replaced by any exponent 1 ≤ p <∞. By an easy randomizationargument, every R-bounded family is γ-bounded and we have γ(T ) ≤ R(T ). If E1

has finite cotype, the notions of R-boundedness and γ-boundedness are equivalent (cf.Lemmas 2.5.1 and 2.5.2) and we have R(T ) hE1 γ(T ). If E1 and E2 are Hilbert spaces,both notions reduce to uniform boundedness and we have γ(T ) = R(T ) = supT∈T ‖T‖.

2.7 Besov spaces

In this section we recall the definition of Besov spaces using the so-called Littlewood-Paley decomposition. The Fourier transform f of a function f ∈ L1(Rd;E) will benormalized as

f(ξ) =1

(2π)d/2

∫Rd

f(x)e−ix·ξ dx, ξ ∈ Rd.

Let φ ∈ S (Rd) be a fixed Schwartz function whose Fourier transform φ is nonneg-ative and has support in ξ ∈ Rd : 1

2≤ |ξ| ≤ 2 and which satisfies∑

k∈Z

φ(2−kξ) = 1 for ξ ∈ Rd \ 0.

Such a function can easily be constructed (cf. [14, Lemma 6.1.7]). Define the sequence(ϕk)k≥0 in S (Rd) by

ϕk(ξ) = φ(2−kξ) for k = 1, 2, . . . and ϕ0(ξ) = 1−∑k≥1

ϕk(ξ), ξ ∈ Rd.


Similar as in the real case one can define S (Rd;E) as the usual Schwartz spaceof rapidly decreasing E-valued smooth functions on Rd. As in the real case this is aFrechet space. Let the space of E-valued tempered distributions S ′(Rd;E) be definedas the continuous linear operators from S (Rd) into E.

For 1 ≤ p, q ≤ ∞ and s ∈ R the Besov space Bsp,q(Rd;E) is defined as the space of

all E-valued tempered distributions f ∈ S ′(Rd;E) for which

‖f‖Bsp,q(Rd;E) :=

∥∥∥(2ksϕk ∗ f

)k≥0

∥∥∥lq(Lp(Rd;E))

is finite. Endowed with this norm, Bsp,q(Rd;E) is a Banach space, and up to an

equivalent norm this space is independent of the choice of the initial function φ. Thesequence (ϕk ∗ f)k≥0 is called the Littlewood-Paley decomposition of f associated withthe function φ.

The following continuous inclusions hold:

Bsp,q1

(Rd;E) → Bsp,q2

(Rd;E), Bs1p,q(Rd;E) → Bs2

p,q(Rd;E)

for all s, s1, s2 ∈ R, p, q, q1, q2 ∈ [1,∞] with q1 ≤ q2, s2 ≤ s1. Also note that

B0p,1(Rd;E) → Lp(Rd;E) → B0

p,∞(Rd;E).

If 1 ≤ p, q < ∞, then Bsp,q(Rd;E) contains the Schwartz space S (Rd;E) as a dense

subspace.Besov spaces on domains S ⊂ Rd can be defined by taking restrictions (cf. [11, 126]).

2.8 Notes and comments

The Pettis measurability theorem (Proposition 2.2.1) can be found in many books.The version we state has been proved in [128, Propositions I.1.9 and I.1.10]. Thevector-valued extension of Doob’s regularization theorem in Proposition 2.2.2 mightbe well-known. The case of Hilbert spaces E has been considered in [72] by Kunita.His proof uses basis expansions.

For more details on vector-valued random sums and Gaussian random variables werefer to the monographs [16, 76].

The class of UMD Banach spaces has been studied by Maurey [85], Pisier [112],Aldous [8] and Burkholder [26] and many others. In [26] a geometric character-ization of UMD spaces has been obtained. UMD spaces play an important role inharmonic analysis for vector-valued functions. This is due to the fact that the Hilberttransform on Lp(R;E) is bounded for some (for all) p ∈ (1,∞) if and only if E is aUMD space. The “if” part has been proved by Burkholder in [27] and the “onlyif” part is due to Bourgain in [17]. Details on UMD spaces can be found in [26, 29]and references given therein.

The decoupling inequality in Theorem 2.4.1 is actually valid for so-called tangentmartingale difference sequences. This has been proved in [88] by McConnell (and

2.8. Notes and comments 29

independently in [57] by Hitczenko). The proofs are based on the existence of acertain biconcave function constructed by Burkholder in [28]. A different proof hasbeen given by Montgomery-Smith in [91]. Here a representation technique has beenused to obtain the result directly from the UMD property. The proof we presented isa minor variation of a part of the argument in [91]. For more information on tangentsequences we refer to the work of Kwapien and Woyczynski (cf. [75, 76] and thereferences therein).

The notions of type and cotype have been studied in many papers. Let us mentionthe paper by Maurey and Pisier [87] and the references therein. For a detaileddiscussion on type and cotype we refer to the monographs [45, 78] and the surveyarticle [86] and the references given therein. Lemmas 2.5.1 and 2.5.2 are special casesof results proved by Kwapien and by Maurey and Pisier (cf. [78, Lemma 4.5and Proposition 9.14]). Martingale type and cotype have been studied by Pisier in[111, 114], where also the relation with uniformly convex and uniformly smooth spaceshas been given.

The notion of R-boundedness was introduced in [15] by Berkson and Gillespieand can be traced back to the work of Bourgain [18]. It was studied thoroughly byClement, de Pagter, Sukochev and Witvliet (see [31, 135]). In [133], Weisfound a relation between R-boundedness and so-called maximal Lp-regularity of solu-tions of evolution equations. By now it is well established that a large class of analyticsemigroups and resolvent operators associated with partial differential equations areR-bounded. For details we refer to [41, 73].

For the definition of the Besov space we followed the approach of Peetre. Thescalar case of this definition has been considered in [126, Section 2.3.2] and for thevector-valued situation we refer to [10, 54, 120, 127].

Part I

Stochastic integration

31

Chapter 3

γ-Spaces

3.1 Introduction

In this chapter (E, ‖ · ‖) is a Banach space and (H, [·, ·]) is a separable Hilbert spacewith dim(H) ≥ 1. The triple (Ω,A,P) denotes an underlying probability space.

We will study the space of γ-radonifying operators γ(H,E) or γ(L2(S;H), E) for ameasure space (S,Σ, µ). These spaces are defined via convergence of certain Gaussiansums and it is an operator ideal. In Section 3.2 we will give some elementary resultson γ-radonifying operators and we define the γ-norm of a function φ : S → E byidentifying φ with an integral operator. This plays a key role in our work and therefore,we give a formal explanation here.

For a function φ : S → E satisfying conditions (that will be made precise in Section3.2) consider the integral operator Iφ : L2(S) → E given by

Iφf =

∫S

φ(s)f(s) ds, f ∈ L2(S)

and define the γ-norm of φ as

‖φ‖γ(S;E) = ‖Iφ‖γ(L2(S),E).

This randomized norm or γ-norm has been introduced by Kalton and Weis in [67].For Hilbert spaces E one has ‖φ‖γ(S,E) = ‖φ‖L2(S;E) and for certain spaces E (forexample Lp-spaces) it is equivalent with a square-function norm. For this reason theγ-norm is sometimes referred to as a generalized square-function.

Many classical results from Hilbert space theory do not generalize to Banach spacesin a direct way. However, in some cases a reformulation with γ-norms allows such ageneralization. An illustrative example of this is the Fourier-Plancherel formula. Fora function φ ∈ L1(R;E)∩L2(R;E) it is not true in general that the Fourier transform

φ satisfies the isometry ‖φ‖L2(R;E) = ‖φ‖L2(R;E). More precisely, if this holds for allfunctions φ, then E is isomorphic to a Hilbert space. However, for general Banachspaces E one still has

‖φ‖γ(R;E) = ‖φ‖γ(R;E). (3.1.1)

33

34 Chapter 3. γ-Spaces

The γ-norms or generalized square-functions turn out to be useful in various fieldsof mathematics. In [49, 67, 66] γ-norms have been used to study the H∞-calculus. In[55] applications to control have been obtained and in [62] the γ-spaces have been usedfor wavelet decompositions. Also in stochastic integration theory (see [23, 100, 117])the generalized square-function is important. This will be explained in Section 3.4.There we give a complete construction of the stochastic integral for operator-valuedfunctions with respect to a cylindrical Brownian motion.

In Section 3.3 some embedding results for γ-spaces will be obtained in the casethat E has type p or cotype q. In Section 3.5 we study measurability properties offunctions with values in γ(H,E) and γ(L2(S;H), E).

3.2 Definitions and properties

Let (γn)n≥1 be a Gaussian sequence.A bounded operator R ∈ B(H,E) is said to be γ-radonifying if there exists an

orthonormal basis (hn)n≥1 of H such that∑

n≥1 γnRhn converges in L2(Ω;E). Wethen define

‖R‖γ(H,E) :=(E

∥∥∥∑n≥1

γnRhn

∥∥∥2) 12.

This number does not depend on the sequence (γn)n≥1 and the basis (hn)n≥1, anddefines a norm on the space γ(H,E) of all γ-radonifying operators from H into E.Endowed with this norm, γ(H,E) is a Banach space, which is separable if E is separa-ble. As explained in Section 2.3, the convergence of

∑n≥1 γnRhn in Lp, in probability

and a.s. are all equivalent.If R ∈ γ(H,E) and x∗ ∈ E∗, then

‖R∗x∗‖ ≤ ‖R‖γ(H,E)‖x∗‖ (3.2.1)

and ‖R‖ ≤ ‖R‖γ(H,E). The finite rank operators in B(H,E) are in γ(H,E), and theyform a dense subset of γ(H,E). If E is a Hilbert space, then γ(H,E) = C2(H,E)isometrically, where C2(H,E) is the space of Hilbert-Schmidt operators.

The following ideal property will be useful.

Proposition 3.2.1 (Ideal property). Let H1 and H2 be separable Hilbert spaces andlet E1 and E2 be Banach spaces. If S ∈ B(H2, H1), R ∈ γ(H1, E1), and T ∈ B(E1, E2),then T R S ∈ γ(H2, E2) and

‖T R S‖γ(H2,E2) ≤ ‖T‖ ‖R‖γ(H1,E1) ‖S‖.

As a consequence of this one has the following extension procedure from [67]. IfH1 and H2 are Hilbert spaces, then every bounded operator T : H1 → H2 induces abounded operator T : γ(H1, E) → γ(H2, E) by the formula

TR := R T ∗

3.2. Definitions and properties 35

and we have

‖T‖B(γ(H1,E),γ(H2,E)) = ‖T‖B(H1,H2). (3.2.2)

We shall frequently use the following convergence result:

Lemma 3.2.2. If the T1, T2, . . . ∈ B(H) and T ∈ B(H) satisfy

(1) supn≥1 ‖Tn‖ <∞,

(2) T ∗h = limn→∞ T ∗nh for all h ∈ H,

then for all R ∈ γ(H,E), R T = limn→∞R Tn in γ(H,E).

Proof. By the estimate ‖R S‖γ(H,E) ≤ ‖R‖γ(H,E)‖S‖ for S ∈ B(H) and (1), itsuffices to consider finite rank operators R ∈ γ(H,E). For such an operator, sayR =

∑kj=1 hj ⊗ xj, we may estimate

‖R (T − Tn)‖γ(H,E) ≤k∑j=1

‖xj‖‖T ∗hj − T ∗nhj‖.

By (2), the right-hand side tends to zero as n→∞.

The next proposition characterizes when an operator R : H → E is γ-radonifyingin the case that E is a Banach function spaces with finite cotype.

Proposition 3.2.3. Let E be a Banach function space on a σ-finite measure space(S,Σ, µ) with finite cotype. Let H be a separable Hilbert space. For an operator R :H → E the following assertions are equivalent:

(1) R ∈ γ(H,E).

(2) For some (for all) orthonormal bases (hn)n≥1 of H, s 7→( ∑

n≥1 |Rhn(s)|2) 1

2is

in E.

(3) There is a g ∈ E such that for all h ∈ H, for almost all s ∈ S, |(Rh)(s)| ≤‖h‖Hg(s).

(4) There is a strongly measurable function κ : S → H such that s 7→ ‖κ(s)‖Hbelongs to E and for almost all s ∈ S, Rh(s) = [κ(s), h].

Moreover, in this case

‖R‖γ(H,E) hE

∥∥∥( ∑n≥1

|Rhn|2) 1

2∥∥∥E

=∥∥‖κ‖H∥∥

E≤ ‖g‖E. (3.2.3)


Proof of Proposition 3.2.3. (1)⇔(2): This follows from the fact that for a Banachfunction space E with finite cotype (cf. [80, Theorem 1.d.6 and Corollary 1.f.9] and[78, Lemma 4.5 and Proposition 9.14]) one has

∥∥∥( N∑n=1

|xn|2) 1

2∥∥∥E

hE

(E

∥∥∥ N∑n=1

γnxn

∥∥∥2

E

) 12

for x1, . . . , xN ∈ E. This also gives the first part of (3.2.3).

(2)⇒(3): Let g ∈ E be defined as g =( ∑

n≥1 |Rhn|2) 1

2. For h =

∑Nn=1 anhn we

have

|Rh(s)| =∣∣∣ N∑n=1

anRhn(s)∣∣∣ ≤ ( N∑

n=1

|an|2) 1

2( N∑n=1

|Rhn(s)|2) 1

2 ≤ ‖h‖Hg(s), s ∈ S.

For general h ∈ H, one can do an approximation argument.(3)⇒(4): Let (fn)n≥1 be such that H0 = fn : n ≥ 1 is dense in H and closed

under taking Q-linear combinations. Let A ∈ Σ be a null-set such that for all s ∈ Aand for all h ∈ H0, |Rh(s)| ≤ g(s)‖f‖H <∞ and H0 3 h 7→ Rh(s) is Q-linear. By theRiesz representation theorem for a fixed s ∈ A, the mapping H0 3 h→ Rh(s) has aunique extension to an element κ(s) ∈ H with Rh(s) = [h, κ(s)] for all h ∈ H0. Byapproximation one can show that for all h ∈ H, for almost all s ∈ S, Rh(s) = [h, κ(s)].For s ∈ A, we have

‖κ(s)‖H = sup‖h‖H≤1,h∈H0

|[h, κ(s)]| = sup‖h‖H≤1,h∈H0

|Rh(s)| ≤ g(s).

Letting κ(s) = 0 for s ∈ A, this gives (4) and the last inequality in (3.2.3).(4)⇒(2): Let (hn)n≥1 be an orthonormal basis for H. Let A ∈ Σ be a null-set

such that for all s ∈ A and all n ≥ 1, Rhn(s) = [hn, κ(s)]. It follows that for s ∈ Awe have ( ∑

n≥1

|Rhn(s)|2) 1

2=

( ∑n≥1

|[hn, κ(s)]|2) 1

2= ‖κ(s)‖H .

This gives (2) and the middle equality of (3.2.3).

Let (S,Σ, µ) be a separable measure space (i.e. Σ is generated by a countablecollection of subsets of S). Let Φ : S → B(H,E) be H-strongly measurable. We saythat Φ belongs to L2(S;H) scalarly if for all x∗ ∈ E∗, Φ∗x∗ ∈ L2(S;H). Such a functionΦ represents an operator R ∈ γ(L2(S;H), E) if for all x∗ ∈ E∗ and f ∈ L2(S;H) wehave

〈Rf, x∗〉 =

∫S

[f(s),Φ∗(s)x∗] dµ(s).

The above notion will be abbreviated by Φ ∈ γ(S;H,E) and we write

‖Φ‖γ(S;H,E) = ‖R‖γ(L2(S;H),E).


If we want to emphasize which measure we use we write γ(S, µ;H,E) for γ(S;H,E).Spaces of the type γ(S;H,E) will be usually referred to as γ-spaces. The number‖Φ‖γ(S;H,E) is called the γ-norm.

Let (S,Σ, µ) be a separable σ-finite measure space. If Φ : S → B(H,E) is an H-strongly measurable function which is in L2(S;H) scalarly, we may define the Pettisintegral operator IΦ ∈ B(L2(S;H), E) as

IΦf =

∫S

Φ(s)f(s) dµ(s), f ∈ L2(S;H). (3.2.4)

This is well-defined (cf. [110, Theorem 3.4]) and notice that supx∗∈BE∗‖Φ∗x∗‖L2(S;H) =

‖IΦ‖. Obviously, Φ ∈ γ(S;H,E) if and only if IΦ ∈ γ(L2(S;H), E). In this case Φrepresents IΦ. If it is not clear which measure on S is used, we will write IµΦ for IΦ.

Notice that the extension procedure explained before (3.2.2) can be applied forinstance to operators T ∈ B(L2(S;H)). In particular the Fourier-Plancherel formulafor γ-spaces (3.1.1) from the introduction may be obtained in this way.

If Φ : S → B(H,E) is of the form Φ =∑N

n=1 1SnRn, where (Sn)Nn=1 are disjoint sets

in Σ with µ(Sn) ∈ (0,∞) and Rn ∈ γ(H,E) for n = 1, . . . , N , then Φ ∈ γ(S;H,E)and

‖Φ‖2γ(S;H,E) = E

∥∥∥ N∑n=1

∑m≥1

γmn(µ(Sn))12Rnhm

∥∥∥2

. (3.2.5)

Here (γmn)m,n≥1 is a doubly indexed Gaussian sequence and (hm)m≥1 is an arbitrary

orthonormal basis for H. Indeed, let fn = (µ(Sn)− 1

21Sn for n = 1, . . . , N and let(fn)n≥N+1 be such that (fn)n≥1 is an orthonormal basis for L2(S). Then (fn⊗hm)m,n≥1

is an orthonormal basis for L2(S;H) and

IΦfn ⊗ hm = (µ(Sn))12Rnhm

for n = 1, . . . , N and m ≥ 1 and zero otherwise. Therefore, (3.2.5) follows from theabove definitions.

The following multiplier result for γ-spaces from [67] will be useful.

Proposition 3.2.4. Let S be a locally compact metric space without isolated pointsand let µ be a positive Borel measure on S. Let E1, E2 be Banach spaces and let Hbe a separable Hilbert space. Let N : S → B(E1, E2) be an E1-strongly measurablefunction whose range is γ-bounded by a constant K. Then for all Φ ∈ γ(S;H,E1) wehave that NΦ belongs to γ(S;H,E2) and

‖NΦ‖γ(S;H,E2) ≤ K‖Φ‖γ(S;H,E1). (3.2.6)

Conversely, if dim(H) ≥ 1 and N : S → B(E1, E2) is an E1-strongly continuousfunction and K is a constant such that (3.2.6) holds for all functions Φ ∈ γ(S;H,E1),then the range of N is γ-bounded with γ(N(s) : s ∈ S) ≤ K.


3.3 Embeddings for spaces of type p and cotype q

In this section we will discuss several embedding results for γ(H,E)-valued functions.Some measurability properties of such functions will be explained in Section 3.5.

The following result is well-known (cf. [59, 117] in the case that H = R and [101]for general H).

Proposition 3.3.1. Let (S,Σ, µ) be a separable σ-finite measure space that does notconsist of finitely many atoms. Then the following assertions hold:

(1) E has type 2 if and only if the mapping Φ 7→ IΦ extends to a continuous embed-ding L2(S; γ(H,E)) → γ(L2(S;H), E).

(2) E has cotype 2 if and only if the mapping IΦ 7→ Φ extends to a continuousembedding γ(L2(S;H), E) → L2(S; γ(H,E)).

The norms of the embeddings in (1) and (2) can be estimated by the type 2 constantT2(E) and the cotype 2 constant C2(E) of E.

The next theorem is an embedding result for spaces with type p and cotype qin the case that S = Rd and µ is the Lebesgue measure. For the definition of thevector-valued Besov spaces Bs

p,q(Rd;E) we refer to Section 2.7.

Theorem 3.3.2. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞.

(1) E has type p if and only if the mapping I : Φ 7→ IΦ extends to a continuousembedding

B( 1

p− 1

2)d

p,p (Rd; γ(H,E)) → γ(L2(Rd;H), E).

(2) E has cotype q if and only if the mapping I : IΦ 7→ Φ extends to a continuousembedding

γ(L2(Rd;H), E) → B( 1

q− 1

2)d

q,q (Rd; γ(H,E)).

The norms of the embeddings in (1) and (2) can be estimated by 2(Tp(E))2 and2(Cq(E))2, where Tp(E) and Cq(E) stand for the type p and cotype q constants ofE.

The proof of Theorem 3.3.2 is based on two lemmas. Recall that the vector-valuedFourier transform is defined in Section 2.7.

Let (S,Σ, µ) be a measure space. For a bounded operator R : L2(S;H) → E anda set S0 ∈ Σ we define R|S0 : L2(S;H) → E by

R|S0f := R(1S0f).

Note that if R ∈ γ(L2(S;H), E), then by the right-ideal property R|S0 is an elementof γ(L2(S;H), E) and

‖R|S0‖γ(L2(S;H),E) ≤ ‖R‖γ(L2(S;H),E).

3.3. Embeddings for spaces of type p and cotype q 39

Lemma 3.3.3. Let (S,Σ, µ) be a separable measure space and let (Sj)j≥1 ⊂ Σ be apartition of S.

(1) Let E have type p ∈ [1, 2]. Then for all R ∈ γ(L2(S;H), E) we have

‖R‖γ(L2(S;H),E) ≤ Tp(E)( ∑j≥1

‖R|Sj‖pγ(L2(S;H),E)

) 1p.

(2) Let E have cotype q ∈ [2,∞]. Then for all R ∈ γ(L2(S;H), E) we have

‖R‖γ(L2(S;H),E) ≥ Cq(E)−1( ∑j≥1

‖R|Sj‖qγ(L2(S;H),E)

) 1q.

Proof. (1): We may assume that µ(Sj) > 0 for all j. Since L2(S) is separable, wemay choose an orthonormal basis (gjk)j,k≥1 for L2(S) in such a way that for each j thesequence (gjk)k≥1 is an orthonormal basis for L2(Sj). Let (hm)m≥1 be an orthonormalbasis for H. Let (γmjk)m,j,k≥1 and (r′j)j≥1 be a triple-indexed Gaussian sequence anda Rademacher sequence on probability spaces (Ω,A,P) and (Ω′,A′,P′), respectively.By a standard randomization argument,

‖R‖γ(L2(S;H),E) =(E

∥∥∥ ∑j,k≥1

∑m≥1

γmjkR(gjk ⊗ hm)∥∥∥2) 1

2

=(E

∥∥∥ ∑j,k≥1

∑m≥1

γmjkR|Sj(gjk ⊗ hm)

∥∥∥2) 12

=(E′

∥∥∥∑j≥1

r′j∑k≥1

∑m≥1


∥∥∥2

L2(Ω;E)

) 12

≤ Tp(L2(Ω;E))

( ∑j≥1

∥∥∥∑k≥1

∑m≥1


∥∥∥pL2(Ω;E)

) 1p

= Tp(E)( ∑j≥1

‖R|Sj‖pγ(L2(S;H),E)

) 1p,

where we used (2.5.1) in the last line.(2): This is proved similarly.

Lemma 3.3.4. Let 1 ≤ p ≤ 2 ≤ q ≤ ∞.

(1) Let E have type p. If Φ ∈ S (Rd; γ(H,E)) satisfies supp Φ ⊂ [−π, π]d, thenΦ ∈ γ(Rd;H,E) and

‖Φ‖γ(Rd;H,E) ≤ Tp(E)‖Φ‖Lp(Rd;γ(H,E)).


(2) Let E have cotype q. If Φ ∈ S (Rd; γ(H,E)) satisfies supp Φ ⊂ [−π, π]d, then

‖Φ‖γ(Rd;H,E) ≥ Cq(E)−1‖Φ‖Lq(Rd;γ(H,E)).

Proof. LetQ := [−π, π]d. We consider the functions gn(x) = (2π)−d/2ein·x with n ∈ Zd,x ∈ Q, which define an orthonormal basis for L2(Q). Let (hm)m≥1 be an orthonormalbasis for H. Let (γmn)m,n≥1 and (r′n)n≥1 be a doubly-indexed Gaussian sequence anda Rademacher sequence on probability spaces (Ω,A,P) and (Ω′,A′,P′), respectively.

(1): As in (3.2.4) define the bounded operators IΦ : L2(Rd;H) → E and IbΦ :

L2(Rd;H) → E by

IΦf :=

∫Rd

Φ(x)f(x) dx, IbΦf :=

∫Rd

Φ(x)f(x) dx.

In case E is a real Banach space we consider its complexification in the second defi-nition. By the assumption on the support of Φ we may identify I

bΦ with a boundedoperator from L2(Q;H) to E of the same norm. Obviously, we have I

bΦ(gn⊗·) = Φ(n).By a standard randomization argument it follows that for any finite subset F ⊂ Zd,(

E∥∥∥ ∑n∈F

∑m≥1

γmnIbΦ(gn ⊗ hm)∥∥∥2) 1

2=

(E

∥∥∥ ∑n∈F

∑m≥1

γmnΦ(n)hm

∥∥∥2) 12

=(E′

∥∥∥ ∑n∈F

rn∑m≥1

γmnΦ(n)hm

∥∥∥2

L2(Ω;E)

) 12

≤ Tp(L2(Ω;E))

( ∑n∈F

∥∥∥ ∑m≥1

γmnΦ(n)hm

∥∥∥pL2(Ω;E)

) 1p

= Tp(E)( ∑n∈F

‖Φ(n)‖pγ(H,E)

) 1p,

where we used (2.5.1) in the last line. We obtain that IbΦ ∈ γ(L2(Q;H), E). By the

identification made above it follows that IbΦ ∈ γ(L2(R;H), E) and

‖IbΦ‖γ(L2(Rd;H),E) = ‖I

bΦ‖γ(L2(Q;H),E) ≤ Tp(E)( ∑n∈Zd

‖Φ(n)‖pγ(H,E)

) 1p.

From (3.2.2) it follows that

‖Φ‖γ(Rd;H,E) = ‖IΦ‖γ(L2(Rd;H),E) = ‖IbΦ‖γ(L2(Rd;H),E) ≤ Tp(E)

( ∑n∈Zd

‖Φ(n)‖pγ(H,E)

) 1p.

For t ∈ R := [0, 1]d put Φt(s) = Φ(s+ t). Then supp Φt ⊂ Q and

‖Φ‖γ(Rd;H,E) = ‖Φt‖γ(Rd;H,E) ≤ Tp(E)( ∑n∈Zd

‖Φt(n)‖pγ(H,E)

) 1p.

3.3. Embeddings for spaces of type p and cotype q 41

By raising both sides to the power p and integrating over R we obtain

‖Φ‖γ(Rd;H,E) ≤ Tp(E)( ∫

R

∑n∈Zd

‖Φt(n)‖pγ(H,E) dt) 1

p= Tp(E)

( ∫Rd

‖Φ(s)‖pγ(H,E) ds) 1

p.

(2): This is proved similarly. Note that by part (1) (with p = 1) we have Φ ∈γ(Rd;H,E).

We are now prepared for the proof of Theorem 3.3.2. Recall that the Schwartzfunctions φ and ϕk, k ≥ 1, are defined in Section 2.7.

Proof of Theorem 3.3.2. (1): First we prove the ‘only if’ part and assume that E hastype p. Let Φ ∈ S (Rd; γ(H,E)) and let Φk := ϕk ∗Φ. Putting Ψk(x) := Φk(2

−kx) wehave Ψk ∈ S (Rd; γ(H,E)) and

supp Ψk ⊂ ξ ∈ Rd : 0 ≤ |ξ| ≤ 2 ⊂ [−π, π]d.

Hence from Lemma 3.3.4 we obtain Φk ∈ γ(Rd;H,E) and

‖Φk‖γ(Rd;H,E) = 2−kd/2‖Ψk‖γ(Rd;H,E)

≤ 2−kd/2Tp(E)‖Ψk‖Lp(Rd;γ(H,E)) = 2kdp− kd

2 Tp(E)‖Φk‖Lp(Rd;γ(H,E)).

Using the Lemma 3.3.3, applied to the decompositions (S2k)k∈Z and (S2k+1)k∈Z ofRd \ 0, we obtain, for all n ≥ m ≥ 0,

∥∥∥ 2n∑k=2m

Φk

∥∥∥γ(Rd;H,E)

≤ (Tp(E))2( n∑j=m

2( 2jdp− 2jd

2)p‖Φ2j‖pLp(Rd;γ(H,E))

) 1p

+ (Tp(E))2( n−1∑j=m

2((2j+1)d

p− (2j+1)d

2)p‖Φ2j+1‖pLp(Rd;γ(H,E))

) 1p.

Estimating sums of the form∑2n+1

k=2m,∑2n

k=2m+1, and∑2n+1

k=2m+1 in a similar way, itfollows that Φ ∈ γ(Rd;H,E) and

‖Φ‖γ(Rd;H,E) ≤ 2(Tp(E))2‖Φ‖B

( 1p−

12 )d

p,p (Rd;γ(H,E)).

Since S (Rd; γ(H,E)) is dense in B( 1

p− 1

2)d

p,p (Rd; γ(H,E)) it follows that the mapping

Φ 7→ IΦ extends to a bounded operator I from B( 1

p− 1

2)d

p,p (Rd; γ(H,E)) into γ(Rd;H,E)of norm ‖I‖ ≤ 2(Tp(E))2.

It remains to check I is injective. Let Φ ∈ B( 1

p− 1

2)d

p,p (Rd; γ(H,E)) be such that IΦ =0. Let x∗ ∈ E∗ and h ∈ H be arbitrary. One easily checks that Φ∗x∗ = (IΦ)∗x∗ = 0in L2(Rd;H). Therefore, 〈Φh, x∗〉 = 0 a.e. Since Φ : Rd → γ(H,E) is stronglymeasurable, this implies that Φ = 0.


Next we prove the ‘if’ part. Since dim(H) ≥ 1 we may assume that H = R so thatγ(H,E) = E canonically. For n ≥ 1, let ψn ∈ S(Rd) be defined as

ψn(ξ) = c2−nd/2φ(2−nξ),

where c := ‖φ‖−1L2(Rd)

. Then (ψ3n)n≥1 is an orthonormal system in L2(Rd). For any

finite sequence (xn)Nn=1 in E we then have, with Φ :=

∑Nn=1 ψ3n ⊗ xn,

‖Φ‖2γ(Rd;E) = E

∥∥∥ N∑n=1

γnxn

∥∥∥2

.

Notice that for k ≥ 1,

‖ϕk ∗ ϕk‖Lp(Rd) = 2kd−1pkd‖φ ∗ φ‖Lp(Rd)

and‖ϕk+1 ∗ ϕk‖Lp(Rd) = 2kd−

1pkd‖ϕ1 ∗ φ‖Lp(Rd).

Therefore, for n = 1, . . . , N ,

‖ϕ3n ∗ Φ‖Lp(Rd;E) = c2−32nd‖ϕ3n ∗ ϕ3n‖Lp(Rd)‖xn‖ = c2( 1

2− 1

p)3nd‖φ ∗ φ‖Lp(Rd)‖xn‖

and similarly,

‖ϕ3n−1 ∗ Φ‖Lp(Rd;E) = c2( 12− 1

p)3nd−(1− 1

p)d‖ϕ1 ∗ φ‖Lp(Rd)‖xn‖

and

‖ϕ3n+1 ∗ Φ‖Lp(Rd;E) = c2( 12− 1

p)3nd‖ϕ1 ∗ φ‖Lp(Rd)‖xn‖.

Finally, for k ≥ 3N + 2 we have ϕk ∗ Φ = 0. Summing up, it follows that there existsa constant C, depending only on p, d and φ such that

‖Φ‖B

( 1p−

12 )d

p,p (Rd;E)≤ C

( N∑n=1

‖xn‖p) 1

p.

By putting things together we see that E has type p, with Gaussian type p constant

T γp (E) ≤ C‖I‖, where I : B( 1

p− 1

2)d

p,p (Rd;E) → γ(Rd;E) is the embedding.(2): This is proved similarly.

As a special case of Theorem 3.3.2, note that for every Banach space E we obtaincontinuous embeddings

B12d

1,1(Rd;E) → γ(L2(Rd), E) → B− 1

2d

∞,∞(Rd;E).

As is easily checked by going through the proofs, these embeddings are contractive.As a consequence of Theorem 3.3.2 one can also obtain embedding result for domainsS ⊂ Rd (see [65]).

3.4. Stochastic integration of operator-valued functions 43

3.4 Stochastic integration of operator-valued func-

tions

In this section we study the stochastic integral∫

R+Φ(t) dWH(t), where Φ : R+ →

B(H,E) and WH is a cylindrical Brownian motion. Identifying B(R, E) with E, as aspecial case we obtain an integration theory for

∫R+φ(t) dW (t) where φ : R+ → E and

W is a standard Brownian motion.A family WH = (WH(t))t∈R+ of bounded linear operators from H to L2(Ω) is called

an H-cylindrical Brownian motion if

(i) WHh = (WH(t)h)t∈R+ is a real-valued Brownian motion for each h ∈ H,

(ii) E(WH(s)g ·WH(t)h) = (s ∧ t) [g, h]H for all s, t ∈ R+, g, h ∈ H.

We always assume that the H-cylindrical Brownian motion WH is adapted to a givenfiltration F , i.e., the Brownian motions WHh are adapted to F for all h ∈ H. No-tice that if (hn)n≥1 is an orthonormal basis for H, then (WHhn)n≥1 are independentstandard real-valued Brownian motions.

Example 3.4.1. Let (Wn)Nn=1 be standard Brownian motions and let (en)

Nn=1 be an

orthonormal basis for RN . The process WH :=∑N

n=1 en⊗Wn is a cylindrical Brownianmotion with H = Rd. In case N = 1, WH = WR can be identified with a standardBrownian motion.

Example 3.4.2. Let W : R+ × Ω → E be an E-valued Brownian motion and letC ∈ B(E∗, E) be its covariance operator. Let HC be the reproducing kernel Hilbertspace or Cameron-Martin space (cf. [16, 37]) associated with C and let iC : HC → Ebe the inclusion operator. Then the mappings

WHC(t) : i∗Cx

∗ 7→ 〈W (t), x∗〉

uniquely extend to an HC- cylindrical Brownian motion WHC.

As in the real-valued case one can construct a stochastic integral∫

R+φ(t) dWH(t)

if φ ∈ L2(R+;H). Moreover, the isometry

E∣∣∣ ∫

R+

φ(t) dWH(t)∣∣∣2 = ‖φ‖2

L2(R+;H) (3.4.1)

holds and one has the following series representation∫R+

φ(t) dWH(t) =∑n≥1

∫R+

[φ(t), hn] dWH(t)hn,

where the series converges in Lp(Ω) for all p ∈ [1,∞) and a.s.


For a step function Φ : R+ → B(H,E) of the form

Φ =N∑n=1

1(tn−1,tn]Rn

where 0 ≤ t0 ≤ . . . ,≤ tn <∞ and R1, . . . , Rn ∈ γ(H,E) we define∫R+

Φ(t) dWH(t) =N∑n=1

∑m≥1

(WH(tn)hm −WH(tn−1)hm

)Rnhm,

where (hm)m≥1 is an orthonormal basis for H. The series converges in L2(Ω;E) andone can check that it does not depend on the choice of the orthonormal basis and⟨ ∫

R+

Φ(t) dWH(t), x∗⟩

=

∫R+

Φ∗(t)x∗ dWH(t), for all x∗ ∈ E∗. (3.4.2)

Furthermore, by (3.2.5) the following isometry holds:

E∥∥∥∫

R+

Φ(t) dWH(t)∥∥∥2

= ‖Φ‖2γ(R+;H,E)) (3.4.3)

for γ(H,E)-valued step functions Φ. A special case of a γ(H,E)-valued step functionis an elementary function, that is a function Φ : R+ → B(H,E) of the form

Φ =N∑n=1

1(tn−1,tn]

M∑m=1

hm ⊗ xmn.

The stochastic integral of such a function Φ is given by∫R+

Φ(t) dWH(t) =N∑n=1

M∑m=1

(WH(tn)hm −WH(tn−1)hm)xmn.

In the next result, we extend the stochastic integral to a larger class of functions.

Proposition 3.4.3. For an H-strongly measurable function Φ : R+ → B(H,E) be-longing to L2(R+;H) scalarly, the following assertions are equivalent:

(1) There exists a sequence (Φn)n≥1 of elementary functions such that:


Φ∗nx

∗ = Φ∗x∗ in L2(R+;H),

(ii) there exists a strongly measurable random variable η : Ω → E such that

η = limn→∞

∫R+

Φn(t) dWH(t) in probability.


(2) There exists a strongly measurable random variable η : Ω → E such that for allx∗ ∈ E∗ we have

〈η, x∗〉 =

∫R+

Φ∗(t)x∗ dWH(t) almost surely.

(3) Φ ∈ γ(R+;H,E).

In this situation the random variables η in (1) and (2) are uniquely determined andequal almost surely. Moreover, η is Gaussian and for all p ∈ [1,∞) we have η ∈Lp(Ω;E) and

(E‖η‖p)1p hp (E‖η‖2)

12 = ‖Φ‖γ(R+;H,E). (3.4.4)

For all p ∈ [1,∞) the convergence in (1), part (ii), is in Lp(Ω;E).

A function Φ satisfying the equivalent conditions of Proposition 3.4.3 will be calledstochastically integrable with respect to WH . It follows from (1), the remarks in Section2.3 and Doob’s maximal inequality (cf. [115, Theorem 70.2]) that (

∫ ·0Φn(t) dWH(t))n≥1

is a Cauchy sequence in Lp(Ω;Cb(R+;E)). Hence, (∫ ·

0Φn(t) dWH(t))n≥1 converges to

some ζ ∈ L2(Ω;Cb(R+;E)). The process ζ is called the stochastic integral of Φ withrespect to WH , notation

ζ =

∫ ·

0

Φ(t) dWH(t).

One easily checks the process ζ is a martingale and a Gaussian process and thatη = limt→∞ ζ(t) a.s. and in Lp(Ω;E) for all p ∈ [1,∞). The second identity in (3.4.4)may be interpreted as an analogue of the Ito isometry.

Proof. (1)⇒(2): This follows from (3.4.1) and (3.4.2).(2)⇒(3): Since η is strongly measurable and Φ is H-strongly measurable, we may

assume that E is separable. As in (3.2.4) let IΦ ∈ B(L2(R+;H), E) be defined as

IΦf =

∫R+

Φ(t)f(t) dt.

One easily checks that IΦI∗Φ is the covariance operator of the Gaussian random variable

η, and it is well-known that this implies that IΦ is γ-radonifying. An alternativeargument can be given as follows.

Let (hn)n≥1 be an orthonormal basis for H, let (gn)n≥1 be an orthonormal basisfor L2(R+) and for m,n ≥ 1 define fmn ∈ L2(R+;H) as fmn(t) = gm(t)hn. For n ≥ 1,let Wn = WHhn. For m,n ≥ 1, define γmn =

∫R+gm(t) dWn(t). Then (γmn)m,n≥1 is a

doubled indexed Gaussian sequence. Let x∗ ∈ E∗ be arbitrary. For fixed x∗ ∈ E∗ wehave ∑

m,n≥1

〈IΦfmn, x∗〉2 = ‖Φ∗x∗‖2L2(R+;H).


It follows that∑

m,n≥1 γmn〈IΦfmn, x∗〉 converges (unconditionally) in L2(Ω) and∑m,n≥1

γmn〈IΦfmn, x∗〉 =∑n≥1

∑m≥1

γmn[fmn,Φ∗x∗]L2(R+;H)

=∑n≥1

∑m≥1

∫R+

gm(t)[gm, 〈hn,Φ∗x∗〉] dWn(t)

=∑n≥1

∫R+

[hn,Φ∗(t)x∗] dWn(t) =

∫R+

Φ∗(t)x∗ dWn(t) = 〈η, x∗〉.

It follows from the Ito-Nisio theorem (cf. [76, Theorem 2.1.1]) that

η =∑m,n≥1

γmnIΦfmn

(unconditionally) converges a.s. This shows that IΦ is γ-radonifying.

(3)⇒(1): Let R ∈ γ(L2(R+;H), E) be such that it is represented by Φ. Let(Dn)n≥1 be an increasing sequence of σ-algebras on R+ that is generated by finitelymany bounded intervals and such that BR+ = σ(Dn, n ≥ 1). For n ≥ 1 let Rn ∈B(L2(R+;H), E) be defined as

Rnf = RE(f |Dn), f ∈ L2(R+;H)

and let Φn : R+ → B(H,E) be defined as the (Pettis) conditional expectation

Φnh = E(Φh|Dn), h ∈ H.

Then it follows from the ideal property that for all n ≥ 1, Rn ∈ γ(L2(R+;H), E)and Φn is an γ(H,E)-valued step function. Moreover, each Rn is represented byΦn and it follows from the martingale convergence theorem and Lemma 3.2.2 thatR = limn→∞Rn in γ(L2(R+;H), E). It follows from (3.4.3) that

E∥∥∥∫

R+

Φn(t) dWH(t)−∫

R+

Φm(t) dWH(t)∥∥∥2

= ‖Rn −Rm‖γ(L2(R+;H),E) → 0

as n,m tend to infinity. On the other hand, for all x∗ ∈ E∗, Φ∗x∗ = R∗x∗ =limn→∞R∗

nx∗ = Φ∗

nx∗ in L2(R+;E). To obtain (i) and (ii) one can do an easy projection

argument for the Hilbert space H.

The uniqueness of η in (1) and (2) follows from the Hahn-Banach theorem. Theother assertions follow from the results in Section 2.3 and approximation arguments.

One may combine Theorem 3.3.2 with Proposition 3.4.3 to obtain the followingsufficient condition for stochastic integrability.


Corollary 3.4.4. Let E be a Banach space with type p ∈ [1, 2]. If Φ ∈ B1p− 1

2p,p (R;E),

then Φ|R+ is stochastically integrable and

E∥∥∥∫

R+


≤ 4T 4p ‖Φ‖2

B1p−

12

p,p (R;E),

where Tp is the type p constant of E.

If E is a Banach function space with finite cotype, we obtain a necessary andsufficient condition for stochastic integrability.

Corollary 3.4.5. Let E be Banach function space with finite cotype over a σ-finitemeasure space (S,Σ, µ). Let Φ : R+ → B(H,E) be an H-strongly measurable functionand assume that there exists a strongly measurable function φ : R+×S → H such thatfor all h ∈ H and t ∈ R+,

(Φ(t)h)(·) = [φ(t, ·), h]H in E.

Then Φ is stochastically integrable if and only if∥∥∥( ∫R+

‖φ(t, ·)‖2H dt

) 12∥∥∥E<∞. (3.4.5)

In this case we have(E

∥∥∥∫R+


E

) 12 hE

∥∥∥( ∫R+

‖φ(t, ·)‖2H dt

) 12∥∥∥E.

Proof. First assume that Φ is stochastically integrable. Let N = n ∈ N : 1 ≤ n ≤dim(H), let (em)m∈N be the standard unit basis for L2(N , τ), where τ denotes thecounting measure on N . Choose orthonormal bases (fn)n≥1 for L2(R+) and (hn)n∈Nfor H. Define Ψ : R+ ×N → E by Ψ(t, n) := Φ(t)hn and define the integral operatorIΨ : L2(R+ ×N, dt× τ) → E by

IΨf :=

∫N

∫R+

f(t, n)Ψ(t, n) dt dτ(n) =∑n∈N

∫R+

f(t, n)Φ(t)hn dt.

Note that the integral on the right-hand side is well defined as a Pettis integral. LetIΦ ∈ γ(L2(R+;H), E) be the operator representing Φ as in Proposition 3.4.3(3). ThenIΨ ∈ γ(L2(R+ ×N, dt× τ), E) and(

E∥∥∥∫

R+


E

) 12

= ‖IΦ‖γ(L2(R+;H),E) = ‖IΨ‖γ(L2(R+×N, dt×τ),E).

Let (rmn) denoting a doubly indexed sequence of Rademacher variables on a probabilityspace (Ω′,A′,P′). By Lemmas 2.5.1 and 2.5.2 and [80, Theorem 1.d.6 and Corollary


1.f.9]

‖IΨ‖γ(L2(R+×N, dt×τ),E) hE

(E′

∥∥∥∑m,n

rmn

∫R+

∑k

Ψ(t, k)em(k)fn(t) dt∥∥∥2

E

) 12

hE

∥∥∥( ∫R+

∑k

∣∣Ψ(t, k)(·)∣∣2 dt) 1

2∥∥∥E

=∥∥∥( ∫

R+

∑k

∣∣(Φ(t)hk)(·)∣∣2 dt) 1

2∥∥∥E

=∥∥∥( ∫

R+

∑k

∣∣[φ(t, ·), hk]H∣∣2 dt) 1

2∥∥∥E

=∥∥∥( ∫

R+

‖φ(t, ·)‖2H dt

) 12∥∥∥E.

For the converse one can read all estimates backwards, but we have to show thatΦ is scalarly L2(R+;H) if (3.4.5) holds. For all x∗ ∈ E∗ we have

‖Φ∗x∗‖2L2(R+;H) =

( ∑m,n

( ∫R+

fn(t)[hm,Φ∗(t)x∗]H dt

)2) 12

=( ∑n,m

( ∫R+

∑k

〈Ψ(t, k), x∗〉em(k)fn(t) dt)2) 1

2

=(E′

∣∣∣ ∑n,m

rmn

∫R+

∑k

〈Ψ(t, k), x∗〉em(k)fn(t) dt∣∣∣2) 1

2

≤(E′

∥∥∥∑n,m

rmn

∫R+

∑k

Ψ(t, k)em(k)fn(t) dt∥∥∥2

E

) 12‖x∗‖.

3.5 Measurability of γ-valued functions

Let (S,Σ, µ) be a separable σ-finite measure space. In this section we will studymeasurability and representability properties of γ-valued functions.

Lemma 3.5.1. For a function X : S → γ(H,E) the following assertions are equiva-lent:

(1) X is strongly measurable.

(2) X is H-strongly measurable.

If these equivalent conditions hold, there exists a separable closed subspace E0 of Esuch that X(s) ∈ γ(H,E0) for all s ∈ S.

Proof. The implication (1)⇒(2) being trivial, we only need to prove (2)⇒(1).Let F = H ⊗E∗. By identifying each h⊗ x∗ ∈ F with the linear form on γ(H,E)

given by 〈T, h⊗ x∗〉 := 〈Th, x∗〉 and noting that

sup‖T‖γ≤1

|〈T, h⊗ x∗〉| = sup‖T‖γ≤1

|〈Th, x∗〉| ≤ ‖h‖ ‖x∗‖

3.5. Measurability of γ-valued functions 49

it follows that we may identify F with a linear subspace of γ(H,E)∗. Clearly, Fseparates the points of γ(H,E). By the Pettis measurability theorem it then sufficesto prove that X is separably valued and that for all h ∈ H and x∗ ∈ E∗ the real-valuedfunction s 7→ 〈X(s), h⊗ x∗〉 is Σ-measurable.

To prove that X is separably valued, let (hn)n≥1 be any sequence whose linearspan is dense in H. Choose E0 to be a separable closed linear subspace of E such thatX(s)hn ⊂ E0 for all s ∈ S and n ≥ 1. Then by approximation we have X(s)h ⊂ E0

for all s ∈ S and h ∈ H. It follows that X(s) ∈ γ(H,E0) for all s ∈ S, where weidentify γ(H,E0) with a closed subspace of γ(H,E) in the canonical way. Since E0 isseparable, so is γ(H,E0), and we conclude that X is separably valued.

The measurability of s 7→ 〈X(s), h ⊗ x∗〉 follows from the weak measurability ofs 7→ X(s)h for each h ∈ H.

Next, we will extend the representability concept for γ-space from Section 3.2 toγ-valued mappings.

A function Φ : S × Ω → B(H,E) belongs to L0(Ω;L2(R+;H)) scalarly if for allx∗ ∈ E∗, we have Φ∗(·, ω)x∗ ∈ L0(Ω;L2(S;H)).

An H-strongly measurable function Φ : S × Ω → B(H,E) is said to represent arandom variable X ∈ L0(Ω; γ(L2(S;H), E)) if it is scalarly in L0(Ω;L2(S;H)) and forall x∗ ∈ E∗, for almost all ω ∈ Ω,

〈X(ω)f, x∗〉 =

∫S

[f(s),Φ∗(s, ω)x∗]H dµ(s) for all f ∈ L2(S;H).

Lemma 3.5.2. Let (S,Σ, µ) be a separable σ-finite measure space. Let Φ : S × Ω →B(H,E) be an H-strongly measurable function.

(1) If Φ(·, ω) represents an element X(ω) ∈ γ(L2(S;H), E) for all ω ∈ Ω, then theinduced mapping X : Ω → γ(L2(S;H), E) is strongly measurable and representedby Φ.

(2) If Φ represents a strongly measurable mapping X : Ω → γ(L2(S;H), E), thenΦ(·, ω) represents X(ω) for almost all ω ∈ Ω.

Motivated by this result we will often identify Φ with X and as in Section 3.2 wewill write Φ ∈ Lp(Ω; γ(S;H,E)) if X ∈ Lp(Ω; γ(L2(S;H), E)) for p = 0 or p ∈ [1,∞).

Proof. (1): By Lemma 3.5.1 it suffices to check that ω 7→ X(ω)f is strongly measur-able for all f ∈ L2(S;H). By the Pettis measurability theorem it suffices to show thatω 7→ X(ω)f weakly measurable and separably-valued.

Fix f ∈ L2(S;H). By assumption, for all ω ∈ Ω and x∗ ∈ E∗ we have Φ∗(·, ω)x∗ ∈L2(S;H) and

〈X(ω)f, x∗〉 =

∫S

[f(s),Φ∗(s, ω)x∗]H dµ(s) =

∫S

〈Φ(s, ω)f(s), x∗〉 dµ(s). (3.5.1)


By the H-strong measurability of Φ and Fubini’s theorem, the right-hand side is ameasurable function of ω. Thus ω 7→ X(ω)f is weakly measurable.

Since t 7→ Φ(t, ω) is H-strongly measurable and belongs to L2(S;H) scalarly, t 7→Φ(t, ω)f(t) is Pettis integrable. It follows that

X(ω)f =

∫S

Φ(s, ω)f(s) dµ(s).

Then by the Hahn-Banach theorem, ω 7→ X(ω)f takes its values in the closed subspacespanned by the range of (t, ω) 7→ Φ(t, ω)f(t), which is separable by the H-strongmeasurability of Φ.

(2): By the Pettis measurability theorem we may assume without loss of generalitythat E is separable. Let (fm)m≥1 be a dense sequence in L2(S;H) and let (x∗n)n≥1 bea sequence in E∗ with weak∗-dense linear span. Choose a null set N ⊂ Ω such that

(i) Φ∗(·, ω)x∗n ∈ L2(S;H) for all x∗n and all ω ∈ N ,

(ii) (3.5.1) holds for all fm, all x∗n, and all ω ∈ N .

Let Γ denote the linear subspace of all x∗ ∈ E∗ for which

(i)′ Φ∗(·, ω)x∗ ∈ L2(S;H) for all ω ∈ N ,

(ii)′ (3.5.1) holds for all f ∈ L2(S;H) and all ω ∈ N .

By a limiting argument we see that x∗n ∈ Γ for all n ≥ 1. Hence Γ is weak∗-dense. Weclaim that Γ is also weak∗-sequentially closed. Once we have checked this, we obtainΓ = E∗ by the Krein-Smulyan theorem (cf. [23, Proposition 1.2]).

To prove the claim, fix ω ∈ N arbitrary. Then, by (3.5.1) for all x∗ ∈ Γ,

‖Φ∗(·, ω)x∗‖L2(S;H) ≤ ‖X(ω)‖γ(L2(S;H),E)‖x∗‖. (3.5.2)

Suppose now that limn→∞ y∗n = y∗ weak∗ in E∗ with each y∗n ∈ Γ. Then the uni-form boundedness principle and (3.5.2) imply that the sequence Φ∗(·, ω)y∗n is boundedin L2(S;H). By a convex combination argument as in [23, Proposition 2.2] we findthat y∗ ∈ Γ. Indeed, since L2(S;H) is reflexive there is a subsequence (nk)k≥1 andan φω ∈ L2(S;H) such that φω = limk→∞ Φ∗(·, ω)y∗nk

weakly in L2(S;H). Thisimplies (cf. [118]) that there exist (z∗i )i≥1 such that each z∗i is in co(y∗k : k ≥ i)and φω = limi→∞ Φ∗(·, ω)z∗i in L2(S;H). Let (in)n≥1 be a subsequence such thatφω = limn→∞ Φ∗(·, ω)z∗in in H a.e. Since y∗ = limi→∞ z∗i weak∗ in E∗, it follows thatfor each h ∈ H a.e.,

[φω, h] = [h, limn→∞

Φ∗(·, ω)z∗in ] = limn→∞

〈Φ(·, ω)h, z∗in〉 = 〈Φ(·, ω)h, y∗〉 = [h,Φ∗(·, ω)y∗].

Therefore, Φ∗y∗ = φω and we find that y∗ ∈ Γ. This proves the claim.

Finally we have the following consequence of the Kahane-Khinchine inequalities.


Proposition 3.5.3 (γ-Fubini isomorphism). Let (S,Σ, µ) be a σ-finite measure spaceand let p ∈ [1,∞) be fixed. The mapping Fub : Lp(S; γ(H, E)) → B(H, Lp(S;E))defined by

(Fub(X)h)(s) := X(s)h, s ∈ S, h ∈ H,defines an isomorphism from Lp(S; γ(H, E)) onto γ(H, Lp(S;E)).

Proof. Let (hn)n≥1 be an orthonormal basis for H and let (γn)n≥1 be a sequence ofindependent standard Gaussian random variables on a probability space (Ω′,F ′,P′).By the Kahane-Khinchine inequalities and Fubini’s theorem, we have for any X ∈Lp(S; γ(H, E)),

‖Fub(X)‖γ(H,Lp(S;E))

=(E′

∥∥∥∑n≥1

γnFub(X)hn

∥∥∥2

Lp(S;E)

) 12 hp

(E′

∥∥∥∑n≥1

γn Fub(X)hn

∥∥∥pLp(S;E)

) 1p

=( ∫

S

E′∥∥∥∑n≥1

γnXhn

∥∥∥p dµ) 1p hp

( ∫S

(E′

∥∥∥∑n≥1

γnXhn

∥∥∥2) p2dµ

) 1p

=( ∫

S

‖X‖pγ(H,E) dµ) 1

p= ‖X‖Lp(S;γ(H,E)).

(3.5.3)By these estimates the range of the operator X 7→ Fub(X) is closed in γ(H, Lp(S;E)).Hence to show that this operator is surjective it is enough to show that its range isdense. But this follows from

Fub( N∑n=1

1Sn ⊗( K∑k=1

hk ⊗ xkn

))=

K∑k=1

hk ⊗( N∑n=1

1Sn ⊗ xkn

),

for all Sn ∈ Σ with µ(Sn) < ∞ and xkn ∈ E, noting that the elements on the right-hand side are dense in γ(H, E).


The γ-norm defined in Section 3.2 has been studied by many authors. It is relatedto almost summing operators which where first introduced by Linde and Pietschin [79]. The γ-norm is also linked to the so-called `-norms of Figiel and Tomczak-Jaegermann used in the study of Banach space geometry in [48]. More details onthe results in Section 3.2 can be found [45, 67, 100].

Proposition 3.2.3 is taken from [98]. It is a generalization of [24, Theorem 2.3] byBrzezniak and van Neerven. It gives an easy way to check when certain Sobolevembeddings are γ-radonifying. This will be used in Chapter 8.

Theorem 3.3.2 is the main result in [65]. It has been formulated there for the casewhere H = R, but the proof for arbitrary H is exactly the same. Several embeddingresults for bounded domains have been obtained in [65] as well. For example it has


been shown that if E has type p ∈ [1, 2], then for all α > 1p− 1

2, Cα([0, 1];E) →

γ(L2(0, 1), E). Conversely, if for α ∈ (0, 12) fixed Cα([0, 1];E) → γ(L2(0, 1), E), then

E has type p for all p ∈ (1, 2) satisfying α < 1p− 1

2. This may be proved with the

Maurey-Pisier theorem. We do not know what happens if α = 1p− 1

2in the above

results.Proposition 3.4.3 has been proved by van Neerven and Weis in [100]. The

equivalence of (2) and (3) is implicitly contained in [23, Theorem 2.5, Proposition 2.8and Theorem 3.3] by Brzezniak and van Neerven. In the case that H = R, onecan identify WH with a standard Brownian motion and Φ : R+ → B(H,E) with afunction φ : R+ → E. In this situation a closely related result was already provedby Rosinski and Suchanecki in [117]. In the case of finite time intervals, theapproximation in Proposition 3.4.3 (1) (i) can be chosen in such a way that for allh ∈ H, Φh = limn→∞ Φnh in measure (cf. [95]).

Via the results in Section 2.7 one can give sufficient conditions for stochastic inte-grability. Corollary 3.4.4 is an example of such a result. Using the embedding resultsfor bounded domains from [65], one can also give sufficient conditions for integrabilityin case Φ is defined on a bounded interval. Another example can be obtained withPropositions 3.3.1 and 3.4.3. For example in type 2 spaces E every φ ∈ L2(R+;E) isstochastically integrable with respect to a standard Brownian motion and

E∥∥∥∫

R+

φ(t) dW (t)∥∥∥2

≤ T 22 ‖φ‖2

L2(R+;E).

This was already known to Hoffmann-Jørgensen, and Pisier in [59]. Proposition3.4.5 is taken from [95] and is an extension of [100, Corollary 2.10], where the caseH = R was considered. From the above result that Cα([0, 1];E) does not embed intoγ(L2(0, 1), E) if 0 < α < 1

p0− 1

2, where

p0 = supp ∈ [1, 2] : E has type p,

one obtains examples of Holder continuous functions that are not stochastically inte-grable. This extends examples of Rosinski and Suchanecki in [117] and Yor in[137].

The results in Section 3.5 are from [96] and will be used to study stochastic inte-grability of processes.

Chapter 4

Integration with respect tocylindrical Brownian motion

4.1 Introduction

In this chapter (Ω,A,P) is a complete probability space andWH is a cylindrical Brown-ian motion with respect to a complete filtration F = (Ft)t≥0 on (Ω,A,P). Let (E, ‖·‖)be a Banach space and let (H, [·, ·]) be a separable Hilbert space.

In Section 3.4 we gave characterizations for the stochastic integrability of functionsΦ : R+ → B(H,E). In this chapter we extend these results to processes Φ : R+×Ω →B(H,E) in the case that E is a UMD space. The approach is based on decouplinginequalities for stochastic integrals. Such inequalities will be obtained in Section 4.3from the results in Section 2.4. The decoupling inequalities also characterize theUMD property of E. In Section 4.4 we first characterize the stochastically integrableprocesses in an Lp-setting. Using localization arguments the results will be extendedto the general setting in Section 4.5. At the end of Section 4.5 we also extend thestochastic integral to processes which are integrable on each finite time interval.

Some applications of the theory in Sections 4.4 and 4.5 are given in Section 4.6. Wegive equivalent conditions for stochastic integrability for UMD Banach function spaces.Some implications will be obtained for spaces with type or cotype. A multiplier resultfor stochastic integrals will be proved. Moreover, we show that certain martingales arealways stochastically integrable. In particular every integral process is stochasticallyintegrable.

In Section 4.7 we will specialize the results of Sections 4.4 and 4.5 in the case thatthe underlying filtration is induced by the cylindrical Brownian motion WH . In lasttwo sections of this chapter we prove two Ito formulas. In Section 4.8 an Ito formulawill be proved for general smooth functions f : R+ × E → F , where E and F areUMD Banach spaces, which is applied to an E-valued integral process ζ given by

ζ = ξ +

∫ ·

0

ψ(s) ds+

∫ ·

0

Φ(s) dWH(s).

53

54 Chapter 4. Integration w.r.t. cylindrical Brownian motion

Here, among other things Φ is assumed to be an element of L2loc(R+; γ(H,E)). In

Section 4.9 we will show that for a special choice of the function f the L2-assumptionon Φ can be omitted.

Sections 4.2-4.7 are based on [95, 96] and Section 4.8 is the first part of [97].For convenience we recall some of the known results (for the case where E = R) that

we will frequently use in this chapter. For p ∈ [1,∞) and p = 0, let LpF(Ω;L2(R+;H))be the closed subspace of measurable and adapted process in Lp(Ω;L2(R+;H)). Itis well-known how to define a stochastic integral

∫ ·0φ(t) dWH(t) as an element of

L0(Ω;Cb(R+)) for a process φ ∈ L0F(Ω;L2(R+;H)). The process

∫ ·0φ(t) dWH(t) is

a real-valued continuous local martingale starting at zero. If τ is a stopping time, thenalmost surely for all t ∈ R+ we have∫ τ∧t

0

φ(s) dWH(s) =

∫ t

0

1[0,τ ](s)φ(s) dWH(s). (4.1.1)

The quadratic variation process of∫ ·

0φ(t) dWH(t) is given by

∫ ·0‖φ(t)‖2

H dt. Recall(cf. [63, Proposition 17.6 and Lemma 17.12]) that for φ and a sequence (φn)n≥1 inL0F(Ω;L2(R+;H)) we have φ = limn→∞ φn in L0(Ω;L2(R+;H)) if and only if∫ ·

0

φ(t) dWH(t) = limn→∞

∫ ·

0

φn(t) dWH(t) in L0(Ω;Cb(R+)). (4.1.2)

Moreover, for φ as above the Burkholder-Davis-Gundy inequalities (cf. [63, Theorem17.7]) imply that for all p ∈ (0,∞) we have

E supt∈R+

∣∣∣ ∫ t

0

φ(s) dWH(s)∣∣∣p hp E

( ∫R+

‖φ(t)‖2H dt

) p2

(4.1.3)

whenever one of the expressions is finite.


A process Φ : R+×Ω → B(H,E) is said to be elementary with respect to the filtrationF if it is of the form

Φ =N∑n=0

1(tn−1,tn]

M∑m=1

hm ⊗ ξmn, (4.2.1)

where 0 ≤ t0 < · · · < tN < ∞ and for each n ≥ 0, ξ1n, . . . , ξMn are Ftn−1-measurable(with the understanding that (t−1, t0] := 0 and Ft−1 := F0) and h1, . . . , hM ∈ H areorthonormal. For such processes Φ and t ∈ R+ we define the stochastic integral as∫ t

0

Φ(s) dWH(s) =N∑n=1

M∑m=1

(WH(tn ∧ t)hm −WH(tn−1 ∧ t)hm

)ξmn, t ∈ R+.

In this way∫ ·

0Φ(s) dWH(s) is an a.s. pathwise continuous E-valued process and an

element of L0(Ω;Cb(R+;E)). We extend this definition by approximation.


Definition 4.2.1. For an H-strongly measurable and adapted process Φ that is scalarlyin L0(Ω;L2(R+;H)) we say that Φ is stochastically integrable with respect to WH ifthere exists a sequence (Φn)n≥1 of elementary processes such that:


Φ∗nx

∗ = Φ∗x∗ in L0(Ω;L2(R+;H)),

(ii) there exists a process ζ ∈ L0(Ω;Cb(R+;E)) such that

ζ = limn→∞

∫ ·

0

Φn(t) dWH(t) in L0(Ω;Cb(R+;E)).

The process ζ is called the stochastic integral of Φ with respect to WH , notation

ζ =

∫ ·

0

Φ(t) dWH(t).

In this way∫ ·

0Φ(t) dWH(t) is uniquely defined up to indistinguishability.

We denote by Mc,loc0 (Ω;E) the space of continuous local martingales starting at 0,

identifying martingales that are indistinguishable. Each M ∈ Mc,loc0 (Ω;E) defines a

random variable with values in C(R+;E). This follows from the Pettis measurabilitytheorem and the fact that C(R+;E) is a complete separable metric space under thetranslation invariant metric

d(f, g) =∑n≥0

2−nsupt∈[n,n+1) ‖f(t)− g(t)‖

1 + supt∈[n,n+1) ‖f(t)− g(t)‖.

Thus we may identify Mc,loc0 (Ω;E) with a linear subspace of L0(Ω;C(R+;E)). If

we want to stress the role of the underlying filtration F we write Mc,loc0 (Ω;E) =

Mc,loc0 (Ω,F ;E).

Proposition 4.2.2. Let Φ : R+ × Ω → E be an H-strongly measurable and adaptedprocess which belongs scalarly to L0(Ω;L2(R+;H)). If there exists a process ζ ∈L0(Ω;Cb(R+;E)) such that for all x∗ ∈ E∗ we have

〈ζ, x∗〉 =

∫ ·

0

Φ∗(t)x∗ dWH(t) in L0(Ω;Cb(R+; R)),

then ζ belongs to Mc,loc0 (Ω;E).

The condition in Proposition 4.2.2 can be seen as a weak stochastic integral orPettis stochastic integral. This condition always holds if Φ is stochastically integrable.Below in Theorem 4.5.9 we will show that for UMD spaces E, both notions are actuallyequivalent.


Proof. Clearly, ζ0 = 0 almost surely and ζ is adapted, so it suffices to show that ζis a local martingale. It is obvious that for all x∗ ∈ E∗, 〈ζ, x∗〉 is a local martingale.Define a sequence of stopping times (τn)n≥1 by

τn := inft ≥ 0 : ‖ζt‖ ≥ n.

For all x∗ ∈ E∗ we have

〈ζτn , x∗〉 =

∫ ·

0

〈Φ(s), x∗〉1[0,τn](s) dWH(s) in Cb(R+; R) almost surely.

Since the local martingale on left-hand side is bounded, the Burkholder-Davis-Gundyinequalities and [63, Corollary 17.8] imply that it is a martingale and for all x∗ ∈ E∗

and 0 ≤ s ≤ t it follows that

〈E(ζτn∧t|Fs), x∗〉 = E(〈ζτn∧t, x∗〉|Fs) = 〈ζτn∧s, x∗〉

almost surely. It follows that for all 0 ≤ s ≤ t we have E(ζτn∧t|Fs) = ζτn∧s a.s., so(ζτn∧t)t∈R+ is a martingale and (ζt)t∈R+ is a local martingale.

The stochastic integral satisfies the following local property.

Proposition 4.2.3. Let Φ : R+ × Ω → B(H,E) be an H-strongly measurable andadapted process which belongs scalarly to L0(Ω;L2(R+;H)) and is stochastically inte-grable with respect to WH . Suppose A ∈ F is a measurable set such that for all x∗ ∈ E∗

we have

Φ∗(t, ω)x∗ = 0 for almost all (t, ω) ∈ R+ × A.

Then almost surely in A, for all t ∈ R+,∫ t

0

Φ(s) dWH(s) = 0.

Proof. Let x∗ ∈ E∗ be arbitrary. By strong measurability it suffices to show thatalmost surely in A for all t ∈ R+,

Mt :=

∫ t

0

Φ∗(s)x∗ dWH(s) = 0.

For the quadratic variation of the continuous local martingale M we have

[M ]∞ =

∫R+

‖Φ∗(s)x∗‖2H ds = 0 a.s. on A.

Therefore, M ≡ 0 a.s. on A.

4.3. Decoupling inequalities 57

4.3 Decoupling inequalities

Let (Ω, A, P) be a copy of (Ω,A,P) and let WH be a copy of WH on (Ω, A, P). Forelementary processes Φ as in (4.2.1), the decoupled stochastic integral defined on Ω× Ωis given by( ∫ t

0

Φ(s) dWH(s))(ω, ω) =

N∑n=1

M∑m=1

((WH(tn ∧ t)− WH(tn−1 ∧ t))hm

)(ω)ξmn(ω),

for t ∈ R+. This is an a.s. continuous E-valued process. For UMD spaces E itsLp-norm is related to the Lp-norm of

∫R+

Φ(t) dWH(t) for all p ∈ (1,∞).

Theorem 4.3.1. Let E be a UMD space and let p ∈ (1,∞). Then for all elementaryprocesses Φ : R+ × Ω → B(H,E) we have

E∥∥∥∫

R+

Φ(t) dWH(t)∥∥∥p hE,p EE

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥p.

In Corollary 5.4.2 we give a converse to Theorem 4.3.1.

Proof. Assume that Φ is as in (4.2.1). The result follows from Theorem 2.4.1 with forn = 1, . . . , N and m = 0, . . . ,M − 1, v(n−1)M+m := ξmn,

η(n−1)M+m = (WH(tn)−WH(tn−1)hm, η(n−1)M+m = (WH(tn)− WH(tn−1))hm,

G(n−1)M+m = σ(Ftn−1 , (WH(tn)−WH(tn−1))hk for 1 ≤ k ≤ m,and

G(n−1)M+m = σ(Ftn−1 , (WH(tn)− WH(tn−1))hk for 1 ≤ k ≤ m.

4.4 Characterizations for the Lp-case

A random variable X : Ω → γ(L2(R+;H), E) is elementary with respect to F if it isrepresented by an elementary process. A random variable X : Ω → γ(L2(R+;H), E)is called strongly progressive with respect to F if it is pointwise limit of a sequence ofelementary random variables Xn : Ω → γ(L2(R+;H), E).

For a random variable X : Ω → γ(L2(R+;H), E) we denote by 〈X, x∗〉 : Ω →L2(R+;H) the random variable defined by

〈X, x∗〉(ω) := X∗(ω)x∗.

We call X scalarly adapted with respect to F if 〈X, x∗〉 : Ω → γ(L2(R+;H),R) 'L2(R+;H) is adapted for all x∗ ∈ E∗. Note that if X is represented by a process Φ,then for all x∗ ∈ E∗ we have

〈X, x∗〉 = Φ∗x∗ almost everywhere on R+ × Ω,


where we identify 〈X, x∗〉 with the H-valued process (t, ω) 7→ (〈X, x∗〉(ω))(t). If Φ isscalarly adapted, then X is scalarly adapted as well.

For p ∈ [1,∞), the closure in Lp(Ω; γ(L2(R+;H), E)) of the elementary elements isdenoted by LpF(Ω; γ(L2(R+;H), E)). The following proposition shows that this spaceconsists precisely of the scalarly adapted elements in Lp(Ω; γ(L2(R+;H), E)). We willalso write LpF(Ω; γ(R+;H,E)) for the adapted processes Φ in Lp(Ω; γ(R+;H,E))

Lemma 4.4.1. Let p ∈ [1,∞) be fixed. For a strongly measurable map X : Ω →γ(L2(R+;H), E) the following assertions are equivalent:

(1) X ∈ LpF(Ω; γ(L2(R+;H), E)).

(2) X ∈ Lp(Ω; γ(L2(R+;H), E)) and X is strongly progressive with respect to F .

(3) X ∈ Lp(Ω; γ(L2(R+;H), E)) and X is scalarly adapted with respect to F .

Proof. The implications (1)⇒(2)⇒(3) follow trivially from the definitions.(3)⇒(1): For T > 0, let P T : L2(R+;H) → L2(R+;H) be defined as P Tf = f1[0,T ].

First assume that P TX = X.For δ ≥ 0 we define the right translate Rδ of an operator R ∈ γ(L2(R+;H), E) by

Rδf := Rfδ, f ∈ L2(R+;H),

where fδ denotes the left translate of f . It follows by the right-ideal property andLemma 3.2.2 that Rδ ∈ γ(L2(R+;H), E) with ‖Rδ‖γ(H,E) ≤ ‖R‖γ(H,E) and that δ 7→ Rδ

is continuous with respect to the γ-norm.Define the right translate Xδ ∈ Lp(Ω; γ(L2(R+;H), E)) by pointwise action, i.e.,

Xδ(ω) := (X(ω))δ. Note that Xδ is strongly measurable by Lemma 3.5.1. By domi-nated convergence, δ 7→ Xδ is continuous into Lp(Ω; γ(L2(R+;H), E)). Thus, for ε > 0fixed, we may choose δ > 0 such that

E‖X −Xδ‖pγ(L2(R+;H),E) < εp. (4.4.1)

Let 0 = t0 < · · · < tN = T be an arbitrary partition of [0, T ] of mesh ≤ δ and letIn = (tn−1, tn] for n = 1, . . . , N . Let Xδ

n denote the restriction of Xδ to In, i.e., Xδn is

the element of Lp(Ω; γ(L2(In;H), E)) defined by

Xδn(ω)g := Xδ(ω)ing, g ∈ L2(In;H),

where in : L2(In;H) → L2(R+;H) is the inclusion mapping. By the scalar adaptivity ofX, the random variable 〈Xδing, x

∗〉 is strongly Ftn−δ-measurable, and therefore Ftn−1-measurable, for all g ∈ L2(In;H) and all x∗ ∈ E∗. Hence by the Pettis measurabilitytheorem and Lemma 3.5.1, Xδ

n ∈ Lp(Ω,Ftn−1 ; γ(L2(In;H), E)). Thus we may pick a

simple random variable Yn ∈ Lp(Ω,Ftn−1 ; γ(L2(In;H), E)) such that

E‖Xδn − Yn‖pγ(L2(In;H),E)) <

εp

Np,

4.4. Characterizations for the Lp-case 59

say Yn =∑Mn

m=1 1Amn ⊗ Smn with Amn ∈ Ftn−1 and Smn ∈ γ(L2(In;H), E)). Bya further approximation as in Proposition 3.4.3 we may assume that the Smn arerepresented by elementary functions Ψmn : [0, T ] → B(H,E) of the form

Ψmn(t) =Jmn∑j=1

1(s(j−1)mn,sjmn](t)Kmn∑k=1

(hk ⊗ xkmn),

where tn−1 ≤ s0mn < · · · < sJmnmn ≤ tn and (hk)k≥1 is a fixed orthonormal basis forH. Define the process Ψ : [0, T ]× Ω → B(H,E) by

Ψ(t, ω) :=Mn∑m=1

1Amn(ω)Ψmn(t), t ∈ In.

It is easily checked that Ψ is elementary. Let Y ∈ Lp(Ω; γ(L2(R+;H), E)) be repre-sented by Ψ. Then Y is elementary and satisfies(

E‖Xδ − Y ‖pγ(L2(R+;H),E)

) 1p < ε. (4.4.2)

Finally, by (4.4.1) and (4.4.2),(E‖X − Y ‖pγ(L2(R+;H),E)

) 1p ≤ 2ε.

This proves that X can be approximated in Lp(Ω; γ(L2(R+;H), E)) by a sequence ofelementary elements.

For the general case let ZT = X P T . By the right-ideal property, we haveZT ∈ Lp(Ω; γ(L2(R+;H), E)). By the above result that we can find elementary Xn ∈L0(Ω; γ(L2(R+), E)) such that

‖Xn − Zn‖Lp(Ω;γ(L2(R+;H),E)) <1

n.

It follows from Lemma 3.2.2 that X = limn→∞ Zn in Lp(Ω; γ(L2(R+), E)) and we mayconclude the result.

Motivated by this result, the elements of LpF(Ω; γ(L2(R+;H), E)) will be called theprogressive elements of Lp(Ω; γ(L2(R+;H), E)).

If Φ is a process representing an elementary element X ∈ Lp(Ω; γ(L2(R+;H), E)),we define the process IWH (X) ∈ Lp(Ω;Cb(R+;E)) as

IWH (X) :=

∫ ·

0

Φ(t) dWH(t).

Note that IWH (X) does not depend on the choice of the representing process Φ.


Proposition 4.4.2 (Ito isomorphism). Let E be a UMD space and fix p ∈ (1,∞).The mapping X 7→ IWH (X) has a unique extension to a bounded operator

IWH : LpF(Ω; γ(L2(R+;H), E)) → Lp(Ω, Cb(R+;E)).

This operator is an isomorphism onto its range and we have the two-sided estimate

E supt∈R+

‖IWH (X)(t)‖p hp ‖X‖pLp(Ω;γ(L2(R+;H),E)).

For all X ∈ LpF(Ω; γ(L2(R+;H), E)), IWH (X) is a martingale,

IWH (X)(∞) := limt→∞

IWH (X)(t) (4.4.3)

exists in Lp(Ω;E) and almost surely and

E‖IWH (X)(∞)‖p = supt∈R+

E‖IWH (X)(t)‖p hp E supt∈R+

‖IWH (X)(t)‖p. (4.4.4)

For Φ ∈ LpF(Ω; γ(R+;H,E)) we also write IWH (Φ) for IWH (X) if Φ representsX ∈ LpF(Ω; γ(L2(R+;H), E)).

Proof. Let X ∈ Lp(Ω; γ(L2(R+;H), E)) be elementary, and let Φ be the elementaryprocess representing X. We may estimate

E‖X‖pγ(L2(R+;H),E)

(i)= E

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥pL2(Ω;E)

(ii)hp E

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥pLp(Ω;E)

(iii)hp,E E

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥p (iv)

hp E supt∈R+

‖IWH (X)(t)‖p,

where in (i) we used Proposition 3.4.3, (ii) follows from the Kahane-Khinchine inequal-ities, in (iii) we used the estimate of Theorem 4.3.1 involving the UMD(p) constantof E and (iv) is due to Doob’s maximal inequality (cf. [115, Theorem 70.2]). Itfollows that the map X 7→ IWH (X) has a unique extension to an isomorphism fromLpF(Ω; γ(L2(R+;H), E)) onto its range, which is a closed subspace of Lp(Ω;Cb(R+;E)).

That IWH (X) is a martingale is obvious for elementary X, and the general casefollows from an approximations argument. The Lp-convergence in (4.4.3) follows fromthe martingale convergence theorem (cf. [46, Corollary V.2]). The a.s. convergence in(4.4.3) is obvious for elementary processes. The general case follows from an approxi-mation argument. Finally, (4.4.4) follows from the contractiveness of the conditionalexpectation (cf. Theorem V.4), Fatou’s lemma and Doob’s maximal inequality.

The next result is the main result of this section. It describes the relationshipbetween the stochastic integral and the operator IWH in the Lp-setting. It is a firstextension of Proposition 3.4.3 to processes.


Theorem 4.4.3. Let E be a UMD space and fix p ∈ (1,∞). For an H-strongly measur-able and adapted process Φ : R+×Ω → B(H,E) which is scalarly in Lp(Ω;L2(R+;H))the following assertions are equivalent:

(1) There exists a sequence (Φn)n≥1 of elementary processes such that:


Φ∗nx

∗ = Φ∗x∗ in Lp(Ω;L2(R+;H)),

(ii) there exists a process ζ ∈ Lp(Ω;Cb(R+;E)) such that

ζ = limn→∞

∫ ·

0

Φn(t) dWH(t) in Lp(Ω;Cb(R+;E)).

(2) There exists a strongly measurable random variable η ∈ Lp(Ω;E) such that forall x∗ ∈ E∗ we have

〈η, x∗〉 =

∫R+

Φ∗(t)x∗ dWH(t) in Lp(Ω).

(3) Φ ∈ Lp(Ω; γ(R+;H,E)).

(4) For almost all ω ∈ Ω, the function Φ(·, ω) is stochastically integrable with respectto an independent H-cylindrical Brownian motion WH , and

ω 7→∫

R+

Φ(t, ω) dWH(t) ∈ Lp(Ω;Lp(Ω;E)).

In this situation Φ is stochastically integrable, Φ is in LpF(Ω; γ(R+;H,E)), and wehave

IWH (Φ) = ζ =

∫ ·

0

Φ(t) dWH(t) in Lp(Ω;Cb(R+;E)), (4.4.5)

η =

∫R+

Φ(t) dWH(t) = limt→∞

ζ(t), (4.4.6)

where the limit exists almost surely and in Lp(Ω;E),

E‖Φ‖pγ(R+;H,E) hp E∥∥∥∫

R+

Φ(t) dWH(t)∥∥∥pLp(Ω;E)

(4.4.7)

and

E‖η‖p hp E supt∈R+

∥∥∥∫ t

0

Φ(t) dWH(t)∥∥∥p hE,p E‖Φ‖pγ(R+;H,E). (4.4.8)

Remark 4.4.4. The estimates (4.4.7) and (4.4.8) also characterizes the UMD prop-erty of E. This follows from (3.4.4) and the fact that the estimate in Theorem 4.3.1characterizes the UMD property. The estimate (4.4.8) can be seen as a vector-valuedstochastic integral version of the Burkholder-Davis-Gundy inequalities.


Proof of Theorem 4.4.3. (1)⇒(3): Let (Xn)n≥1 in Lp(Ω; γ(L2(R+;H), E)) be sucheach Xn is represented by Φn. By property (ii) and Proposition 4.4.2 for all m,n ≥ 1,

‖Xn −Xm‖pLp(Ω;γ(L2(R+;H),E)) = E∥∥∥∫ ·

0

Φn(t)− Φm(t) dWH(t)∥∥∥p.

This shows that (Xn)n≥1 is a Cauchy sequence in Lp(Ω; γ(L2(R+;H), E)). Let X ∈Lp(Ω; γ(L2(R+;H), E)) be the limit. Since each Xn is elementary we have X ∈LpF(Ω; γ(L2(R+;H), E)), and with property (i) it follows that

〈X, x∗〉 = limn→∞

〈Xn, x∗〉 = lim

n→∞Φ∗nx

∗ = Φ∗x∗ in Lp(Ω;L2(R+;H)).

Hence, Φ represents X.(3)⇒(1): By Lemma 4.4.1 we have Φ ∈ LpF(Ω; γ(R+;H,E)). Thus we may choose a

sequence (Φn)n≥1 of elementary processes with limn→∞ Φn = Φ in Lp(Ω; γ(R+;H,E)).The sequence (Φn)n≥1 has properties (i) and (ii). Indeed, property (i) is clear. Property(ii), with ζ = IWH (Φ), follows from Proposition 4.4.2, since

limn→∞

∫ ·

0

Φn(t) dWH(t) = limn→∞

IWH (Φn) = IWH (Φ) in Lp(Ω;Cb(R+;E)).

Moreover (4.4.5) and the second two-sided estimate (4.4.8) follow from Proposition4.4.2.

(3)⇔(4): This equivalence and (4.4.7) follow from Lemma 3.5.2, Proposition 3.4.3and the Kahane-Khinchine inequalities.

(1)⇒(2): From Proposition 4.4.2 we obtain the second equality in (4.4.6). Now (1)and the Burkholder-Davis-Gundy inequalities (4.1.3) imply⟨ ∫

R+

Φ(t) dWH(t), x∗⟩

= limn→∞

⟨ ∫R+

Φn(t) dWH(t), x∗⟩

= limn→∞

∫R+

Φ∗n(t)x

∗ dWH(t) =

∫R+

Φ∗(t)x∗ dWH(t) = 〈η, x∗〉

in Lp(Ω). This proves (2) and the first equality in (4.4.6). The first two-sided estimatein (4.4.8) follows from Proposition 4.4.2.

(2)⇒(3): We denote by BF the closed unit ball of a Banach space F . Since Φ isH-strongly measurable, without loss of generality we may assume that E is separable.Since E is reflexive, E∗ is separable as well and we may fix a dense sequence (x∗n)n≥1

in BE∗ . Define the closed linear subspaces Fn of E by

Fn :=n⋂i=1

ker(x∗i ).

Let En be the quotient space E/Fn, and let Qn : E → En be the quotient map. Thendim(En) <∞ and there is a canonical isomorphism E∗

n ' F⊥n , where F⊥

n = x∗ ∈ E∗ :


x∗ = 0 on Fn. Moreover, each En is a UMD space with UMD(p) constant that canbe estimated by the UMD(p) constant of E. We proceed in several steps.

Step 1 – For every finite-dimensional subspace G of E and every ε > 0 there existsan index N ≥ 1 such that

‖x‖ ≤ (1 + ε)‖QNx‖ ∀x ∈ G. (4.4.9)

To show this, it suffices to consider x ∈ BG. Since BG is compact we can find elementsy∗1, . . . , y

∗n ∈ E∗ with ‖y∗i ‖ ≤ 1 such that

‖x‖ ≤(1 +

ε

2

)sup

1≤i≤n|〈x, y∗i 〉|, ∀x ∈ BG.

Since (x∗i )i≥1 is norm dense in B∗E, we may approximate the y∗i to obtain an index N

such that‖x‖ ≤

(1 + ε

)sup

1≤j≤N|〈x, x∗j〉|, ∀x ∈ BG.

It follows that for all x ∈ BG,

‖x‖ ≤ (1 + ε) infy∈FN

sup1≤j≤N

|〈x− y, x∗j〉| ≤ (1 + ε) infy∈FN

‖x− y‖ = (1 + ε)‖QNx‖.

This proves (4.4.9).Step 2 – Let the processes Φn : R+ × Ω → B(H;En) be given by Φn(t, ω)h :=

QnΦ(t, ω)h. Clearly Φn belongs to Lp(Ω;L2(R+;H)) scalarly. Moreover, Φ representsan element Xn ∈ Lp(Ω; γ(L2(R+;H), En)), since for the finite-dimensional spaces Enwe have γ(L2(R+;H), En) ' B(L2(R+;H), En). It is easily checked that IWHXn =Qnη. Hence,

E‖Xn‖pγ(L2(R+;H),En) hE,p E‖IWHXn‖pEn= E‖Qnη‖pEn

≤ E‖η‖p, (4.4.10)

where we used that for the UMD constants we have βp(En) ≤ βp(E).For 1 ≤ m ≤ n let Qnm : En → Em be given by QnmQnx := Qmx. Then ‖Qnm‖ ≤ 1

and Xm = QnmXn almost surely. It follows that for almost all ω ∈ Ω for all 1 ≤ m ≤ n,‖Xm(ω)‖γ(L2(R+;H),Em) ≤ ‖Xn(ω)‖γ(L2(R+;H),En). By Fatou’s lemma and (4.4.10),

E supn≥1

‖Xn‖pγ(L2(R+;H),En) = E limn→∞

‖Xn‖pγ(L2(R+;H),En)

≤ lim infn→∞

E‖Xn‖pγ(L2(R+;H),En) .p,E E‖η‖p.(4.4.11)

Step 3 – Let N0 be a null set such that for all ω ∈ N0 we have

C(ω) := supn≥1

‖Xn(ω)‖γ(L2(R+;H),En) <∞.

By Lemma 3.5.2, for each n ≥ 1 we can find a null set Nn of that for all ω ∈ Nn andx∗ ∈ E∗

n, 〈Xn(ω), x∗〉 = Φ∗n(·, ω)x∗ in L2(R+;H). Let N := N0 ∪ (

⋃n≥1Nn). We claim

that for all ω ∈ N and all x∗ ∈ E∗, Φ∗(·, ω)x∗ ∈ L2(R+;H).


Fix ω ∈ N . First let x∗ be a linear combination of the elements x∗1, . . . , x∗n. Then

x∗ ∈ F⊥n and hence, for all t ∈ R+, Φ∗(t, ω)x∗ = Φ∗

n(t, ω)x∗. It follows that

‖Φ∗(·, ω)x∗‖L2(R+;H) = ‖〈Xn(ω), x∗〉‖L2(R+;H)

≤ ‖Xn(ω)‖γ(L2(R+;H),En)‖x∗‖ ≤ C(ω)‖x∗‖.

Next let x∗ ∈ E∗ be arbitrary; we may assume that x∗ ∈ B∗E. Since (x∗k)k≥1 is norm

dense in BE∗ we can find a subsequence (kn)n≥1 such that x∗ = limn→∞ x∗knstrongly.

It follows that for all m,n ≥ 1 we have

‖Φ∗(·, ω)(x∗kn− x∗km

)‖L2(R+;H) ≤ C(ω)‖x∗kn− x∗km

‖.

We deduce that (Φ∗(·, ω)x∗kn)n≥1 is a Cauchy sequence in L2(R+;H), and after passing

to an almost everywhere convergent limit we find that the limit equals Φ∗(·, ω)x∗.Hence, Φ∗(·, ω)x∗ = limn→∞ Φ∗(·, ω)x∗kn

in L2(R+;H). Since ω ∈ N was arbitrary,this proves the claim.

Step 4 – By Step 3, for ω ∈ N fixed we may define the Pettis integral operatorX(ω) : L2(R+;H) → E by

X(ω)f :=

∫R+

Φ(t, ω)f(t) dt.

We claim that X(ω) ∈ γ(L2(R+;H), E) and

‖X(ω)‖γ(L2(R+;H),E) ≤ supn≥1

‖Xn(ω)‖γ(L2(R+;H),En). (4.4.12)

To prove this, let the random variables ρn(ω) ∈ Lp(Ω′;E) be given by

ρn(ω) :=n∑i=1

γi

∫R+

Φ(t, ω)fi(t) dt,

where (γi)i≥1 is a Gaussian sequence on Ω and (fi)i≥1 is an orthonormal basis forL2(R+;H).

Let ε > 0 be arbitrary and fixed. Since ρn(ω) takes its values in a finite-dimensionalsubspace of E, it follows from Step 1 that there is an index Nn such that

E‖ρn(ω)‖2 ≤ (1 + ε)2E‖QNnρn(ω)‖2.

Clearly,

E‖QNnρn(ω)‖2 = E∥∥∥ n∑i=1

γi

∫R+

ΦNn(t, ω)fi(t) dt∥∥∥2

≤ ‖XNn(ω)‖2γ(L2(R+;H),ENn ),

and it follows that

supn≥1

E‖ρn(ω)‖2 ≤ (1 + ε)2 supN≥1

‖XN(ω)‖2γ(L2(R+;H),EN ).

4.5. Characterizations for the localized case 65

Since E does not contain a copy of c0, the theorem of Hoffmann-Jørgensen andKwapien (cf. [78, Theorem 9.29]) assures that X(ω) ∈ γ(L2(R+;H), E) and

‖X(ω)‖2γ(L2(R+;H),E) = sup

n≥1E‖ρn(ω)‖2 ≤ (1 + ε)2 sup

N≥1‖XN(ω)‖2

γ(L2(R+;H),EN ).

Since ε > 0 was arbitrary, the claim follows.Step 5 – To finish the proof, we note that X : Ω → γ(L2(R+;H), E) is almost

surely equal to a strongly measurable random variable by Lemma 3.5.2. It follows from(4.4.11) and (4.4.12) that X ∈ Lp(Ω; γ(L2(R+;H), E)). By definition X is representedby Φ and hence (3) follows.

4.5 Characterizations for the localized case

It will be convenient to introduce a continuous process

ξX : R+ × Ω → γ(L2(R+;H), E)

associated with a strongly measurable random variable X : Ω → γ(L2(R+;H), E) asfollows. For t ∈ R+ we define the γ(L2(R+;H), E)-valued random variable ξX(t) :Ω → γ(L2(R+;H), E) by

ξX(t, ω)f := (X(ω))(1[0,t]f), f ∈ L2(R+;H).

Note that by Lemma 3.2.2, limt→∞ ξX(t) = X in γ(L2(R+;H), E) a.s. The strongmeasurability of ξX as a γ(L2(R+;H), E)-valued random variable follows from Lemma3.5.1.

Lemma 4.5.1. Let X ∈ L0(Ω; γ(L2(R+;H), E)). The process ξX defined above hascontinuous trajectories. If X is scalarly adapted, then ξX is strongly measurable andadapted.

Proof. For ω ∈ Ω fixed, the right-ideal property implies that

‖ξX(t, ω)‖γ(L2(R+;H),E) ≤ ‖X(ω)‖γ(L2(R+;H),E).

By Lemma 3.2.2, t 7→ ξX(t, ω) is continuous for all ω ∈ Ω.Suppose next that X is scalarly adapted. Fix t ∈ R+. For f ∈ L2(R+;H) and

x∗ ∈ E∗, 〈ξX(t)f, x∗〉 = 〈Xf1[0,t], x∗〉 is Ft-measurable, so the Pettis measurability

theorem and Lemma 3.5.1 imply that ξX(t) is strongly Ft-measurable.If we combine the pathwise continuity with the adaptedness, the strong measura-

bility follows as well.

Remark 4.5.2. The process (t, ω) 7→ ‖ξX(t, ω)‖2γ(L2(R+;H),E) is nonnegative and in-

creasing. For X ∈ LpF(Ω; γ(L2(R+;H), E)) we may think of this process as an analogueof the quadratic variation process.


Let L0F(Ω; γ(L2(R+;H), E)) be the closure of

X ∈ L0(Ω; γ(L2(R+;H), E)) : X is elementary

For a stopping time τ and a scalarly adapted X ∈ L0(Ω; γ(L2(R+;H), E)) we definethe γ(L2(R+;H), E)-valued random variable ξX(τ) : Ω → γ(L2(R+;H), E) by

(ξX(τ))(ω)f := ξX(τ(ω), ω)f = X(ω)(1[0,τ ]f), f ∈ L2(R+;H).

The random variable ξX(τ) is well-defined since ξX has continuous paths and is adaptedby Lemma 4.5.1. By the right-ideal property,

‖ξX(τ)(ω)‖γ(L2(R+;H),E) ≤ ‖X(ω)‖γ(L2(R+;H),E).

In particular, if X ∈ LpF(Ω; γ(L2(R+;H), E)) for some p ∈ [1,∞), then Lemma 4.4.1implies that ξX(τ) ∈ LpF(Ω; γ(L2(R+;H), E)).

We can now localize Lemma 4.4.1.

Lemma 4.5.3. For a random variable X : Ω → γ(L2(R+;H), E) the following asser-tions are equivalent:

(1) X ∈ L0F(Ω; γ(L2(R+;H), E));

(2) X is strongly progressive with respect to F ;

(3) X is scalarly adapted with respect to F .

Proof. The implications (1)⇒(2)⇒(3) follow trivially from the definitions.(3)⇒(1): Define a sequence of stopping times (τn)n≥1 by

τn := inft ≥ 0 : ‖ξX(t)‖γ(L2(R+;H),E) ≥ n.

It follows that ξX(τn) ∈ Lp(Ω; γ(L2(R+;H), E)) for all p ∈ (1,∞), and since

〈ξX(τn), x∗〉 = 〈X, x∗〉1[0,τn]

is progressive for each x∗ ∈ E∗ it follows from Lemma 4.4.1 that each ξX(τn) ∈L0F(Ω; γ(L2(R+;H), E)). By Lemma 3.2.2 we have

X = limn→∞

ξX(τn) in γ(L2(R+;H), E) a.s.

As a consequence, X ∈ L0F(Ω; γ(L2(R+;H), E)).

The following lemma is a slight generalization of a stopping time argument in [88,Lemma 3.3]. For the convenience of the reader we include the details.


Lemma 4.5.4. Let p ∈ [1,∞). Let E1 and E2 be Banach spaces and let φ : R+×Ω →E1 and ψ : R+ × Ω → E2 be continuous adapted processes. Assume furthermore thatψ0 = 0 and that limt→∞ φ(t) and limt→∞ ψ(t) exist a.s. If there exists a constant C ≥ 0such that for all stopping times τ we have

E‖φτ‖pE1≤ C E‖ψτ‖pE2

(4.5.1)

whenever these norms are finite, then for all δ > 0 and ε > 0 we have

P(

supt∈R+

‖φt‖E1 > ε)≤ Cδp

εp+ P

(supt∈R+

‖ψt‖E2 ≥ δ). (4.5.2)

Proof. Let δ, ε > 0 be arbitrary. Define stopping times µ and ν by

µ(ω) := inft ∈ R+ : ‖φt(ω)‖E1 ≥ ε, ν(ω) := inft ∈ R+ : ‖ψt(ω)‖E2 ≥ δ,

and put τ := µ ∧ ν. Then τ is a stopping time and E‖φτ‖pE1≤ εp, E‖ψτ‖pE2

≤ δp. ByChebyshev’s inequality, (4.5.1), and pathwise continuity we have

P(

supt∈R+

‖φt‖E1 > ε, supt∈R+

‖ψt‖E2 < δ)≤ P

(‖φτ‖E1 ≥ ε

)≤ 1

εpE‖φτ‖pE1

≤ C

εpE‖ψτ‖pE2

≤ Cδp

εp,

where the last inequality uses the fact that ψ0 = 0. This implies

P(

supt∈R+

‖φt‖E1 > ε)≤ Cδp

εp+ P

(supt∈R+

‖φt‖E1 > ε, supt∈R+

‖ψt‖E2 ≥ δ).

Clearly (4.5.2) follows from this.

Lemma 4.5.5. Let E be a UMD space and let p ∈ (1,∞). For an element X inLpF(Ω; γ(L2(R+;H), E)) and a stopping time τ , we have

IWH (ξX(τ)) = IWH (X)τ almost surely. (4.5.3)

Proof. For elementary X, (4.5.3) is obvious. For general X ∈ LpF(Ω; γ(L2(R+;H), E))we do an approximation argument. Choose a sequence of elementary elements suchthat limn→∞Xn = X in LpF(Ω; γ(L2(R+;H), E)). Then ξX(τ) = limn→∞ ξXn(τ) inLpF(Ω; γ(L2(R+;H), E)) and it follows from Proposition 4.4.2 that

IWH (ξX(τ)) = limn→∞

IWH (ξXn(τ)) and IWH (ξX) = limn→∞

IWH (ξXn)

in Lp(Ω;Cb(R+;E)). In particular,

IWH (ξX)τ = limn→∞

IWH (ξXn)τ

in Lp(Ω;Cb(R+;E)). The general case of (4.5.3) now follows from the fact that (4.5.3)holds for all Xn.


Lemma 4.5.6. Let E be a UMD space and let p ∈ (1,∞). There is a constant Cp,Esuch that for all X ∈ LpF(Ω; γ(L2(R+;H), E)) and for all δ > 0 and ε > 0 we have

P(

supt∈R+

‖(IWH (X)t‖ > ε)≤ Cp,Eδ

p

εp+ P

(‖X‖γ(L2(R+;H),E) ≥ δ

)(4.5.4)

and

P(‖X‖γ(L2(R+;H),E) > ε

)≤ Cp,Eδ

p

εp+ P

(supt∈R+

‖(IWH (X))t‖ ≥ δ). (4.5.5)

Proof. For all ω ∈ Ω and t ∈ R+,

‖(ξX(t))(ω)‖γ(L2(R+;H),E) ≤ ‖X(ω)‖γ(L2(R+;H),E)

with equality for t→∞, and therefore,

‖X(ω)‖γ(L2(R+;H),E) = supt∈R+

‖(ξX(t))(ω)‖γ(L2(R+;H),E).

Hence by Lemma 4.5.4 it suffices to prove that for every stopping time τ we have

E‖IWH (X)τ‖p hp,E E‖ξX(τ)‖pγ(L2(R+;H),E)

provided both norms are finite. But this follows from Proposition 4.4.2 and Lemma4.5.5.

We will now extend Proposition 4.4.2.

Proposition 4.5.7 (Ito homeomorphism). Let E be a UMD space. The map-ping X 7→ IWH (X) has a unique extension to a homeomorphism from the spaceL0F(Ω; γ(L2(R+;H), E)) onto a closed subspace of L0(Ω;Cb(R+;E)). For every X ∈

L0F(Ω; γ(L2(R+;H), E)), IWH (X) is an element of Mc,loc

0 (Ω,F ;E) and the estimates(4.5.4) and (4.5.5) hold.

For Φ ∈ L0F(Ω; γ(R+;H,E)) we write IWH (Φ) for IWH (X) if Φ represents X ∈

L0F(Ω; γ(L2(R+;H), E)).

Proof. Fix X ∈ L0F(Ω; γ(L2(R+;H), E)) and define a sequence of stopping times

(τn)n≥1 byτn := inft ≥ 0 : ‖ξX(t)‖γ(L2(R+;H),E) ≥ n.

Clearly each ξX(τn) is in Lp(Ω; γ(L2(R+;H), E)) for every p ∈ (1,∞). Since

〈ξX(τn), x∗〉 = 〈X, x∗〉1[0,τn]

is adapted for every x∗ ∈ E∗, an application of Lemma 4.4.1 shows thatXn := ξX(τn) ∈LpF(Ω; γ(L2(R+;H), E)) for every p ∈ (1,∞).


By Proposition 4.4.2 we can define a sequence (Mn)n≥1 in Mc,loc0 (Ω;E) by

Mn := IWH (ξXn).

Since limn→∞Xn = X it follows from Lemma 4.5.6, applied to the differences Xm−Xn,that (Mn)n≥1 is a Cauchy sequence in L0(Ω;Cb(R+;E)). It follows that (Mn)n≥1

converges to M ∈ L0(Ω;Cb(R+;E)). As a process, M = (Mt)t∈R+ is adapted and

M0 = 0 almost surely. To show that M ∈Mc,loc0 (Ω;E) it is now enough to show that

it is a local martingale. We claim that,

M τm = Mm in Cb(R+;E) a.s.

This will complete the proof, since it shows that M is a local martingale with localizingsequence (τm)m≥1. To prove the claim we fix m ≥ 1. It follows from Lemma 4.5.5 thatfor all n ≥ m ≥ 1, a.s.

(Mn)τm = (IWH (ξXn))τm = IWH (ξXn(τm)) = Mm. (4.5.6)

in Cb(R+;E). By passing to a subsequence we may assume that limn→∞Mn = M inCb(R+;E) almost surely. Then also limn→∞(Mn)τm = M τm in Cb(R+;E) a.s., and theclaim now follows by letting n tend to infinity in (4.5.6). It follows that IWH (X) := Mis well-defined. At the same time, this argument shows that (4.5.4) extends to allX ∈ L0

F(Ω; γ(L2(R+;H), E)). This in turn shows that IWH is continuous.Next, we extend (4.5.5) to arbitrary X ∈ L0

F(Ω; γ(L2(R+;H), E)). Let M =IWH (ξX) and define a sequence of stopping times (τn)n≥1 as

τn = inft ≥ 0 : ‖ξX(t)‖ ≥ n.

By the above results we have IWH (ξXn) = M τn . Applying (4.5.5) to each Xn andletting n tend to infinity one obtains (4.5.5) for X. From this, we deduce that IWH

has a continuous inverse. This also shows that the mapping IWH has a closed rangein L0(Ω;Cb(R+;E)).

Remark 4.5.8. Lemma 4.5.5 extends to arbitrary X ∈ L0F(Ω; γ(L2(R+;H), E)). This

may be proved similarly as in Lemma 4.5.5, but now using Proposition 4.5.7 for theapproximation argument.

The next theorem is the main result of this section.

Theorem 4.5.9. Let E be a UMD space. For an H-strongly measurable and adaptedprocess Φ : R+ × Ω → B(H,E) which is scalarly in L0(Ω;L2(R+;H)) the followingassertions are equivalent:

(1) Φ is stochastically integrable.


(2) There exists a process ζ ∈ L0(Ω;Cb(R+;E)) such that for all x∗ ∈ E∗ we have

〈ζ, x∗〉 =

∫ ·

0

Φ∗(t)x∗ dWH(t) in L0(Ω;Cb(R+)).

(3) Φ ∈ γ(R+;H,E) almost surely.

(4) For almost all ω ∈ Ω, Φ(·, ω) is stochastically integrable with respect to WH .

In this situation Φ ∈ L0F(Ω; γ(R+;H,E)) and

IWH (Φ) = ζ =

∫ ·

0

Φ(t) dWH(t) in L0(Ω;Cb(R+;E)), (4.5.7)

E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p hp,E E‖Φ‖pγ(R+;H,E). (4.5.8)

The estimate (4.5.8) should be understood in the sense that the left-hand side isfinite if and only if the right-hand side is finite, in which case the estimates hold withconstants only depending on p and E. It can be seen as a vector-valued stochasticintegral version of the Burkholder-Davis-Gundy inequalities (see Remark 4.5.2).

Proof. (1)⇔(3): Notice that if (3) holds, then it follows from Lemma 3.5.2 that Φrepresents an element in L0(Ω; γ(L2(R+;H), E)). The equivalence and (4.5.7) may beproved in the same way as Theorem 4.4.3 using Lemma 4.5.3 and Proposition 4.5.7.

(3)⇒(4): It follows from Lemma 3.5.2 that for almost all ω ∈ Ω, Φ(·, ω) ∈γ(R+;H,E). The result now follows from Proposition 3.4.3.

(4)⇒(3): Let N be a null set such that Φ(·, ω) is stochastically integrable withrespect to WH for all ω ∈ N . Proposition 3.4.3 assures that for such ω we may defineX(ω) ∈ γ(L2(R+;H), E) defined by

X(ω)f =

∫R+

Φ(t, ω)f(t) dt.

An application of Lemma 3.5.2 shows that up to a null set, the resulting randomvariable X : Ω → γ(L2(R+;H), E) is strongly measurable. This proves (2).

(1)⇒(2): This may be proved as in Theorem 4.4.3, using (4.1.2).(2)⇒(1): It follows from Proposition 4.2.2 that ζ ∈ Mc,loc

0 (Ω;E). Let (τn)n≥1 bedefined as

τn = n ∧ inft ≥ 0 : ‖ζ(t)‖ ≥ n.

It follows from the assumptions and (4.1.1) that for all n and all x∗ ∈ E∗ we have

〈ζτn , x∗〉 =

∫ ·

0

1[0,τn](t)Φ∗(t)x∗ dWH(t) in Cb(R+;E) a.s.


By (4.1.3), each 1[0,τn]Φ is scalarly in L2(Ω;L2(R+;H)) and it follows that

〈ζ(τn), x∗〉 =

∫R+

1[0,τn](t)Φ∗(t)x∗ dWH(t) in L2(Ω).

By Theorem 4.4.3, each 1[0,τn]Φ is stochastically integrable with integral and it is easyto check that

ζτn =

∫ ·

0

1[0,τn](t)Φ∗(t)x∗ dWH(t).

Therefore, we can find elementary processes (Φn)n≥1 such that∥∥∥ζτn − ∫ ·

0

Φn(t) dWH(t)∥∥∥L0(Ω;Cb(R+;E))

<1

n.

It follows that∥∥∥ζ − ∫ ·

0


≤ ‖ζ − ζτn‖L0(Ω;Cb(R+;E)) +∥∥∥ζτn − ∫ ·

0


≤ ‖ζ − ζτn‖L0(Ω;Cb(R+;E)) +2

n.

The latter clearly converges to 0 as n tends to infinity. This gives (ii). Now choosex∗ ∈ E∗ arbitrary. Since∫ ·

0

Φ∗(t)x∗ dWH(t) = limn→∞

∫ ·

0

Φ∗n(t)x

∗ dWH(t) in L0(Ω;C(R+))

an applications of (4.1.2) gives (i).Finally, we show that (4.5.8) holds. Let X ∈ L0

F(Ω; γ(L2(R+;H), E)) be such thatΦ represents X. First assume that the left-hand side is finite. Define a sequence ofstopping times (τn)n≥1 by

τn = inft ≥ 0 : ‖ξX(t)‖γ(L2(R+;H),E) ≥ n.

Observe that ξX(τn) ∈ LpF(Ω; γ(L2(R+;H), E)) and that it is represented by Φ1[0,τn].It follows that Φ1[0,τn] satisfies the equivalent conditions in Theorem 4.4.3. Combiningthe identity ∫ τn∧t

0

Φ(s) dWH(s) =

∫ t

0

1[0,τn](s)Φ(s) dWH(s), t ∈ R+

which follows for instance from Theorem 4.5.9(1), with the dominated convergencetheorem (here we use the assumption) and Fatou’s lemma, we obtain

E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p = lim

n→∞E supt∈R+

∥∥∥∫ t

0

1[0,τn](s)Φ(s) dWH(s)∥∥∥p

hp,E lim inf ‖ξX(τn)‖pLp(Ω;γ(L2(R+;H),E)) ≥ ‖X‖pLp(Ω;γ(L2(R+;H),E)).


This shows that X ∈ Lp(Ω; γ(L2(R+;H), E)) and the result now follows from Theorem4.4.3. If the right-hand side is finite, then the result can be deduced from Theorem4.4.3.

Corollary 4.5.10 (Series expansion). Let E be a UMD space and let Φ : R+ ×Ω → B(H,E) be an H-strongly measurable and adapted process which is scalarly inL0(Ω;L2(R+;H)). If Φ is stochastically integrable with respect to WH , then for allh ∈ H the process Φh : R+ × Ω → E is stochastically integrable with respect to WHh.Moreover, if (hn)n≥1 is an orthonormal basis for H, then∫ ·

0

Φ(t) dWH(t) =∑n≥1

∫ ·

0

Φ(t)hn dWH(t)hn,

with unconditional convergence in L0(Ω;Cb(R+;E)).

Proof. Let PN be the orthogonal projection in H onto the span of the vectors (hn)Nn=1.

Let X ∈ Lp(Ω; γ(L2(R+;H), E)) be the element represented by Φ. By the right-idealproperty we have

‖X PN‖γ(L2(R+;H),E) ≤ ‖X‖γ(L2(R+;H),E)

almost surely. Here we think of PN as an operator on γ(L2(R+;H), E) defined by(PNS)f := S(PNf) with (PNf)(t) := PN(f(t)). By an approximation argument onecan show that

limN→∞

‖X −X PN‖γ(L2(R+;H),E) = 0,

almost surely. Since ΦPN is represented by X PN , the result follows from Proposition4.5.7 and Theorem 4.5.9. The convergence of the series is unconditional since anypermutation of (hn)n≥1 is again an orthonormal basis for H.

One can extend the stochastic integral to a larger class of processes. This will beused in Sections 4.6 and 4.7.

We say that a sequence X := (Xn)n≥1 is in L(WH ;E) if for all n ≥ 1, Xn ∈L0F(Ω; γ(L2(R+;H), E)) and Xm = Xn Qm for all 1 ≤ m ≤ n, where Qn : Ω →B(L2(R+;H)) are defined as Qnf = 1[0,τn]f for some increasing sequence of stoppingtimes (τn)n≥1 with limn→∞ τn = ∞ a.s.

Let E be a UMD space. We extend IWH , to the space L(WH ;E) in the followingway:

IWH (X)(t) = IWH (X Qn)(t), for 0 ≤ t ≤ τn.

One can check that this is well-defined and that IWH maps into the space Mc,loc0 (Ω;E).

For an H-strongly measurable and adapted process Φ : R+ × Ω → B(H,E) whichis in L0(Ω;L2

loc(R+;H)) scalarly, we say that Φ represents X ∈ L(WH ;E) if for alln ≥ 1, Φ1[0,τn] is represented by Xn.

For an H-strongly measurable and adapted process Φ : R+ × Ω → B(H,E) whichis in L0(Ω;L2

loc(R+;H)) scalarly, we say that Φ is locally stochastically integrable if for

4.6. Criteria for stochastic integrability 73

all T > 0, Φ1[0,T ] is stochastically integrable. For such processes Φ, we may extendthe stochastic integral as∫ t

0

Φ(s) dWH(s) =

∫ t

0

Φ(s)1[0,n] dWH(s) for t ∈ [0, n).

This is well-defined and∫ ·

0Φ(s) dWH(s) is in L0(Ω;C(R+;E)). For UMD spaces E,

one can check that by Theorem 4.5.9, Φ is locally stochastically integrable if and onlyif Φ represents an element in L(WH ;E).

4.6 Criteria for stochastic integrability

By combining Corollary 3.4.5 with Theorem 4.5.9 and recalling the fact that UMDspaces have finite cotype, we obtain:

Theorem 4.6.1. Let E be UMD Banach function space over a σ-finite measure space(S,Σ, µ). Let Φ : R+×Ω → B(H,E) be an H-strongly measurable and adapted processand assume that there exists a strongly measurable function φ : R+×Ω×S → H suchthat for all h ∈ H and t ∈ R+,

(Φ(t)h)(·) = [φ(t, ·), h]H in E.

Then Φ is stochastically integrable if and only if∥∥∥( ∫R+

‖φ(t, ·)‖2H dt

) 12∥∥∥E<∞ a.s.

In this case, we have for all p ∈ (1,∞),

E supt∈R+

∥∥∥∫ t

0

Φ(t) dWH(t)∥∥∥p hp,E E

∥∥∥( ∫R+

‖φ(t, ·)‖2H dt

) 12∥∥∥pE.

As a consequence of Proposition 3.3.1 and Theorem 4.5.9 we obtain the followingresult in case of type 2 and cotype 2:

Theorem 4.6.2. Let E be a UMD space and let p ∈ (1,∞).

(1) If E has type 2, then every H-strongly measurable and adapted process Φ ∈L2(R+; γ(H,E)) a.s. is stochastically integrable and for all p ∈ (1,∞) we have

E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p .p,E E‖Φ‖pL2(R+;γ(H,E)),

whenever the right-hand side is finite.


(2) If E has cotype 2, then every H-strongly measurable stochastically integrableprocess Φ belongs to L2(R+; γ(H,E)) a.s. and for all p ∈ (1,∞) we have

E‖Φ‖pL2(R+;γ(H,E)) .p,E E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p,


A similar result for type p and cotype q spaces can be obtained with Theorem3.3.2.

We continue with a multiplier result which extends Proposition 3.2.4.

Proposition 4.6.3. Let E1 and E2 be UMD spaces, fix p ∈ (1,∞). and let N :R+ × Ω → B(E1, E2) be an E1-strongly measurable and adapted process such that

K := ess supω∈Ω

γ(N(t, ω) : t ∈ R+) <∞.

If Φ : R+ ×Ω → B(H,E1) is an H-strongly measurable process which is stochasticallyintegrable with respect to WH , then the process NΦ : R+ × Ω → B(H,E2) defined by(NΦ)(t)h := N(t)(Φ(t)h) is stochastically integrable with respect to WH as well andfor all p ∈ (1,∞),

E supt∈R+

∥∥∥∫ t

0

N(s)Φ(s) dWH(s)∥∥∥p .p,E1,E2 K

p E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p,


Proof. It follows from Theorem 4.4.3 that for a.a. ω ∈ Ω, Φ(·, ω) is stochasticallyintegrable with respect to WH . Combining Propositions 3.2.4 and 3.4.3, we see thatfor a.a. ω ∈ Ω the function N(·, ω)Φ(·, ω) is stochastically integrable with respect toWH and∥∥∥∫

R+

N(t, ω)Φ(t, ω) dWH(t)∥∥∥pLp(Ω;E2)

.p Kp∥∥∥∫

R+

Φ(t, ω) dWH(t)∥∥∥pLp(Ω;E1)

.

Since NΦ is H-strongly measurable and adapted, the result follows from Theorems4.4.3 and 4.5.9.

Next we show that certain Bochner integral processes are locally stochasticallyintegrable.

Proposition 4.6.4. Let E be a UMD Banach space. Let Ψ : R+ × Ω → γ(H,E) beH-strongly measurable and adapted. Assume that Ψ ∈ L1

loc(R+; γ(H,E)) almost surelyand define the process Φ : R+ × Ω → γ(H,E) as

Φ(t) =

∫ t

0

Ψ(s) ds.


Then Φ is locally stochastically integrable and for all T ∈ R+ and p ∈ (1,∞),

E supt∈[0,T ]

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p .p,E

√T E‖Ψ‖pL1(0,T ;γ(H,E)). (4.6.1)

Proof. Clearly, Φ is H-strongly measurable and adapted and scalarly in L2(0, T ;H)almost surely. The γ-norm of Φ can be estimated using an argument of Kalton andWeis [67]. Namely it holds that a.s.

‖Φ‖γ(0,T ;H,E) =∥∥∥t 7→ ∫ T

0

1[0,t](s)Ψ(s) ds∥∥∥γ(0,T ;H,E)

≤∫ T

0

‖t 7→ 1[0,t](s)Ψ(s)‖γ(0,T ;H,E) ds ≤∫ T

0

√T − s ‖Ψ(s)‖γ(H,E) ds.

Therefore, the result follows from Theorem 4.5.9.

In the last part of this section we will show that certain operator-valued localmartingales are locally stochastically integrable. In particular, we show that the in-tegral process

∫ ·0Φ(s) dWH(s) is locally stochastically integrable with respect to an

F -adapted standard Brownian motion W .When E is a real Hilbert space and p ∈ (1,∞), the answer is clearly affirmative

and by the Burkholder-Davis-Gundy inequalities we have for all T ∈ [0,∞)(E

∥∥∥∫ T

0

∫ t

0

Φ(s) dWH(s) dW (t)∥∥∥p) 1

p hp

∥∥∥∫ ·

0

Φ(s) dWH(s)∥∥∥Lp(Ω;L2(0,T ;E))

≤√T

(E supt∈[0,T ]

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p) 1

p hp

√T‖Φ‖Lp(Ω;L2(0,T ;E)),

(4.6.2)

whenever the right-hand side is finite. More generally, every B2(H,E)-valued Lp-martingale, where E is a Hilbert space, is locally stochastically integrable, and anestimate can be given using Doob’s inequality. In the following we shall generalize theseobservations to γ(H,E)-valued local Lp-martingales, where E is a UMD space. We willsay that a processM : R+×Ω → γ(H,E) is an Lp-martingale ifM(t) ∈ Lp(Ω; γ(H,E))for all t ∈ R+ and E(M(t)|Fs) = M(s) in Lp(Ω; γ(H,E)) for all 0 ≤ s ≤ t < ∞. Wesay that M : R+ × Ω → γ(H,E) is a local Lp-martingale if there exists an increasingsequence of stopping times (τn)n≥1 with limn→∞ τn = ∞ a.s. such that each M τn is anLp-martingale.

Theorem 4.6.5. Let E be a UMD space and fix p ∈ (1,∞). Let M : R+×Ω → γ(H,E)be an Lp-martingale and assume that M(0) = 0, then M is locally stochastically inte-grable with respect to WH . Moreover, for all T ∈ R+,(

E supt∈[0,T ]

∥∥∥∫ t

0

M(s) dWH(s)∥∥∥p) 1

p.p,E

√T

(E‖M(T )‖pγ(H,E)

) 1p .


Proof. Recall from Proposition 2.2.2 that we may assume that M has cadlag paths.We check that M belongs to L0(Ω;L2(0, T ;H)) scalarly. Let x∗ ∈ E∗ be fixed. Wemay apply Doob’s maximal inequality to obtain

E‖M∗x∗‖pL2(0,T ;H) ≤ Tp2 E sup

t∈[0,T ]

‖M∗(t)x∗|p .p Tp2 E‖M∗(T )x∗‖pH .

Let B = Lp0(Ω,A;E) be the closed subspace in Lp(Ω;E) of all random variableswith zero mean, and define the bounded and strongly left continuous function N :R+ → B(B) by

N(t)ξ := E(ξ|Ft), ξ ∈ B, t ∈ [0, T ].

Since E is a UMD space, by a result of Bourgain [18] the set N(t) : t ∈ R+ isR-bounded, and therefore γ-bounded, with γ-bound depending only on p and E. Thecorresponding estimate is sometimes called the vector-valued Stein inequality.

By the Fubini isomorphism (see Proposition 3.5.3) we may identify the randomvariables M(t) ∈ Lp(Ω; γ(H,E)) with operators M(t) ∈ γ(H,Lp(Ω;E)). Recall thatfor all t ∈ [0, T ], for all h ∈ H, for almost all ω ∈ Ω, (M(t)h)(ω) = M(t, ω)h Define aconstant function G : [0, T ] → B(H,B) by

G(t) := M(T ), t ∈ [0, T ].

Clearly G represents the element RG ∈ γ(L2(0, T ;H), B) given by

RGf =

∫ T

0

M(T )f(t) dt, f ∈ L2(0, T ;H)

and ‖RG‖γ(L2(0,T ;H),B =√T E‖M(T )‖γ(H,E). Since for all t ∈ [0, T ], M(t) = N(t)M(T )

in B, we may apply the above multiplier result to conclude that M represents anelement R ∈ γ(L2(0, T ;H), B) with

‖R‖γ(L2(0,T ;H),B) .p,E ‖RG‖γ(L2(0,T ;H),B).

Using the γ-Fubini isomorphism we define X = Fub−1(R). Recall that for all f ∈L2(0, T ;H), for almost all ω ∈ Ω, (Rf)(ω) = X(ω)f .

We claim that X is represented by M . Once we know this, it follows with Theorem4.4.3 that(

E∥∥∥∫ T

0

M(t) dWH(t)∥∥∥p) 1

p hp,E

(E‖M‖pγ(0,T ;H,E)

) 1p =

(E‖X‖pγ(L2(0,T ;H),E)

) 1p

hp ‖R‖γ(L2(0,T ;H),B) .p,E

√T

(E‖M(T )‖pγ(H,E)

) 1p .

Let f ∈ L2(0, T ;H), x∗ ∈ E∗ be arbitrary. We have to show that [M∗x∗, f ]L2(0,T ;H) =〈Xf, x∗〉 almost surely. It suffices to check that

E(1A[M∗x∗, f ]L2(0,T ;H)) = E(1A〈Xf, x∗〉)


for all A ∈ FT . By the Fubini theorem we have

E(1A[M∗x∗, f ]L2(0,T ;H)) =

∫Ω

∫ T

0

〈M(t, ω)f(t), x∗〉1A(ω) dt dP (ω)

=

∫ T

0

∫Ω

〈M(t, ω)f(t), x∗〉1A(ω) dP (ω) dt

=

∫ T

0

〈M(t)f(t),1A ⊗ x∗〉 dt

= 〈Rf,1A ⊗ x∗〉 = E(〈Xf, x∗〉1A).

This proves the claim.

Next we prove a localized version of this result.

Theorem 4.6.6. Let E be a UMD space. Let M : R+ × Ω → γ(H,E) be a local Lp-martingale for some p ∈ (1,∞] satisfying M(0) = 0. Then M is locally stochasticallyintegrable with respect to WH .

If moreover M : R+ × Ω → γ(H,E) is a continuous local martingale, then for allp ∈ (1,∞), T ∈ R+ and δ, ε > 0 we have

P(

supt∈[0,T ]

∥∥∥∫ t

0

Ms dWH(s)∥∥∥ > ε

)≤ Cp,Eδ

p

εpT

12 + P( sup

t∈[0,T ]

‖Mt‖γ(H,E) ≥ δ), (4.6.3)

where Cp,E is a constant which depends only on p and E.

Proof. Let T ∈ R+ be arbitrary. Let (τn)n≥1 be increasing, limn→∞ τn = ∞ and suchthat each M τn is an Lp-martingale. By Theorem 4.6.5 M τn is locally stochasticallyintegrable and hence by Theorem 4.5.9 M τn ∈ γ(0, T ;H,E) a.s. Since limn→∞ τn = ∞this implies M ∈ γ(0, T ;H,E) a.s. We may apply Theorem 4.5.9 to conclude that Mis locally stochastically integrable with respect to WH .

For the proof of the second part we proceed in two steps.Step 1 – We first assume that M is a continuous Lp-martingale for some p ∈ (1,∞).

Then M is locally stochastically integrable by the above considerations. To prove(4.6.3), by Lemma 4.5.4 it is enough to show that

E∥∥∥∫ τ

0

Mt dWH(t)∥∥∥p .p,E T

p2 E‖Mτ‖pγ(H,E) (4.6.4)

for all stopping times with values in [0, T ]. But, using (4.5.8) and Theorem 4.6.5 weobtain

E∥∥∥∫ τ

0

Mt dW (t)∥∥∥p = E

∥∥∥∫ T

0

Mt1[0,τ ](t) dW (t)∥∥∥p = E

∥∥∥∫ T

0

Mt∧τ1[0,τ ](t) dW (t)∥∥∥p

= ‖M τ1[0,τ ]‖Lp(Ω;γ(0,T ;H,E)) ≤ ‖M τ‖Lp(Ω;γ(0,T ;H,E)) .p,E Tp2 E‖Mτ‖pγ(H,E).

This proves the result for continuous Lp-martingales M .


Step 2 – Let M be a continuous local martingale. Choose a localizing sequence(τn)n≥1 such that each M τn is a continuous Lp-martingale and τn a.s. Then M τn islocally Lp-stochastically integrable. By Theorem 4.5.9 M τn ∈ γ(0, T ;E) a.s. Sincefor almost all ω ∈ Ω, we have M τn(·, ω) = M(·, ω) for n large. It follows that M ∈γ(0, T ;E) a.s. By Step 1 (4.6.3) holds for all M τn . We obtain (4.6.3) for M by lettingn tend to infinity.

As a consequence we have the following generalization of (4.6.2).

Corollary 4.6.7. Let E be a UMD space. Let WH and W be an H-cylindrical Brown-ian motion and a Brownian motion, respectively, both adapted to the filtration F . Ifthe H-strongly measurable and adapted process Φ : R+ × Ω → B(H,E) is locally sto-chastically integrable with respect WH , then the integral process

(∫ t

0Φ(s) dWH(s)

)t∈[0,T ]

is locally stochastically integrable with respect to W and for all p ∈ (1,∞) and T ∈ R+

we have (E

∥∥∥∫ T

0

∫ t

0

Φ(s) dWH(s) dW (t)∥∥∥p) 1

p.p,E

√T ‖Φ‖Lp(Ω;γ(0,T ;H,E)).

4.7 Brownian filtrations

In this section several results are improved under the condition that F = FWH is theaugmented filtration induced by WH , that is the augmentation of the filtration

σ(WH(s)h : 0 ≤ s ≤ t, h ∈ H), t ≥ 0.

In particular, we show that the martingale representation theorem for E-valuedlocal martingales holds.

Theorem 4.7.1 (Representation of UMD-valued Brownian local martingales). LetE be a UMD space. Then every E-valued local martingale M := (Mt)t∈R+ adaptedto the augmented filtration FWH has a continuous version and there exists a uniqueX ∈ L(WH ;E) such that

M = M0 + IWH (X). (4.7.1)

For the proof, we need a representation theorem for FWH∞ -measurable random vari-

ables. It can be seen as an extension of Proposition 4.7.2 in the case of a Brownianfiltration.

Proposition 4.7.2. Let E be a UMD space and fix p ∈ (1,∞). For every ξ ∈Lp(Ω,FWH

∞ ;E) with Eξ = 0 there is a unique X ∈ LpF(Ω; γ(L2(R+;H), E)) such that

IWH (X)∞ = ξ a.s.

4.7. Brownian filtrations 79

Proof. Let Lp0(Ω,F∞;E) be the closed subspace of Lp(Ω;E) consisting of all F∞-measurable random variables with mean zero. It follows from Proposition 4.4.2 thatwe may define a mapping IWH

∞ : LpF(Ω; γ(L2(R+;H), E)) → Lp0(Ω,F∞;E) as

IWH∞ (X) = IWH (X)(∞).

In this way IWH∞ is an isomorphism onto its closed range in Lp0(Ω,F∞;E). Now for the

result it suffices to show that IWH∞ has dense range in Lp0(Ω,FWH

∞ ;E).

Let (hk)k≥1 be a fixed orthonormal basis for H. For K = 1, 2 . . . let F (K)∞ be denote

for the augmented σ-algebra generated by

WH(t)hk : t ∈ R+, 1 ≤ k ≤ K.

Since FWH∞ is generated by the σ-algebras F (K)

∞ , by the martingale convergence the-orem and approximation we may assume that η is in Lp0(Ω,FK

∞;E) and of the form∑Nn=1(1An − P (An))⊗ xn with An ∈ FK

∞ and xn ∈ E. From linearity and the identity

IWH∞ (φ⊗ x) = (IWH

∞ (φ))⊗ x, φ ∈ LpF(Ω;L2(R+;H)),

it is clear that it even suffices to show that 1An−P (An) is given by IWH∞ (φ) for some φ ∈

LpF(Ω;L2(R+;H)). But by the Ito representation theorem for Brownian martingales(cf. [63, Lemma 18.11], [68, Theorem 3.4.15]), there exists φ ∈ L2

F(Ω;L2(R+;H)) suchthat 1An − P (An) =

∫R+φ(t) dW (t). The quadratic variation of

∫ ·0φ(t) dW (t) is given

by∫ ·

0φ2(t) dt (cf. [63, Theorem 17.11]). Now (4.1.3) and Doob’s maximal inequality

imply that φ ∈ LpF(Ω;L2(R+;H)).

Proof of Theorem 4.7.1. We may take M0 = 0.Step 1: First assume that there is a T > 0 such that MT = Mt ∈ L1(Ω;E) for

all t > T . Then clearly, M∞ = MT exists in L1(Ω,FWH∞ ;E). We show that M has a

continuous version.Choose (ξn)n≥1 in L2(Ω,FWH

∞ ;E) such that M∞ = limn→∞ ξn. By Proposition4.7.2 the martingales (Mn)n≥1 defined as Mn

t = E(ξn|FWHt ) a.s. have bounded and

continuous paths. It follows from Doob’s maximal inequality (cf. the continuous timeversion of [63, Proposition 7.15] or [115, Theorem 70.1]) that for all ε > 0 and n,m ≥ 1,

P((Mn −Mm)∗ > ε) ≤ ε−1E‖ξn − ξm‖.

This shows that (Mn)n≥1 is a Cauchy sequence in L0(Ω;Cb(R+;E)). Its limit is therequired version of M .

Step 2: Under the assumption of Step 1, we show that there is a unique X ∈L0F(Ω; γ(L2(R+;H), E)) such that IWH (X) = M .

The uniqueness of X follows from the injectivity of IWH . For each n ≥ 1 define astopping time τn as

τn = inft ≥ 0 : ‖Mt‖ ≥ n ∧ n.


By Proposition 4.7.2 there is a sequence (Xn)n≥1 in L0F(Ω; γ(R+;H,E)) such that

IWH (Xn) = M τn .

Clearly, (M τn)n≥1 converges to M in L0(Ω;Cb(R+;E)). It follows from Proposition4.5.7 that (Xn)n≥1 is a Cauchy sequence in L0

F(Ω; γ(L2(R+;H), E)) and therefore itconverges to some X ∈ L0

F(Ω; γ(L2(R+;H), E)). It follows from Proposition 4.5.7 thatIWH (X) = M .

Step 3: We prove the general case. The uniqueness follows from the injectivity ofIWH . Fix N Let (τn)n≥1 be a localizing sequence for M . For n ≥ 1 the martingaleNn = Mn∧τn satisfies the above properties with T = n. Since the result holds foreach Nn, we obtain a sequence (Xn)n≥1 in L0

F(Ω; γ(L2(R+;H), E)) such that Nn =IWH (Xn). By the uniqueness it follows that for all 1 ≤ m ≤ n, Xn Qm = Xm, whereQm = 1[0,τm]1[0,m]. We conclude that we may take X = (Xn)n≥1.

As a consequence of Theorems 4.6.2 and 4.7.1 we have the following result forspaces with cotype 2.

Corollary 4.7.3. Let E be a UMD space with cotype 2. Then every E-valued localmartingale M := (Mt)t∈R+ adapted to the augmented filtration FWH has a continuousversion and there exists an a.e. unique locally stochastically integrable Φ : R+ × Ω →B(H,E) such that

M = M0 +

∫ ·

0

Φ(t) dWH(t).

In particular, Φ ∈ L2(R+; γ(H,E)) a.s.

To end this section we have the following extension of Theorems 4.4.3 and 4.5.9 incase the filtration is the Brownian filtration.

Proposition 4.7.4. If the filtration F is assumed to be the augmented Brownianfiltration FWH , then Theorems 4.4.3 and 4.5.9 (1)⇔(2) are true for arbitrary Banachspaces E.

Proof. (1)⇒(2): This is clear.The proof of Theorem 4.4.3 (2)⇒(1) is similar to the proof of Proposition 4.7.2.

Therefore, we only give a sketch of the proof. For K = 1, 2, . . . let F (K)∞ be the σ-

algebra generated by the Brownian motions WHhk, 1 ≤ k ≤ K. Choose a sequenceof simple random variables (ηn)n≥1 in Lp(Ω,F (n)

∞ ;E) with mean zero and such thatη = limn→∞ ηn. This is possible by the martingale convergence theorem and thePettis measurability theorem. By the martingale representation theorem for finite-dimensional spaces, for all n ≥ 1 there exists a stochastically integrable process Φn suchthat ηn =

∫R+

Φn(t) dWH(t). The sequence (Φn)n≥1 satisfies (i) and (ii) of condition

(1) of Theorem 4.4.3. Indeed, (ii) follows from Doob’s maximal inequality and (i)follows from the (4.1.3). The processes Φn need not be elementary, but since each Φn

takes values in a finite-dimensional subspace of E one can approximate the Φn withelementary processes to complete the proof.

Theorem 4.5.9 (2)⇒(1) may be proved as before.

4.8. The Ito formula 1 81

4.8 The Ito formula 1

The next lemma defines a trace which will be needed in the statement of the Itoformula.

Lemma 4.8.1. Let E,F,G be Banach spaces and let (hn)n≥1 be an orthonormal basisof H. Let R ∈ γ(H,E), S ∈ γ(H,F ) and T ∈ B(E,B(F,G)) be given. Then the sum

TrR,ST :=∑n≥1

(TRhn)(Shn) (4.8.1)

converges in G and does not depend on the choice of the orthonormal basis (hn)n≥1.Moreover,

‖TrR,ST‖ ≤ ‖T‖‖R‖γ(H,E)‖S‖γ(H,F ). (4.8.2)

In particular, Lemma 4.8.1 applies to T ∈ B(E,B(E∗,R)) given by (Tx)(x∗) =〈x, x∗〉 and R ∈ γ(H,E) and S ∈ γ(H,E∗). Moreover, in [67] it has been shown thatone has S∗R ∈ C1(H) with

‖S∗R‖C1(H) ≤ ‖R‖γ(H,E)‖S‖γ(H,E∗) and Tr(S∗R) = TrR,ST.

Here C1(H) denotes the trace class operators.

Proof. First assume that S =∑N

n=1 hn⊗ yn for y1, . . . , yN ∈ F . Then the convergence

of the series in (4.8.1) is obvious. Letting ξR =∑N

n=1 γnRhn and ξS =∑N

n=1 γnyn weobtain

‖TrR,ST‖ = ‖ET (ξR)(ξS)‖ ≤ E‖T (ξR)(ξS)‖ ≤ ‖T‖E‖ξR‖‖ξS‖≤ ‖T‖(E‖ξR‖2)

12 (E‖ξS‖2)

12 ≤ ‖T‖‖R‖‖S‖.

Now let S ∈ γ(H,F ) be arbitrary. For each N ≥ 1, let PN ∈ B(H) denote theorthogonal projection on spanhn : n ≤ N. Letting, Sn = S Pn, we have S =limn→∞ Sn in γ(H,F ). For all m,n ≥ 1, we have

‖TrR,SnT − TrR,SmT‖ = ‖TrR,Sn−SmT‖ ≤ ‖T‖‖R‖γ(H,E)‖Sn − Sm‖γ(H,F ).

Therefore, (TrR,SnT )n≥1 is a Cauchy sequence in G, and it converges. Clearly, for all

N ≥ 1, TrR,SNT =

∑Nn=1(TRhn)(Shn). Now the convergence of (4.8.1) and (4.8.2)

follow.Next, we show that the trace is independent of the choice of the orthonormal

basis. Let (en)n≥1 be another orthonormal basis for H. For R =∑M

m=1 hm ⊗ xm for

x1, . . . , xM ∈ E and S =∑N

n=1 hn ⊗ yn as before, we have∑k≥1

T (Rek)(Sek) =∑k≥1

∑m≥1

∑n≥1

[ek, hm][ek, hn]T (Rhm)(Shn)

=M∑m=1

N∑n=1

∑k≥1

[ek, hm][ek, hn]T (Rhm)(Shn)

=M∑m=1

N∑n=1

δmnT (Rhm)(Shn) = TrR,ST.


The general case follows from an approximation argument as before.

If E = F we shall write TrR := TrR,R.A function f : R+ ×E → F is said to be of class C1,2 if f is continuously differen-

tiable in the first variable and twice continuously Frechet differentiable in the secondvariable. The derivatives with respect to the first and second variable will be denotedby D1f and D2f , respectively. We can now formulate the main theorem of this section.

Theorem 4.8.2 (Ito formula). Let E and F be UMD spaces. Assume that f : R+ ×E → F is of class C1,2. Let Φ : R+ × Ω → B(H,E) be an H-strongly measurableand adapted process which is locally stochastically integrable with respect to WH andassume that the paths of Φ belong to L2

loc(R+; γ(H,E)) a.s. Let ψ : R+ × Ω → Ebe strongly measurable and adapted with paths in L1

loc(R+;E) a.s. Let ξ : Ω → E bestrongly F0-measurable. Define ζ : R+ × Ω → E by

ζ = ξ +

∫ ·

0

ψ(s) ds+

∫ ·

0

Φ(s) dWH(s).

Then s 7→ D2f(s, ζ(s))Φ(s) is locally stochastically integrable and almost surely wehave for all t ∈ R+,

f(t, ζ(t))− f(0, ξ) =

∫ t

0

D1f(s, ζ(s)) ds+

∫ t

0

D2f(s, ζ(s))ψ(s) ds

+

∫ t

0

D2f(s, ζ(s))Φ(s) dWH(s)

+1

2

∫ t

0

TrΦ(s)

(D2

2f(s, ζ(s)))ds.

(4.8.3)

Notice that a.s. ζ has continuous paths. Since f is of class C1,2 we obtain thata.s. D1f(·, ζ), D2f(·, ζ) and D2f(·, ζ) are locally bounded. It follows that the first twointegrals are a.s. pathwise well-defined as a Bochner integral. Also the last integral isa.s. pathwise defined as a Bochner integral. To see this, notice that by Lemma 4.8.1a.s., we have∫ t

0

∥∥TrΦ(s)

(D2

2f(s, ζ(s)))∥∥ ds ≤ ∫ t

0

‖D22f(s, ζ(s))‖‖Φ(s)‖2

γ(H,E) ds

≤ sups∈[0,t]

‖D22f(s, ζ(s))‖‖Φ‖2

L2(0,t;γ(H,E)).

The next remark follows from Theorem 4.6.2.

Remark 4.8.3. Assume that the conditions of Theorem 4.8.2 are fulfilled.

(i) If, in addition to, we assume that E has type 2, then the assumption that Φ islocally stochastically integrable is automatically fulfilled since we assumed Φ ∈L2

loc(R+; γ(H,E)) a.s.


(ii) If E has cotype 2, then the assumption that Φ ∈ L2loc(R+; γ(H,E)) is automati-

cally fulfilled since we assumed Φ is locally stochastically integrable.

As a consequence of Theorem 4.8.2 we obtain:

Corollary 4.8.4. Let E1, E2 and F be UMD Banach spaces and let f : E1×E2 → Fbe a bilinear map. Let (hn)n≥1 be an orthonormal basis of H. For i = 1, 2 let Φi :R+ × Ω → B(H,Ei), ψi : R+ × Ω → Ei and ξi : Ω → Ei satisfy the assumptions ofTheorem 4.8.2 and define

ζi(t) = ξi +

∫ t

0

ψi(s) ds+

∫ t

0

Φi(s) dWH(s).

Then, almost surely for all t ∈ R+,

f(ζ1(t), ζ2(t))− f(ζ1(0), ζ2(0)) =

∫ t

0

f(ζ1(s), ψ2(s)) + f(ψ1(s), ζ2(s)) ds

+

∫ t

0

f(ζ1(s),Φ2(s)) + f(Φ1(s), ζ2(s)) dWH(s)

+

∫ t

0

∑n≥1

f(Φ1(s)hn,Φ2(s)hn) ds.

In particular, for a UMD space E, taking E1 = E, E2 = E∗, F = R and f(x, x∗) =〈x, x∗〉, it follows that almost surely for all t ∈ R+,

〈ζ1(t), ζ2(t)〉−〈ζ1(0), ζ2(0)〉

=

∫ t

0

〈ζ1(s), ψ2(s)〉+ 〈ψ1(s), ζ2(s)〉 ds

+

∫ t

0

〈ζ1(s),Φ2(s)〉+ 〈Φ1(s), ζ2(s)〉 dWH(s)

+

∫ t

0

∑n≥1

〈Φ1(s)hn,Φ2(s)hn〉 ds.

(4.8.4)

For the proof of Theorem 4.8.2 we need two lemmas.

Lemma 4.8.5. Let E be a UMD space. Let Φ : R+ × Ω → B(H,E) be H-stronglymeasurable and adapted. Assume that Φ is stochastically integrable with respect to WH

and Φ ∈ L0(Ω;L2(R+; γ(H,E))). Then there exists a sequence of elementary processes(Φn)n≥1 such that

Φ = limn→∞

Φn in L0(Ω;L2(R+; γ(H,E))) ∩ L0(Ω; γ(L2(R+;H), E)).

Proof. Let (hn)n≥1 be an orthonormal basis forH and denote by Pn the projection ontothe span of h1, . . . , hn in H. Let (Dn)n≥1 be an increasing sequence of σ-algebras on


R+ generated by finitely many bounded intervals and such that BR+ = σ(Dn, n ≥ 1).Define Ψn : R+ × Ω → γ(H,E) as

Ψn(t, ω)h := E(Rδn(Φ(·, ω)Pnh)|Dn),

where Rδn : L2(R+;E) → L2(R+;E) denotes right translation over δn and the δn ↓ 0are chosen in such a way that the Ψn are adapted. By the martingale convergence theo-rem and Lemma 3.2.2, Φ = limn→∞ Ψn in L0(Ω;L2(R+; γ(H,E))) and Φ = limn→∞ Ψn

in L0(Ω; γ(L2(R+;H), E)).

The processes Ψn are not elementary in general but of the form

Ψn =Kn∑k=1

1(tk,n,tk+1,n] ⊗ ξk,n,

where each ξk,n is an Ftk,n-measurable E-valued random variable. Approximating

each ξk,n in probability by a sequence of Ftk,n-simple random variables we obtain

a sequence of elementary processes (Ψn,m)m≥1 such that limm→∞ Ψn,m = Ψn bothin L0(Ω;L2(R+; γ(H,E))) and in L0(Ω; γ(L2(R+;H), E)). For an appropriate subse-quence (mn)n≥1, the elementary processes Φn,mn have the required properties.

The next lemma may be proved similarly.

Lemma 4.8.6. Let E be a Banach space, and let ψ ∈ L0(Ω;L1(R+;E)) be a stronglymeasurable and adapted process. Then there exists a sequence of elementary processes(ψn)n≥1 such that ψ = limn→∞ ψn in L0(Ω;L1(R+;E)).

Proof of Theorem 4.8.2. It is enough to prove the result on arbitrary finite intervals.Therefore we may assume that ψ ∈ L0(Ω;L1(R+;E)) and that Φ is stochasticallyintegrable with respect to WH and Φ ∈ L0(Ω;L2(R+; γ(H,E))). The rest of the proofis divided into several steps.

Step 1 – Reduction to the case F = R. Assume that the theorem holds in the caseF = R. For all x∗ ∈ E∗ we have a.s.

‖(D2f(·, ζ)Φ

)∗x∗‖2

L2(R+;H) = ‖Φ∗(D2f(·, ζ))∗x∗‖2L2(R+;H)

≤∫ t

0

‖Φ(s)‖2γ(H,E)‖(D2f(s, ζ(s)))∗x∗‖2 ds

≤ ‖Φ‖2L2(R+;γ(H,E)) sup

s∈R+

‖(D2f(s, ζ(s)))∗x∗‖2,

hence D2f(·, ζ)Φ is scalarly in L0(Ω;L2(R+;H)). Fix x∗ ∈ E∗. Since differentiating f


and applying x∗ are commuting operations we may deduce that a.s. for all t ∈ [0, T ],

〈f(t, ζ(t)),x∗〉 − 〈f(0, ξ), x∗〉

=⟨ ∫ t

0

D1f(s, ζ(s)) ds, x∗⟩

+⟨ ∫ t

0

D2f(s, ζ(s))ψ(s) ds, x∗⟩

+

∫ t

0

(D2f(s, ζ(s))Φ(s)

)∗x∗ dWH(s)

+1

2

⟨ ∫ t

0

Tr(D2

2f(s, ζ(s))(Φ(s),Φ(s)))ds, x∗

⟩.

An application of Theorem 4.5.9 shows that D2f(·, ζ)Φ is stochastically integrable and(4.8.3) holds. It follows that it suffices to consider F = R.

Step 2 – Reduction to elementary processes. Assume that the theorem holds forelementary processes. By path continuity it suffices to show that for all t ∈ R+ almostsurely (4.8.3) holds. Define the sequence (ζn)n≥1 in L0(Ω;Cb(R+;E)) by

ζn(t) = ξn +

∫ t

0

ψn(s) ds+

∫ t

0

Φn(s) dWH(s),

where (ξn)n≥1 is a sequence of F0-measurable simple functions with ξ = limn→∞ ξnalmost surely and (Φn)n≥1 and (ψn)n≥1 are chosen from Lemma 4.8.5 and 4.8.6. ByTheorem 4.5.9 we have ζ = limn→∞ ζn in L0(Ω;Cb(R+;E)). We may choose a subse-quence which we again denote by (ζn)n≥1 such that

ζ = limn→∞

ζn in Cb(R+;E) a.s. (4.8.5)

Thus, in order to prove (4.8.3) holds for the triple (ξ, ψ,Φ) it suffices to show that allterms in (4.8.3) depend continuously on (ξ, ψ,Φ).

For the left hand side of (4.8.3) it follows from (4.8.5) that

limn→∞

f(t, ζn(t))− f(0, ζn(0)) = f(t, ζ(t))− f(0, ζ(0)) almost surely.

Let K : Ω → R+ be defined as

K = supt∈R+

‖f(t, ζ(t))‖ ∨ ‖D1f(t, ζ(t))‖ ∨ ‖D2f(t, ζ(t))‖ ∨ ‖D22f(t, ζ(t))‖.

By Lemma 4.8.6, (4.8.5) and dominated convergence, a.s. we obtain

limn→∞

∫ t

0

D1f(s, ζn(s)) ds =

∫ t

0

D1f(s, ζ(s)) ds,

limn→∞

∫ t

0

D2f(s, ζn(s))ψn(s) ds =

∫ t

0

D2f(s, ζ(s))ψ(s) ds.


For the stochastic integral term in (4.8.3), by (4.1.2) it is enough to show that a.s.,

limn→∞

‖D2f(·, ζ)Φ−D2f(·, ζn)Φn‖L2(0,t;H) = 0. (4.8.6)

HereD2f(·, ζ) andD2f(·, ζn) stand forD2f(·, ζ(·)) andD2f(·, ζn(·)), respectively. But,by Lemma 4.8.5 we have

limn→∞

‖D2f(·, ζn)(Φ− Φn)‖L2(0,t;H) ≤ K limn→∞

‖Φ− Φn‖L2(0,t;B(H,E))

≤ K limn→∞

‖Φ− Φn‖L2(0,t;γ(H,E)) = 0,

and, by (4.8.5) and dominated convergence,

limn→∞

‖(D2f(·, ζ)−D2f(·, ζn))Φ‖L2(0,t;H) = 0 a.s.

Together these estimates give (4.8.6).For the last term in (4.8.3) we have

‖TrΦ(D22f(·, ζ))− TrΦn(D2

2f(·, ζn))‖L1(0,t)

≤ ‖TrΦ(D22f(·, ζ))− TrΦ(D2

2f(·, ζn))‖L1(0,t)

+ ‖TrΦ(D22f(·, ζn))− TrΦn(D2

2f(·, ζn))‖L1(0,t).

The first term tends a.s. to 0 by Lemma 4.8.1, (4.8.5) and dominated convergence. Forthe second term, by Lemma 4.8.1, the Cauchy-Schwartz inequality and Lemma 4.8.5we have

‖TrΦ(D22f(·, ζn))− TrΦn(D2

2f(·, ζn))‖L1(0,t)

≤ ‖TrΦ(D22f(·, ζn))− TrΦ,Φn(D2

2f(·, ζn))‖L1(0,t)

+ ‖TrΦ,Φn((D22f(·, ζn))− TrΦn(D2

2f(·, ζn))‖L1(0,t)

≤ K‖Φ‖L2(0,t;γ(H,E))‖Φ− Φn‖L2(0,t;γ(H,E))

+K‖Φn‖L2(0,t;γ(H,E))‖Φ− Φn‖L2(0,t;γ(H,E)),

which tends to 0 a.s. as well.Step 3 – If ξ is simple, ψ and Φ are elementary, they take their values in a finite-

dimensional subspace E0 ⊂ E and Φ is supported on a finite-dimensional subspaceH0 ⊂ H. Since E0 is isomorphic to some Rn and H0 is isomorphic to some Rm, (4.8.3)follows from the corresponding real-valued Ito formula.

4.9 The Ito formula 2

In Section 4.8 we proved an Ito formula under the assumption that

Φ ∈ L2loc(R+; γ(H,E)) a.s. (4.9.1)


In some sense this assumption is unnatural, because the well-definedness of ζ does notrequire such an assumption. In view of Proposition 4.6.3 one could try to replace theassumption on Φ by a γ-boundedness assumption on derivatives of f . This works forthe well-definedness of the terms in the integral in the Ito formula. However, to do theapproximation argument as in the proof of Theorem 4.8.2 one also needs a continuityproperty in a γ-bounded sense.

The general statement using γ-boundedness we could show is rather technical. Forthis reason we have chosen to prove Ito’s formula without the assumption (4.9.1) ina special case. Namely, we show it in case where f is as in (4.8.4). In this casethe derivatives of f are easily calculated and it is possible to check the γ-boundednessconditions by hand. Before we state the main result we need some results about traces.

Recall that for an H-strongly measurable function Φ : R+ → B(H,E) which isscalarly in L2(R+;H) we define the Pettis integral operator IΦ ∈ B(L2(R+;E), H)by (3.2.4). Below Lemma 4.8.1 we discussed some relations between γ-radonifyingoperators and trace class operators. The next lemma gives some more results in thecase that the Hilbert space is L2(R+;H).

Lemma 4.9.1. Let Φ1 ∈ γ(R+;H,E) and Φ2 ∈ γ(R+;H,E∗). Then for each ortho-normal basis (hn)n≥1 of H,∫

R+

∑n≥1

|〈Φ1(t)hn,Φ2(t)hn〉| dt ≤ ‖Φ1‖γ(R+;H,E)‖Φ2‖γ(R+;H,E),

hence∑

n≥1〈Φ1(t)hn,Φ2(t)hn converges absolutely for almost all t ∈ R+. Moreover,I∗Φ2

IΦ1 ∈ B(L2(R+;H)) is a trace class operator and we have

Tr(I∗Φ2IΦ1) =

∫R+

∑n≥1

〈Φ1(t)hn,Φ2(t)hn〉 dt. (4.9.2)

We will use the short-hand notation Tr(Φ1,Φ2) for the expressions in (4.9.2).The above lemma may be proved in the same way as in [67], where the case H = R

has been considered. It can also be seen as a consequence of the result in [67] if oneidentifies L2(R+;H) with L2(R+ × N).

If Φ1 and Φ2 take values in respectively γ(H,E) and γ(H,E∗), then we alreadyexplained below Lemma 4.8.1 that for all t ∈ R+, Φ∗

2(t)Φ1(t) is a trace class operator.Moreover,

Tr(Φ∗2(t)Φ1(t)) =

∑n≥1

〈Φ1(t)hn,Φ2(t)hn〉.

In this situation (4.9.2) becomes,

Tr(I∗Φ2IΦ1) =

∫R+

Tr(Φ∗2(t)Φ1(t)) dt.

We can now extend (4.8.4).


Theorem 4.9.2. Let E be a UMD space. Let (hn)n≥1 be an orthonormal basis for H.Let Φ1 : R+×Ω → B(H,E) and Φ2 : R+×Ω → B(H,E∗) be H-strongly measurable andadapted and locally stochastically integrable. Let ψ1 : R+×Ω → E and ψ2 : R+×Ω →E∗ be strongly measurable and adapted and locally integrable. Let ξ1 : Ω → E andξ2 : Ω → E∗ be strongly F0-measurable. For i = 1, 2 define

ζi(t) = ξi +

∫ t

0

ψi(s) ds+

∫ t

0

Φi(s) dWH(s).

Then, almost surely for all t ∈ R+,

〈ζ1(t), ζ2(t)〉 − 〈ξ1, ξ2〉 =

∫ t

0

〈ζ1(s), ψ2(s)〉+ 〈ψ1(s), ζ2(s)〉 ds

+

∫ t

0

〈ζ1(s),Φ2(s)〉+ 〈Φ1(s), ζ2(s)〉 dWH(s)

+

∫ t

0

∑n≥1

〈Φ1(s)hn,Φ2(s)hn〉 ds.

(4.9.3)

The first integral is defined as a Lebesgue integral for a.s. For the well-definedness ofthe last integral we refer to Lemma 4.9.1. We interpret 〈ζ1(s),Φ2(s)〉 and 〈Φ1(s), ζ2(s)〉as Φ∗

2(s)ζ1(s) and Φ∗1(s)ζ2(s) for s ∈ R+ respectively. Below we will show that Φ∗

2ζ1and Φ∗

1ζ2 are both in L2loc(R+;H) a.s.

To prove the theorem we need the following lemma.

Lemma 4.9.3. Let E be a UMD space. Let ξ : Ω → E be F0-measurable and such thatξ ∈ L4(Ω;E). Let Φ1 ∈ L4

F(Ω; γ(R+;H,E)) and ψ ∈ L4F(Ω;L1(R+;E)) and define the

Ito process ζ : R+ × Ω → E as

ζ(t) = ξ +

∫ t

0

ψ(s) ds+

∫ t

0

Φ1(s) dWH(s).

Then it follows that for all Φ2 ∈ L4F(Ω; γ(R+;H,E∗), the H-valued process Φ∗

2ζ isagain stochastically integrable and

‖Φ∗2ζ‖L2(R+×Ω;H) .E‖Φ2‖L4(Ω;γ(R+;H,E))

(‖ξ‖L4(Ω;E)

+ ‖ψ‖L4(Ω;L1(R+;E)) + ‖Φ1‖L4(Ω;γ(R+;H,E))

).

(4.9.4)

Proof. Clearly, Φ∗2ζ is strongly measurable and adapted, so we only have to show that

(4.9.4) holds. By the triangle inequality,

‖Φ∗2ζ‖L2(R+×Ω;H) ≤

3∑i=1

‖Φ∗2ζi‖L2(R+×Ω;H),


where ζ1 = ξ, ζ2 =∫ ·

0ψ(s) ds and ζ3 =

∫ ·0Φ(s)WH(s). Below the terms are estimated

separately. For i = 1, by (3.2.1) and the Cauchy Schwartz inequality,

‖Φ∗2ζ1‖2

L2(R+×Ω;H) ≤ E‖Φ2‖2γ(R+;H,E∗)‖ξ‖2 ≤ ‖Φ2‖2

L4(Ω;γ(R+;H,E∗))‖ξ‖2L4(Ω;E).

For i = 2 let N : R+ × Ω → B(E∗,R) be defined as

N(t)x∗ =

∫ t

0

〈ψ(s), x∗〉 ds.

Then by [73, Example 2.18], for each ω ∈ Ω, N(t, ω) : t ∈ R+ is R-bounded andtherefore γ-bounded by ‖ψ(·, ω)‖L1(R+;E) a.s. By Fubini’s theorem and Proposition3.2.4 we obtain that

‖Φ∗2ζ2‖2

L2(R+×Ω;H) = E‖NΦ2‖2L2(R+;H) = E‖NΦ2‖2

γ(R+;H)

≤ E‖ψ‖2L1(R+;E)‖Φ2‖2

γ(R+;H,E∗)

≤ ‖Φ2‖2L4(Ω;γ(R+;H,E∗))‖ψ‖2

L4(Ω;L1(R+;E)).

For i = 3, let η = limt→∞ ζ3(t) which exists in L4(Ω;E) and a.s. by Theorem 4.4.3.Then for all t ∈ R+, a.s. ζ3(t) = E(η|Ft) and it follows from the contractiveness of theconditional expectation that for all t ∈ R+,

‖Φ∗2(t)ζ3(t)‖2

L2(Ω;H) = ‖Φ∗2(t)E(η|Ft)‖2

L2(Ω;H)

= ‖E(Φ∗2(t)η|Ft)‖2

L2(Ω;H) ≤ ‖Φ∗2(t)η‖2

L2(Ω;H).

Via Fubini’s theorem an integration over R+ gives,

‖Φ∗2ζ3‖2

L2(R+×Ω;H) ≤ ‖Φ∗2η‖2

L2(R+×Ω;H).

It follows that

‖Φ∗2ζ3‖2

L2(R+×Ω;H) ≤ ‖Φ∗2η‖2

L2(Ω;L2(R+;H))

= E‖Φ∗2η‖2

L2(R+;H)) ≤ E‖Φ2‖2γ(R+;H,E∗)‖η‖2

≤ ‖Φ2‖2L4(Ω;γ(R+;H,E))‖η‖L4(Ω;E)

.E ‖Φ2‖2L4(Ω;γ(R+;H,E))‖Φ1‖2

L4(Ω;γ(R+;H,E)),

where the last inequality follows from Theorem 4.4.3.

Proof of Theorem 4.9.2. It suffices to show that the result holds for t ∈ [0, T ], whereT ∈ R+, so we may assume that ψi = 0 and Φi = 0 on (T,∞) for i = 1, 2.

Step 1 – Reduction to the bounded case.By a stopping time argument one may reduce to the case where for a fixed constant

K we have that a.s.

‖ψ1‖L1(R+;E), ‖Φ1‖γ(R+;H,E), supt∈R+

‖ζ1(t)‖,

‖ψ2‖L1(R+;E∗), ‖Φ2‖γ(R+;H,E∗), supt∈R+

‖ζ2(t)‖(4.9.5)


are bounded by a constant K ≥ 0.Indeed, assume that the theorem holds for this case. Let ξ, ψ, Φ and ζ be arbitrary.

For i = 1, 2, define sequences of stopping times (τni )n≥1 as

τn1 = inft ∈ R+ :

∫ t

0

‖ψ1(s)‖ ds ≥ n,∥∥∥∫ t

0

Φ1(s) dWH(s)∥∥∥ ≥ n,

or ‖Φ11[0,t]‖γ(R+;H,E) ≥ n,

τn2 = inft ∈ R+ :

∫ t

0

‖ψ2(s)‖ ds ≥ n,∥∥∥∫ t

0

Φ2(s) dWH(s)∥∥∥ ≥ n,

or ‖Φ21[0,t]‖γ(R+;H,E∗) ≥ n.

For n ≥ 1, let τn = τn1 ∧ τn2 . For n ≥ 1 and i = 1, 2 define

ζni (t) = ξni +

∫ t

0

1[0,τn](s)ψi(s) ds+

∫ t

0

1[0,τn](s)Φi(s) dWH(s),

where ξni = ζi(0)1‖ζi(0)‖≤n, for i = 1, 2. Since ‖ξni ‖ ≤ n and by the definition of thestopping time, we have supt∈R+

‖ζni (t)‖ ≤ 3n. By the assumption for the bounded caseit follows that 1[0,τn]Φ

∗i ζnj ∈ L2(R+;H) almost surely for i 6= j and

〈ζn1 (t),ζn2 (t)〉 − 〈ξn1 , ξn2 〉

=

∫ t

0

1[0,τn](s)〈ζn1 (s), ψ2(s)〉+ 1[0,τn](s)〈ψ1(s), ζn2 (s)〉 ds

+

∫ t

0

1[0,τn](s)〈ζn1 (s),Φ2(s)〉+ 1[0,τn](s)〈Φ1(s), ζn2 (s)〉 dWH(s)

+

∫ t

0

1[0,τn](s)∑k≥1

〈Φ1(s)hk,Φ2(s)hk〉 ds.

(4.9.6)

If we let n tend to infinity, (4.9.6) converges to (4.9.3). Indeed, notice that for a.a.ω ∈ Ω fixed for all n large enough, τn(ω) = ∞ and ζni (·, ω) = ζni (·, ω) for i = 1, 2. Theconvergence of all terms, except the stochastic integral term, follow immediately fromthis, since they are defined pathwise a.s. Since the stochastic integral is not definedpathwise we consider this term separately. By assumption for each n ≥ 1 we have

1[0,τn]Φ∗2ζn1 ,1[0,τn]Φ

∗1ζn2 ∈ L2(R+;H) a.s.

Since a.s. for all n ≥ 1 large enough and all s ∈ R+,

1[0,τn](s)Φ1(s)∗ζn2 (s) = Φ1(s)

∗ζ2(s) and 1[0,τn](s)Φ2(s)∗ζn1 (s) = Φ∗

2(s)ζ1(s).

It follows that Φ∗1ζ2 and Φ∗

2ζ1 are in L2(R+;H) a.s. and

Φ∗1ζ2 = lim

n→∞1[0,τn]Φ

∗1ζn2 and Φ∗

2ζ1 = limn→∞

1[0,τn]Φ∗2ζn1 in L2(R+;H) a.s.


It follows from (4.1.2) that∫ ·

0

〈Φ1(s), ζ2(s)〉 dWH(s) = limn→∞

∫ ·

0

〈1[0,τn](s)Φ1(s), ζn2 (s)〉 dWH(s) in L0(Ω;Cb(R+)),

and∫ ·

0

〈ζ1(s),Φ2(s)〉 dWH(s) = limn→∞

∫ ·

0

〈ζn1 (s),1[0,τn](s)Φ2(s)〉 dWH(s) in L0(Ω;Cb(R+)).

We may pass to a subsequence to obtain that (4.9.3) from (4.9.6).Step 2 – Reduction to the finite-dimensional case.Assume that (4.9.3) holds in the case where the processes take values in finite-

dimensional subspaces of E and E∗. We show that the general case of (4.9.3) follows.By Step 1 it suffices to consider the case where (4.9.5) holds. Choose simple (ξ1

n)n≥1

in L4(Ω,F0;E) and (ξ2n)n≥1 in L4(Ω,F0;E

∗) such that ξ1 = limn→∞ ξn1 in L4(Ω;E) andξ2 = limn→∞ ξn2 in L4(Ω;E∗). Choose step functions (ψ1

n)n≥1 in L4(Ω;L1(R+;E)) and(ψ2

n)n≥1 in L4(Ω;L1(R+;E∗)) such that

ψ1 = limn→∞

ψn1 in L4(Ω;L1(R+;E)) and ψ2 = limn→∞

ψn2 in L4(Ω;L1(R+;E∗)).

By Lemma 4.4.1 we can find elementary (Φn1 )n≥1 in L4(Ω; γ(R+;H,E)) and (Φn

2 )n≥1

in L4(Ω; γ(R+;H,E∗)) such that

Φ1 = limn→∞

Φn1 in L4(Ω; γ(R+;H,E)) and Φ2 = lim

n→∞Φn

2 in L4(Ω; γ(R+;H,E∗)).

For each n ≥ 1 and i = 1, 2 define

ζni (t) = ξni +

∫ t

0

ψni (s) ds+

∫ t

0

Φni (s) dWH(s).

By Theorem 4.4.3 we have ζ1 = limn→∞ ζn1 in L4(Ω;Cb(R+;E)) and ζ2 = limn→∞ ζn2in L4(Ω;Cb(R+;E∗)). By passing to a subsequence we may assume that all the aboveconvergence holds a.s. as well.

By assumption (4.9.3) holds for the approximating sequences, i.e. for all n ≥ 1,

〈ζn1 (t), ζn2 (t)〉 − 〈ξn1 , ξn2 〉 =

∫ t

0

〈ζn1 (s), ψn2 (s)〉+ 〈ψn1 (s), ζn2 (s)〉 ds

+

∫ t

0

〈ζn1 (s),Φn2 (s)〉+ 〈Φn

1 (s), ζn2 (s)〉 dWH(s)

+

∫ t

0

∑k≥1

〈Φn1 (s)hk,Φ

n2 (s)hk〉 ds.

(4.9.7)

Next, we let n tends to infinity for each term separately. Obviously,

〈ζ1(t), ζ2(t)〉 = limn→∞

〈ζn1 (t), ζn2 (t)〉 a.s., 〈ξ1, ξ2〉 = limn→∞

〈ξn1 , ξn2 〉 a.s.


and by the dominated convergence theorem we have∫ t

0

〈ζ1(s), ψ2(s)〉+ 〈ψ1(s), ζ2(s)〉 ds = limn→∞

∫ t

0

〈ζn1 (s), ψn2 (s)〉+ 〈ψn1 (s), ζn2 (s)〉 ds a.s.

For the stochastic integral term it follows from Lemma 4.9.3 that

Φ∗2ζ1 = lim

n→∞(Φn

2 )∗ζn1 in L2(R+ × Ω;H),

Φ∗1ζ2 = lim

n→∞(Φn

1 )∗ζn2 in L2(R+ × Ω;H).

We may pass to a subsequence to obtain∫ t

0

〈ζ1(s),Φ2(s)〉 dWH(s) = limn→∞

∫ t

0

〈ζn1 (s),Φn2 (s)〉 dWH(s) a.s.,

∫ t

0

〈Φ1(s), ζ2(s)〉 dWH(s) = limn→∞

∫ t

0

〈Φn1 (s), ζn2 (s)〉 dWH(s) a.s.

Finally, for the last term of (4.9.7) the convergence follows from Lemma 4.9.1.Step 3 – The proof may be completed as in Theorem 4.8.2.


Theorem 4.3.1 was already proved by Garling in [50] by different methods. Hehas used the result to prove the boundedness of the Hilbert transform on Lp(R;E) forp ∈ (1,∞). The proof given here is more direct from the UMD property. Theorem 4.3.1is the main ingredient in the proofs of the characterizations of stochastic integrabilitygiven in Theorems 4.4.3 and 4.5.9. These two theorems are the main results in [96].For some implications in these theorems only the upper estimate or lower estimate ofTheorem 4.3.1 is needed. We will return to this point in Chapter 5.

In the case that H = R and WH is a standard Brownian motion Theorem 4.5.9 (4)⇒ (1) has been obtained by McConnell in [88]. His proof is based on the extensionof Theorem 2.4.1 which is explained in Section 2.8. Related to this is the proof ofLemma 4.5.6 which is similar to [88, Lemma 3.3]. Our approach is somewhat simpler,as it allows the use of F -stopping times rather than the F ⊗F -stopping times used in[88].

Theorem 4.4.3 (1) ⇔ (2) is rather surprising. It shows essentially that a “Bochnertype” integrability property is equivalent to a “Pettis type” integrability property. Theproof of this goes through (2) ⇒ (3). This proof simplifies considerably if one assumesthat E has a Schauder basis. To get around such an assumption, an approximationargument with quotient maps has been done. Also Theorem 4.5.9 (1) ⇔ (2) is suchan equivalence. However, here we need the whole integral process in (2).

In the case of finite time intervals, the approximation in Definition 4.2.1 and The-orem 4.4.3 can be chosen in such a way that for all h ∈ H, Φh = limn→∞ Φnh inmeasure (cf. [95]).


Since (4.1.3) holds for all p ∈ (0,∞), one might expect that (4.5.8) extends to all0 < p <∞. We do not know whether this is true.

The results for stochastic integration of B(H,E)-valued processes with respect toan H-cylindrical Brownian motion WH can be applied to obtain corresponding resultsfor B(E)-valued processes with respect to an E-valued Brownian motion W . Indeed,the stochastic integral of such a process Ψ is defined as∫ ·

0

Ψ(t) dW (t) :=

∫ ·

0

Ψ(t) iC dWHC(t),

where C ∈ B(E∗, E) is the covariance operator of W and WHCthe HC-cylindrical

Brownian motion canonically associated with W (cf. Example 3.4.2). For more detailswe refer to the discussion in [100, Section 5].

Theorem 4.6.1 is an extension of Corollary 3.4.5 to processes. Theorem 4.6.2 ac-tually holds for martingale (co)-type 2 spaces. This is more general than the classof UMD spaces with type 2, since there are martingale (co)type 2 spaces which arenot UMD (see [17]). For information on integration in martingale type 2 spaces werefer to the works of Brzezniak in [19], Dettweiler in [42] and Neidhardt in[102]. Similar consequences can be given for UMD spaces with type p or cotype q viaTheorem 3.3.2.

Theorem 4.6.5 is partly based on a result of Bourgain from [18]. It is a vector-valued generalization of a result in [123] by Stein and sometimes referred to as thevector-valued Stein inequality. A detailed proof may be found in [31, Proposition 3.8]by Clement, de Pagter, Sukochev and Witvliet.

The proof of Theorem 4.7.1 is modelled after [63, Theorem 18.10].Theorem 4.8.2 has been proved in [97]. In the case that E and F are martingale

type 2 spaces this result has been proved in [102] (also see [21]). Also Corollary 4.8.4for martingale type 2 spaces E1, E2 and F can be found in [21, Corollary 2.1]. However,we want to emphasize that we do not know how one can obtain (4.8.4) with martingaletype 2 methods, since E and E∗ have martingale type 2 if and only if E is isomorphicto a Hilbert space. Applications of Corollary 4.8.4 are given in Chapter 10.

The Ito formula is a fundamental tool in stochastic calculus. It has many ap-plications that we did not cover. An interesting consequence of the formula is thatthe process t 7→ f(t, ζ(t)) is again locally stochastically integrable. This follows fromProposition 4.6.4 and Theorem 4.6.6.

In our setup, at first sight it seems somewhat unnatural to assume that Φ is locallystochastically integrable and in L2

loc(R+; γ(H,E)) in Theorem 4.8.2. However, alreadyin the case where F = R, it is not obvious how to proceed if one leaves out the L2-assumption. It is possible to prove the result for more general Φ under conditions onf as γ-boundedness of derivatives of f . However, in the approximation procedure inthe proof we also need certain continuity properties, which then would translate incontinuity properties in γ-bounds. Since this leads to difficult statements, we choseto prove the formula for the important case where f : E × E∗ → R is given byf(x, x∗) = 〈x, x∗〉. For this special choice we were able to check the γ-boundedness


properties by hand and obtain the Ito formula without any L2-assumption. This isthe contents of Theorem 4.9.2 and it extends Corollary 4.8.4.

Chapter 5

Integration in randomized UMDspaces

5.1 Introduction

Let (E, ‖ · ‖) be a Banach space and let (H, [·, ·]) be a separable Hilbert space. Let(Ω,A,P) be a complete probability space and WH be a cylindrical Brownian motionwith respect to a complete filtration F = (Ft)t≥0 on (Ω,A,P).

In Section 5.2 we consider a randomized version of the UMD property. Basically,the ±1 in the definition of UMD are replaced by a Rademacher sequence. This defin-ition has been introduced by Garling in [51]. The Rademacher sequence can be puton the left or right-hand side. This corresponds to different properties. The propertiesare called UMD− and UMD+ below. In this way the UMD property is split into twoparts.

Our motivation to consider the randomized UMD spaces comes from Section 5.3.Here we will explain that one sided versions of the decoupling inequalities of Theorem4.3.1 hold for UMD− and UMD+ spaces. In this way it is possible to extend some ofthe stochastic integration theory of Chapter 4 to a larger setting. This is interestingsince this may be applied, for instance, to L1-spaces which are not UMD spaces.

A natural question is whether the one-sided versions of the decoupling inequalitiesof Theorem 4.3.1 also imply the UMD− and UMD+ property. This question is partlyanswered in Section 5.4.

This chapter is based on [129].

5.2 Randomized UMD spaces

Let (rn)n≥1 be a Rademacher sequence on a different probability space (Ω, A, P). Recallthat E denotes expectations on Ω.

Definition 5.2.1. Let E be a Banach space and let p ∈ (1,∞).

95

96 Chapter 5. Integration in randomized UMD spaces

(1) The space E is a UMD−(p) space if for every N ≥ 1, every martingale differencesequence (dn)

Nn=1 in Lp(Ω, E), we have

(E

∥∥∥ N∑n=1

dn

∥∥∥p) 1p

.E,p

(EE

∥∥∥ N∑n=1

rndn

∥∥∥p) 1p. (5.2.1)

(2) The space E is a UMD+(p) space if for every N ≥ 1, every martingale differencesequence (dn)

Nn=1 in Lp(Ω, E), we have

(EE

∥∥∥ N∑n=1

rndn

∥∥∥p) 1p

.E,p

(E

∥∥∥ N∑n=1

dn

∥∥∥p) 1p. (5.2.2)

The smallest constant such that (5.2.1) and (5.2.2) hold will be denoted by β−p (E)and β+

p (E) respectively.The corresponding notions where only Paley-Walsh martingale difference sequences

are denoted by UMD−PW and UMD+

PW. The corresponding constants will be denotedby β−PW,p(E) and β+

PW,p(E)In [51] Garling showed that if E is a UMD±(p) space for some p ∈ (1,∞), then

E is a UMD±(p) space for all p ∈ (1,∞). Thus, both definitions are independent ofp ∈ (1,∞) and spaces with this property will be referred to as UMD− and UMD+

spaces. Similarly, one can show that UMD±PW(p) is p-independent and therefore, is

denoted by UMD±PW.

Some results on UMD− and UMD+ spaces obtained in [51] are:

• If E is a UMD+ space, then its dual E∗ is a UMD− space. If E∗ is a UMD+

space, then its predual E is a UMD− space.

• Every UMD− space has finite cotype. Every UMD+ space is super-reflexive.

• E is a UMD space if and only if it is both UMD− and UMD+.

Similar results hold for UMD−PW and UMD+

PW spaces.In the definition of UMD+ we had to exclude p = 1, since already R fails the

corresponding property UMD+(1). The following lemma gives a ‘maximal’ version ofthe UMD− property which does extend to p = 1. It will allow us to build UMD−

spaces from given UMD− spaces in Proposition 5.2.3.

Lemma 5.2.2. Let p ∈ [1,∞) be arbitrary and fixed. A Banach space E has theUMD−-property if and only if for all N ≥ 1, for all E-valued martingale differencesequences (dn)

Nn=1 we have

E sup1≤n≤N

∥∥∥ n∑k=1

dk

∥∥∥p .E,p EE∥∥∥ N∑k=1

rkdk

∥∥∥p.

5.2. Randomized UMD spaces 97

Proof. By [51, Theorem 2], E is a UMD− space if and only if for all N ≥ 1 and for allE-valued martingale difference sequence (dn)

Nn=1,

E sup1≤n≤N

∥∥∥ n∑k=1

dk

∥∥∥p .E,p EE sup1≤n≤N

∥∥∥ n∑k=1

rkdk

∥∥∥p.But by the Levy-Octaviani inequalities for symmetric random variables [76, Section1.1] applied for fixed ω ∈ Ω, we have

EE∥∥∥ N∑k=1

rkdk

∥∥∥p ≤ EE sup1≤n≤N

∥∥∥ n∑k=1

rkdk

∥∥∥p ≤ 2p EE∥∥∥ N∑k=1

rkdk

∥∥∥pand the result follows.

Proposition 5.2.3. Let (S,Σ, µ) be a σ-finite measure space and let E be a UMD−

space. Then for all p ∈ [1,∞), the space Lp(S;E) is a UMD− space.

Proof. We start with the preliminary observation that if G is a sub-σ-algebra of A andf ∈ Lp(Ω;Lp(S;E)) is A-measurable and satisfies E(f |G) = 0, then for µ-almost alls ∈ S the function fs(·) := (f(·))(s) belongs to Lp(Ω;E), is A-measurable and satisfiesE(fs|G) = 0. This follows from an easy application of the Fubini theorem.

Fix an arbitrary p ∈ [1,∞) and let (dn)Nn=1 be a martingale difference sequence

in Lp(Ω;Lp(S;E)). Identifying each dn with an element in Lp(S;Lp(Ω;E)) it followsfrom the observation just made that for µ-almost all s ∈ S, (dn(s))

Nn=1 is a martingale

difference sequence with values in E. Hence for µ-almost all s ∈ S we have

E supn≤N

∥∥∥ n∑k=1

dk(s)∥∥∥p .E,p EE

∥∥∥ N∑k=1

rkdk(s)∥∥∥p,

where we used Lemma 5.2.2. From this we obtain

E supn≤N

∥∥∥ n∑k=1

dk

∥∥∥pLp(S;E)

= E supn≤N

∫S

∥∥∥ n∑k=1

dk(s)∥∥∥pdµ(s) ≤

∫S

E supn≤N

∥∥∥ n∑k=1

dk(s)∥∥∥pdµ(s)

.E,p

∫S

EE∥∥∥ N∑k=1

rkdk(s)∥∥∥pdµ(s) = EE

∥∥∥ N∑k=1

rkdk

∥∥∥pLp(S;E)

.

It follows from Lemma 5.2.2 that Lp(S;E) is a UMD− space.

Apart from the trivial case where (S,Σ, µ) consists of finitely many atoms, thespace L1(S) is an example of a UMD− space that is not UMD.


5.3 One-sided estimates for stochastic integrals

Let H be a separable Hilbert space with dim(H) ≥ 1 and let E be a Banach. Let WH

and WH be cylindrical Brownian motions as in Chapter 4. For the randomized UMDspaces UMD− and UMD+ there are more precise statements than the one in Theorem4.3.1. However, we only know a proof under mild assumptions on the probability spaceand the filtration A. Namely, we can only follow the arguments in [50, Theorem 2] ifthe filtration is the Brownian filtration. Below we reformulated the result of [50] foroperator valued processes. The filtration is slightly more general than the Brownianfiltration, using a product structure.

Let (Ωi,Ai,Pi), i = 1, 2 be probability spaces. In the next Theorem we assume WH

is defined on Ω1 and FWH is the augmented filtration induced by WH . Furthermore,Ω = Ω1 × Ω2, A = A1 ⊗ A2, and P = P1 ⊗ P2. Recall that WH is defined on theprobability space (Ω, A, P).

Theorem 5.3.1. Under the above assumption on the filtration the following resultshold:

(1) If E is a UMD− space and p ∈ [1,∞), then there exists a constant cp such thatfor all elementary processes Φ : R+ × Ω → B(H,E) we have

E∥∥∥∫

R+

Φ(t) dWH(t)∥∥∥p ≤ cppEE

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥p. (5.3.1)

(2) If E is a UMD+ space and p ∈ (1,∞), then for all elementary processes Φ :R+ × Ω → B(H,E) we have

EE∥∥∥∫

R+

Φ(t) dWH(t)∥∥∥p ≤ cppE

∥∥∥∫R+

Φ(t) dWH(t)∥∥∥p. (5.3.2)

Proof. First we argue for fixed ω2 ∈ Ω2. In that case, for p ∈ (1,∞) both assertionsfollow from the proofs in [50]. For p = 1 one can also argue as in [50] using Lemma 5.2.2.The estimates (5.3.1) and (5.3.2) follows from Fubini’s theorem and an integration overΩ2.

Many results from Chapter 4 generalize to UMD− spaces. We state the mostimportant ones. The proofs are based on (5.3.1) and the same arguments as before.

The following version of Proposition 4.4.2 holds.

Proposition 5.3.2 (Ito map). Let E be a UMD− space and fix p ∈ (1,∞). Themapping X 7→ IWH (X) has a unique extension to a bounded operator

IWH : LpF(Ω; γ(L2(R+;H), E)) → Lp(Ω, Cb(R+;E)).

For all X ∈ LpF(Ω; γ(L2(R+;H), E)), IWH (X) is a martingale and (4.4.3) and (4.4.4)still hold.

5.4. Necessity of the randomized UMD property 99

The following version of Proposition 4.5.7 remains true for UMD− space.

Proposition 5.3.3. Let E be a UMD− space. The mapping X 7→ IWH (X) has aunique extension to a continuous mapping from the space L0

F(Ω; γ(L2(R+;H), E)) intoL0(Ω;Cb(R+;E)). For every X ∈ L0

F(Ω; γ(L2(R+;H), E)), IWH (X) is an element ofMc,loc

0 (Ω,F ;E) and the estimate (4.5.4) holds.

Theorem 4.5.9 has the following one sided extension.

Theorem 5.3.4. Let E be a UMD− space. Let Φ : R+ × Ω → B(H,E) be an H-strongly measurable and adapted process which is scalarly in L0(Ω;L2(R+;H)). Thefollowing implications of Theorem 4.5.9 still hold: (3) ⇒ (1) ⇒ (2) and (3) ⇔ (4).In the situation of (3), Φ ∈ L0

F(Ω; γ(R+;H,E)) and

IWH (Φ) = ζ =

∫ ·

0

Φ(t) dWH(t) in L0(Ω;Cb(R+;E)), (5.3.3)

E supt∈R+

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p .p,E E‖Φ‖pγ(R+;H,E). (5.3.4)

As a consequence, there also is a one sided extension of Theorem 4.6.1. By Propo-sition 5.2.3, this in particular applies to the Banach function space E = L1.

The Ito formula from Section 4.8 has the following extension.

Theorem 5.3.5 (Ito formula). Let E be a UMD− space and let F be a UMD spaces.Under the same assumptions on f , Φ, ψ and ξ as in Theorem 4.8.2, (4.8.3) still holds.

For the Ito formula of Section 4.9 we have the following result.

Theorem 5.3.6. Let E be a UMD− space. Under the same assumption on Φi, ψi, ξifor i = 1, 2 as in Theorem 4.9.2, but with Φ2 = 0, it holds that

〈ζ1(t), ζ2(t)〉 − 〈ξ1, ξ2〉 =

∫ t

0

〈ζ1(s), ψ2(s)〉+ 〈ψ1(s), ζ2(s)〉 ds

+

∫ t

0

〈Φ1(s), ζ2(s)〉 dWH(s).

(5.3.5)

5.4 Necessity of the randomized UMD property

In [50] a converse to Theorem 4.3.1 has been obtained. Here we prove a converse toTheorem 5.3.1. In particular, we recover a result in [50] in an indirect way.

Theorem 5.4.1. Let E be a Banach space and let p ∈ (1,∞). The following state-ments hold:

(1) If there exists a constant cp > 0 such that (5.3.1) holds for all elementaryprocesses, then E is a UMD−

PW space;


(2) If there exists a constant cp > 0 such that (5.3.2) holds for all elementaryprocesses, then E is a UMD+

PW space.

As a consequence of Theorem 5.4.1 we recover a converse to Theorem 4.3.1.

Corollary 5.4.2. Let E be a Banach space and let p ∈ (1,∞). If the assertion ofTheorem 4.3.1 holds, i.e. there exists a constants c−p , c

+p > 0 such that (5.3.1) and

(5.3.2) hold for all elementary processes, then E is a UMD space.

Proof. By (1) and (2) of Theorem 5.4.1, E is a UMD−PW and UMD+

PW space. Therefore,for p ∈ (1,∞), an arbitrary Paley-Walsh martingale difference sequence (dn)

Nn=1 in

Lp(Ω;E) and an arbitrary sequence (εn)Nn=1 in −1, 1, it follows by a randomization

argument that

E∥∥∥ N∑n=1

εndn

∥∥∥p ≤ (β−PW,p(E))pEE∥∥∥ N∑n=1

εnrndn

∥∥∥p= (β−PW,p(E))pEE

∥∥∥ N∑n=1

rndn

∥∥∥p≤ (β−PW,p(E))p(β+

PW,p(E))pE∥∥∥ N∑n=1

dn

∥∥∥p.This shows that E is a UMDPW(p) space, hence E is a UMD space (see Section2.4).

Since dim(H) ≥ 1, for the proof of Theorem 5.4.1 we may assume H = R, i.e.WH and WH are standard Brownian motions and they will be denoted by W and Wrespectively. Moreover B(H,E) is identified with E and processes Φ : R+ × Ω →B(H,E) are identified with processes φ : R+ × Ω → E.

First we prove some lemmas. The next lemma is well-known and follows from thestrong Markov property.

Lemma 5.4.3. Let τ0 = 0 and define inductively

τn = inft ≥ τn−1 : |Wt −Wτn−1| = 1, 1 ≤ n ≤ N.

Then (τn)Nn=1 is an increasing sequence of stopping times and (∆τn,∆Wn)

Nn=1 is an

i.i.d. sequence of random vectors, where

∆τn = τn − τn−1, ∆Wn = Wτn −Wτn−1 , 1 ≤ n ≤ N.

Moreover (∆Wn)Nn=1 is a Rademacher sequence adapted to (Fτn)Nn=1.

The following lemma gives some important properties of the independent Brownianmotion W at random times. Such stopped Brownian motions W are not Gaussianrandom variables in general, but they inherit some important properties.


Lemma 5.4.4. For 1 ≤ n ≤ N , let ∆Wn = Wτn − Wτn−1. Then (∆Wn)Nn=1 is an

i.i.d. sequence of symmetric random variables, which is independent of (∆Wn)Nn=1.

Furthermore, each ∆Wn has finite moments of all orders.

Proof. For all 1 ≤ n ≤ N , ˜∆Wn is symmetric, because ∆Wn(ω, ·) is symmetric for eachω ∈ Ω. It follows from the strong Markov property of (W, W ) that (∆Wn,∆Wn)

Nn=1 is

an i.i.d. sequence. So in order to prove the independence of (∆Wn)Nn=1 and (∆Wn)

Nn=1,

it is enough to show that ∆W1 = Wτ1 and ∆W1 = Wτ1 are independent. The followingargument has been shown to us by Tuomas Hytonen. For every Brownian motion Bon Ω we introduce the following two stopping times:

τB± = inft ≥ 0 : Bt = ±1.

Note that τ1 = τW− ∧ τW+ and for the Brownian motion −W , we have τ−W+ = τW−and τ−W− = τW+ . Let B ∈ R be a Borel measurable set. Since (W, W ) is identically

distributed with (−W, W ) it follows that

P(Wτ1 = 1, Wτ1 ∈ B) = P(τW+ < τW− , Wτ1 ∈ B)

= P(τ−W− < τ−W+ , Wτ1 ∈ B) = P(Wτ1 = −1, Wτ1 ∈ B).

Clearly,

P(Wτ1 = 1, Wτ1 ∈ B) + P(Wτ1 = −1, Wτ1 ∈ B) = P(Wτ1 ∈ B).

Hence

P(Wτ1 = 1, Wτ1 ∈ B) =1

2P(Wτ1 ∈ B) = P(Wτ1 = 1)P(Wτ1 ∈ B).

The same holds for −1. This proves the independence.For 0 < p <∞ we have

EE|∆Wn|p = EE|Wτ1|p = gppEτp/21 ,

where gpp is the p-th moment of a standard Gaussian random variable and the statementfollows from the elementary fact that τ1 has finite moments of all orders.

Below we will consider adapted and measurable processes φ : R+ × Ω → E thattake values in a finite-dimensional subspace of E. Since finite-dimensional subspacesof E are isomorphic to Rn for some integer n, one may construct a stochastic integralfor such processes φ that satisfy φ ∈ L2(R+, E) a.s. By the Burkholder-Davis-Gundyinequalities, for all p ∈ (1,∞) and for φ as above we have

∫R+φ(t) dW (t) ∈ Lp(Ω;E) if

φ ∈ Lp(Ω;L2(R+;E)). In this case, the decoupled stochastic integral∫

R+φ(t) dW (t) is

defined pathwise as an element of Lp(Ω;Lp(Ω;E)). Moreover, if (5.3.1) or (5.3.2) holdsfor all elementary processes one may extend this to all processes as above. In fact, in[50] the inequalities (5.3.1) and (5.3.2) have been proved for this class of processes.

The next lemma is a variation of an example in [51]. We include a proof forconvenience.


Lemma 5.4.5. Let E = c0 and p ∈ [1,∞). There does not exist a constant cp > 0such that for all elementary processes φ, (5.3.1) holds.

Proof. Assume that there exists a constant cp > 0 such that for all elementaryprocesses φ, (5.3.1) holds. Then we may extend (5.3.1) to all measurable and adaptedprocesses φ ∈ Lp(Ω;L2(0,∞;E)) that take values in a finite-dimensional subspace ofE. For each N ≥ 1, we will construct a process φ as above and such that(

E∥∥∥∫

R+

φ(t) dW (t)∥∥∥p)1/p

= N and(EE

∥∥∥∫R+

φ(t) dW (t)∥∥∥p)1/p

≤ Kp

√N.

Here Kp > 0 is some universal constant. This gives a contradiction.We use the notation of Lemmas 5.4.3 and 5.4.4. Fix an integer N ≥ 1. Let

D = −1, 1N , and for each e = (en)Nn=1 ∈ D define the process φe : R+ × Ω → R by

φe(t) =

en1Ae,n for t ∈ (τn−1, τn], n = 1, . . . , N,

0 for t = 0 or t > τN ,

where Ae,1 = Ω and for 2 ≤ n ≤ N ,

Ae,n = ∆W1 = e1, . . . ,∆Wn−1 = en−1.

Then each φe is stochastically integrable with∫R+

φe(t) dW (t) =N∑n=1

∆Wnen1Ae,n .

Define φ : R+ × Ω → l∞(D) by φ = (φe)e∈D. Then φ is stochastically integrable and

for almost all ω ∈ Ω and e ∈ D, we have∣∣∣(∫

R+φ(t) dW (t)

)(ω)(e)

∣∣∣ ≤ N . For almost

all ω ∈ Ω and e = (∆Wn(ω))Nn=1, we have∣∣∣(∫

R+φ(t) dW (t)

)(ω)(e)

∣∣∣ = N. This shows

that (E

∥∥∥∫R+

φ(t) dW (t)∥∥∥pl∞(D)

)1/p

= N, for all p ∈ [1,∞).

On the other hand, we have∫R+

φ(t) dW (t) =N∑n=1

∆Wnvn,

where for 1 ≤ n ≤ N , vn = (en1Ae,n)e∈D.For ω ∈ Ω and e ∈ D let k(ω, e) be 0 if ∆W1(ω) 6= e1 and let k(ω, e) be the

maximum of all integers n ≤ N such that ∆Wi(ω) = ei for all i ≤ n if ∆W1(ω) = e1.

For almost all ω ∈ Ω and for all e ∈ D,(∑N

n=1 ∆Wnvn

)(ω)(e) is equal to

−∆Wk(ω,e)+1(ω, ·)∆Wk(ω,e)+1(ω) +∑k(ω,e)

n=1 ∆Wn(ω, ·)∆Wn(ω), if k(ω, e) < N,∑Nn=1 ∆Wn(ω, ·)∆Wn(ω), if k(ω, e) = N.


Of course we have for all k ≤ N ,

−Wk∆Wk +k−1∑n=1

∆Wn∆Wn = 2k−1∑n=1

∆Wn∆Wn −k∑

n=1

∆Wn∆Wn.

We obtain that for almost all ω ∈ Ω,

∥∥∥∫R+

φ(t, ω) dW (t)∥∥∥l∞(D)

≤ 3 supk≤N

∣∣∣ k∑n=1

∆Wn(ω, ·)∆Wn(ω)∣∣∣.

Since for almost all ω ∈ Ω, (∆Wn(ω, ·))Nn=1 is a sequence of independent centeredGaussian random variables on Ω, we have by the Levy-Octaviani inequalities for inde-pendent symmetric random variables (see [76, Section 1.1]) for almost all ω ∈ Ω,

E supk≤N

∣∣∣ k∑n=1

∆Wn(ω, ·)∆Wn(ω)∣∣∣p ≤ 2pE

∣∣∣ N∑n=1

∆Wn(ω, ·)∆Wn(ω)∣∣∣p

= 2pE∣∣∣ N∑n=1

∆Wn(ω, ·)∣∣∣p = 2pE|WτN (ω, ·)|p = 2pgppτN(ω)p/2.

Here gpp is the p-th moment of a standard Gaussian random variable. We may concludethat (

EE∥∥∥∫

R+

φ(t) dW (t)∥∥∥pl∞(D)

)1/p

≤ 6gp(Eτ p/2N )1/p.

Recall that the sequence (τn− τn−1)Nn=1 is identically distributed. For p = 2 we obtain

(Eτ p/2N )1/p = (EτN)1/2 =(E

N∑n=1

τn − τn−1

)1/2

=( N∑n=1

E(τn − τn−1))1/2

=( N∑n=1

Eτ1)1/2

=√N

√Eτ1.

For 1 ≤ p < 2 we have by Holder’s inequality,

(Eτ p/2N )1/p ≤ (EτN)1/2 =√N

√Eτ1.

Finally for p > 2, by the triangle inequality in Lp/2(Ω),

(Eτ p/2N )1/p =(E

( N∑n=1

τn − τn−1

)p/2)1/p

≤( N∑n=1

(E(τn − τn−1)p/2)2/p

)1/2

=( N∑n=1

(Eτ p/21 )2/p)1/2

=√N(Eτ p/21 )1/p.


By Lemma 5.4.4 this proves that for all p ∈ [1,∞) and some universal constant Kp,(EE

∥∥∥∫R+

φ(t) dW (t)∥∥∥p)1/p

≤ Kp

√N.

Since l∞(D) can be identified isometrically with a finite-dimensional subspace of c0,this completes the proof.

Corollary 5.4.6. Let E be a Banach space. If there exists a constant cp > 0 such thatfor all elementary processes (5.3.1) holds, then E has finite cotype.

Proof. It follows from the above example that c0 is not finitely representable in E.Hence the Maurey-Pisier Theorem (see [87]) implies that E has finite cotype.

Proof of Theorem 5.4.1. It is enough to consider martingales that starts at zero (see[26, Remark 1.1 ]). Let (rn)

Nn=1 be a Rademacher sequence on the probability space

(Ω,A,P) and let (dn)Nn=1 be an E-valued martingale difference sequence with respect

to the filtration (σ(r1, r2, . . . , rn))Nn=0. We may write dn = rnfn(r1, . . . , rn−1) for n =

1, . . . , N , for some fn : −1, 1n−1 → E. Let (rn)Nn=1 be a Rademacher sequence on

the probability space (Ω, A, P).(1): We will show that there exists a constant C−

p > 0 only depending on E suchthat

E∥∥∥ N∑n=1

dn

∥∥∥p ≤ (C−p )pEE

∥∥∥ N∑n=1

rndn

∥∥∥p. (5.4.1)

We use the notation of Lemmas 5.4.3 and 5.4.4. Define a process φ : R+ × Ω → E by

φ(t) =

fn(∆W1, . . . ,∆Wn−1) for t ∈ (τn−1, τn], n = 1, . . . , N

0 for t = 0 or t > τN .

The process φ is stochastically integrable and we have

E∥∥∥∫

R+

φ(t) dW (t)∥∥∥p = E

∥∥∥ N∑n=1

∆Wnfn(∆W1, . . . ,∆Wn−1)∥∥∥p

= E∥∥∥ N∑n=1

rnfn(r1, . . . , rn−1)∥∥∥p = E

∥∥∥ N∑n=1

dn

∥∥∥p.Also, we have

EE∥∥∥∫

R+

φ(t) dW (t)∥∥∥p = EE

∥∥∥ N∑n=1

∆Wnfn(∆W1, . . . ,∆Wn−1)∥∥∥p.

By Lemmas 2.5.2, 5.4.4 and Corollary 5.4.6, we have

EE∥∥∥ N∑n=1

∆Wnxn

∥∥∥p ≤ KpEE∥∥∥ N∑n=1

rnxn

∥∥∥p,


where (xn)Nn=1 is a sequence in E and Kp > 0 is some constant depending only on E

and p. By conditioning (cf. [63, Lemma 3.11]) this result extends to

EE∥∥∥ N∑n=1

∆WnXn

∥∥∥p ≤ KpEE∥∥∥ N∑n=1

rnXn

∥∥∥p, (5.4.2)

where (Xn)Nn=1 is a sequence of E-valued random variables independent of (∆Wn)

Nn=1

and independent of (rn)Nn=1. By Lemmas 5.4.3 and 5.4.4, we may apply (5.4.2) to the

random variables Xn = fn(∆W1, . . . ,∆Wn−1) for 1 ≤ n ≤ N to obtain:

EE∥∥∥ N∑n=1

∆Wnfn(∆W1, . . . ,∆Wn−1)∥∥∥p ≤ Kp

pEE∥∥∥ N∑n=1

rnfn(∆W1, . . . ,∆Wn−1)∥∥∥p

= KppEE

∥∥∥ N∑n=1

rnfn(r1, . . . , rn−1)∥∥∥p (i)

= KppEE

∥∥∥ N∑n=1

rnrnfn(r1, . . . , rn−1)∥∥∥p

= KppEE

∥∥∥ N∑n=1

rndn

∥∥∥p.In (i), we used that (r1, . . . , rN , r1, . . . , rN) and (r1, . . . , rN , r1r1, . . . , rN rN) are identi-cally distributed. By assumption we have

E∥∥∥∫

R+

φ(t) dW (t)∥∥∥p ≤ cppEE

∥∥∥∫R+

φ(t) dW (t)∥∥∥p.

We may conclude that (5.4.1) holds with constant cpKp.(2): We will show that there exists a constant C+

p > 0 only depending on E suchthat

EE∥∥∥ N∑n=1

rndn

∥∥∥p ≤ (C+p )pE

∥∥∥ N∑n=1

dn

∥∥∥p. (5.4.3)

Let φ be as before. By Lemmas 2.5.1, 5.4.3 and 5.4.4 and the same arguments asbefore we have

EE∥∥∥ N∑n=1

rndn

∥∥∥p = EE∥∥∥ N∑n=1

rnfn(r1, . . . , rn−1)∥∥∥p

= EE∥∥∥ N∑n=1

rnfn(∆W1, . . . ,∆Wn−1)∥∥∥p

≤ 1

(EE|W1|)pEE

∥∥∥ N∑n=1

∆Wnfn(∆W1, . . . ,∆Wn−1)∥∥∥p.

By assumption we have

EE∥∥∥∫

R+

φ(t) dW (t)∥∥∥p ≤ cppE

∥∥∥∫R+

φ(t) dW (t)∥∥∥p.


We may conclude that (5.4.3) holds with constantcp

EE|W1|.


The properties UMD− and UMD+ were introduced by Garling in [51] under theabbreviations LERMT (Lower Estimates for Random Martingale Transforms) andUERMT (Upper Estimates for Random Martingale Transforms). We preferred thenotation UMD− and UMD+, since it emphasizes the relation with UMD. Here thesuperscript − stands for Lower and the superscript + stands for Upper. Most of theresult in Section 5.2 are from [51]. One can give alternative proofs and more equivalentconditions of UMD− and UMD+ using the extrapolation techniques of Geiss in [52].This allows for instance to replace the left-hand side of (5.2.1) and (5.2.2) with acertain BMO-norm and the right-hand side of (5.2.1) and (5.2.2) with an L∞-normwithout changing the geometric properties. The same holds for UMD spaces. Noticethat the BMO-norm is smaller than all Lp-norms and the L∞-norm is larger than allLp-norms.

It was shown in [51] that l1, and hence every Banach space finitely representablein l1, is a UMD− space. In particular, for every σ-finite measure space (S,Σ, µ) thespace L1(S) is a UMD− space. This follows from Proposition 5.2.3 as well.

It is still open whether there are UMD+ spaces that are not UMD. The corre-sponding operator version of this problem was solved positively by Geiss in [53]. Hepointed out to us some delicate problems for the corresponding Paley-Walsh versionsof UMD− and UMD+. Namely, it is not known whether UMD−

PW implies UMD− orUMD+

PW implies UMD+. Recall this is indeed the case for UMD spaces.

In Section 5.2 it is stated that if E is a UMD+ space, then E∗ is a UMD− space. Ofcourse such a duality results does not hold for UMD− spaces E, since L1 is a UMD−

space and its dual L∞ is not a UMD+ space. However, it is interesting to see whetherthere are conditions under which such a results is true. For instance, if a superreflexivespace E is a UMD− space, does it follow that E∗ is a UMD+ space?

Other interesting problems are: Is there a martingale type 2 space without UMD−?We mention that in [51, Theorem 4], for each 1 ≤ p < 2, Garling constructed amartingale type p space which is not a UMD− space. Does the space of trace classoperators C1 satisfy UMD−?

The simple proof of Theorem 4.3.1 does not work for the corresponding one sidedversion in Theorem 5.3.1. Therefore, we need the argument of Garling from [50]. Itis not clear to us if the assumption on the filtration is needed. At the end of Section5.3 we extended some of the results in Chapter 4 to UMD− spaces. In [94] this hasbeen used to obtain a stochastic Fubini theorem.

We do not know whether Theorems 4.6.5 and 4.6.6 extend to UMD− spaces. Thedifficulty is that the proof uses the vector-valued Stein inequality from [18] which isbased on UMD property once again. Actually, one only needs a very special caseof this inequality. It is however, still possible to extend Corollary 4.6.7 to UMD−


spaces. We do not include the details on this, but a proof can be obtained using semi-R-boundedness of certain conditional expectations. Semi-R-boundedness has beenstudied in [58] by Hoffmann, Kalton and Kucherenko. It is defined in the sameway as R-boundedness, but with xn = anx for n = 1, . . . , N for certain scalars (an)

Nn=1

and x ∈ E.Theorem 5.4.1 is almost a converse to Theorem 5.3.1. It gives the limitations of the

stochastic integration theory of Chapter 4. Some of the techniques used in the proofare taken from [51]. We do not know whether Theorem 5.4.1 can be strengthened toassertions on UMD± spaces. Recently however, in [32] Sonja Cox has showed thatthe result of Theorem 5.3.1 holds true for UMD±

PW spaces.

Chapter 6

Integration with respect tocontinuous local martingales

6.1 Introduction

Let (E, ‖ · ‖) be a Banach space. Let (Ω,A,P) be a complete probability space witha filtration F := (Ft)t∈R+ that satisfies the usual conditions. In this chapter we studystochastic integration of strongly progressively measurable processes φ : R+ × Ω → Ewith respect to a real-valued continuous local martingale M with M(0) = 0. Theresults below extend the results of Chapter 4 in the case that the cylindrical Brownianmotion is a standard Brownian motion.

In Section 6.3 we characterize the strongly progressively measurable processes φ :R+×Ω → E that are stochastically integrable with respect toM . The characterizationis again given in terms of γ-norms. At some level it is surprising that integration ofa process with respect a general continuous local martingale can be checked usingGaussian sums. However, seen as a generalized square-function, this result is verynatural. The proofs are based on time change techniques that also appear in the caseof real-valued processes φ. In the E-valued case some extra difficulties occur.

In Section 6.4 we give several criteria for stochastic integrability and in Section 6.5we will extend the Ito formula to the above setting.


For more details on progressive measurability, continuous local martingales andstochastic integration of real-valued processes we refer to [63, Chapters 7 and 17] and[68, Chapter 3]. Below we repeat some of the important results that will be usedfrequently.

The quadratic variation process of a real-valued continuous local martingale M willbe denoted by [M ]. This is an adapted, continuous and increasing process and we canassociate a random Lebesgue-Stieltjes measure with it, which we will also denote by[M ](·, ω) for ω ∈ Ω. It is well-known how to define a stochastic integral

∫ ·0φ(t) dM(t)

as an element of L0(Ω;Cb(R+)) for a progressively measurable process φ : R+×Ω → Rwith φ ∈ L2(R+, [M ]) a.s. The process

∫ ·0φ(t) dM(t) is a real-valued continuous local

109

110 Chapter 6. Integration w.r.t. continuous local martingales

martingale starting at zero. If τ is a stopping time, then almost surely for all t ∈ R+

we have ∫ τ∧t

0

φ(s) dM(s) =

∫ t

0

1[0,τ ](s)φ(s) dM(s). (6.1.1)

The quadratic variation process of∫ ·

0φ(t) dM(t) is given by

∫ ·0|φ(t)|2 d[M ](t).

As in Chapter 4 recall that for progressively measurable processes φ and (φn)n≥1

which are a.s. in L2(R+, [M ]) it holds that φ = limn→∞ φn in L0(Ω;L2(R+, [M ])) ifand only if ∫ ·

0

φ(t) dM(t) = limn→∞

∫ ·

0

φn(t) dM(t) in L0(Ω;Cb(R+)). (6.1.2)

Moreover, for φ as above the Burkholder-Davis-Gundy inequalities (cf. [63, Theorem17.7]) imply that for all p ∈ (0,∞),

E supt∈R+

∣∣∣ ∫ t

0

φ(s) dM(s)∣∣∣p hp E

( ∫R+

|φ(t)|2 d[M ](t)) p

2(6.1.3)

whenever one of the expressions is finite.


We say that φ : R+ × Ω → E is an elementary process if it is of the form

φ =N∑n=1

1(tn−1,tn]ξn,

where 0 ≤ t0 ≤ t1, . . . ,≤ tN <∞ and for each n = 1, . . . , N , ξn are Ftn−1-measurableE-valued random variables. For such φ we define the stochastic integral as an elementof L0(Ω;Cb(R+;E)) as∫ t

0

φ(s) dM(s) =N∑n=1

(M(tn ∧ t)−M(tn−1 ∧ t))ξn.

We will extend this definition of the stochastic integral below.It is immediate that the stochastic integral definition can be extended to progres-

sively measurable processes φ : R+ × Ω → E that take values in a finite-dimensionalsubspace of E and satisfy φ ∈ L2(R+, [M ];E) almost surely For more general φ thestochastic integral is constructed in Theorem 6.3.1.

We say that φ : R+ × Ω → E is scalarly in L2(R+, [M ]) a.s. if for all x∗ ∈ E∗, foralmost all ω ∈ Ω, 〈φ(·, ω), x∗〉 ∈ L2(R+, [M ](·, ω)).

Definition 6.2.1. For a strongly progressive process φ that is scalarly in L2(R+, [M ])a.s. we say that φ is stochastically integrable with respect to M if there exists asequence (φn)n≥1 of elementary processes such that:

6.3. Characterizations of integrability 111


〈φn, x∗〉 = 〈φ, x∗〉 in L0(Ω;L2([M ])),

(ii) there exists a process ζ ∈ L0(Ω;Cb(R+;E)) such that

ζ = limn→∞

∫ ·

0

φn(t) dM(t) in L0(Ω;Cb(R+;E)).

The process ζ is called the stochastic integral of φ with respect to M , notation

ζ =

∫ ·

0

φ(t) dM(t).

In this way∫ ·

0φ(t) dM(t) is uniquely defined up to indistinguishability. As in Propo-

sition 4.2.2 one can show that ζ is a continuous local martingale that starts at 0.For a process φ : R+ × Ω → E which is scalarly in L2(R+, [M ]) a.s. and a family

X = (X(ω) : ω ∈ Ω) with X(ω) ∈ γ(L2(R+, [M ](·, ω);E) for almost all ω ∈ Ω, we saythat φ represents X if for all x∗ ∈ E∗, for almost all ω ∈ Ω,

〈φ(·, ω), x∗〉 = X∗(ω)x∗ in L2(R+, [M ](·, ω)).

In the case that M is a Brownian motion the above notion of representability reducesto the one in Section 3.5, since for almost all ω ∈ Ω, [M ](t, ω) = t.

The following relation between the above two representability concepts can beproved as Lemma 3.5.2 (2).

Lemma 6.2.2. Let E be a separable Banach space. Let φ : R+ × Ω → E be stronglymeasurable. For each ω ∈ Ω, let X(ω) ∈ γ(L2(R+, [M ](·, ω)), E). If X is representedby φ, then for almost all ω ∈ Ω, X(ω) is represented by φ(·, ω). In particular, φ ∈γ(R+, [M ];E) almost surely.

6.3 Characterizations of integrability

In this section we will prove the following characterization. It is a generalization ofTheorem 4.5.9 in the case where H = R.

Theorem 6.3.1. Let E be a UMD space. For a strongly progressively measurableprocess φ : R+ × Ω → E which is scalarly in L2(R+, [M ]) a.s. the following assertionsare equivalent:

(1) φ is stochastically integrable.

(2) There exists a process ζ ∈ L0(Ω;Cb(R+;E)) such that for all x∗ ∈ E∗ we have

〈ζ, x∗〉 =

∫ ·

0

〈φ(t), x∗〉 dM(t) in L0(Ω;Cb(R+)).


(3) φ ∈ γ(R+, [M ];E) almost surely;

Furthermore, for all p ∈ (1,∞) we have

E supt∈R+

‖ζ(t)‖p hp,E E‖φ‖pγ(R+,[M ];E). (6.3.1)

The estimate (6.3.1) can be seen as a vector-valued stochastic integral version ofthe Burkholder-Davis-Gundy inequalities.

For the proof of Theorem 6.3.1 we need some additional results. We start with alemma on time changes in spaces of γ-radonifying operators.

We say that A : R+ → R is increasing if for all t ≥ s ≥ 0, A(t) ≥ A(s). IfA : R+ → R with A(0) = 0 is increasing and continuous we may associate a Lebesgue-Stieltjes measure with it. If there is no confusion possible, we will write A for thismeasure as well.

Lemma 6.3.2. Let E be a Banach space. Let A : R+ → R with A(0) = 0 be increasingand continuous. Let S := limt→∞A(t) ≤ ∞ and define τ : R+ → R as

τ(s) =

inft ≥ 0 : A(t) > s for 0 ≤ s < S,

∞ for s ≥ S.

Let φ : R+ → E be strongly measurable and let ψ : R+ → E be defined as

ψ(s) =

φ(τ(s)) for 0 ≤ s < S,

0 for s ≥ S.

Then we have φ ∈ γ(R+, A;E) if and only if ψ ∈ γ(R+;E). In that case,

‖φ‖γ(R+,A;E) = ‖ψ‖γ(R+;E). (6.3.2)

Some basic properties of A and τ can be found in [68, Section 3.4.B]. In particular,recall the substitution rule. For a strongly measurable f : R+ → E we have f ∈L1(R+;E) if and only if f τ ∈ L1(0, S;E) and in that case∫

R+

f(t) dA(t) =

∫[0,S)

f(τ(s)) ds. (6.3.3)

Proof. First notice that for all s ∈ R+, A(τ(s)) = s ∧ S and for all t ∈ R+,

τ(A(t)) = supr ≥ t : A(r) = A(t).

Let (fn)n≥1 be an orthonormal basis for L2(0, S). For each n ≥ 1, let fAn : R+ → Rbe defined as fAn (t) = fn(A(t)). We claim that (fAn )n≥1 is an orthonormal basis forL2(R+, A). First of all it follows from (6.3.3) that for all m,n ≥ 1∫

R+

fAm(t)fAn (t) dA(t) =

∫[0,S)

fm(s)fn(s) ds = δmn.


Hence (fAn )n≥1 is an orthonormal system. Let f ∈ L2(R+, A) be such that for alln ≥ 1, ∫

R+

f(t)fAn (t) dA(t) = 0.

We have to show that f = 0, A-almost everywhere. Take any representative of f anddenote it again by f . Define f : R+ → R as f(t) = f(τ(A(t))). Since τ(A(t)) 6= t ispossible only if A is constant near t, we have f = f , A-almost everywhere. It followsfrom (6.3.3) that for all n ≥ 1,∫

[0,S)

f(τ(s))fn(s) ds =

∫R+

f(s)fAn (s) dA(s) =

∫R+

f(s)fAn (s) dA(s) = 0.

Since (fn)n≥1 is an orthonormal basis for L2(0, S), we obtain that f τ = 0, λ-almosteverywhere, where λ is the Lebesgue measure on [0, S). From (6.3.3), it follows that∫

R+

1f(t) 6=0 dA(t) =

∫R+

1f(t) 6=0 dA(t) =

∫[0,S)

1f(τ(s)) 6=0 ds = 0,

and hence f(t) = 0, A-almost everywhere. We may conclude that the claim is true.”⇒” Let IAφ ∈ γ(L2(R+, A), E) be the integral operator associated with φ (cf.

(3.2.4)). It follows from (6.3.3) that for all x∗ ∈ E∗ we have

‖〈ψ, x∗〉‖L2(0,S) = ‖〈φ, x∗〉‖L2(R+,A), (6.3.4)

so ψ is scalarly in L2(0, S). Hence, we may define the operator Iψ ∈ B(L2(0, S), E) as

Iψf =

∫[0,S)

ψ(s)f(s) ds, f ∈ L2(0, S).

From (6.3.3) we deduce that for all n ≥ 1 and x∗ ∈ E∗,

〈Iψfn, x∗〉 =

∫[0,S)

fn(s)〈ψ(s), x∗〉 ds =

∫R+

fAn (t)〈φ(t), x∗〉 dt = 〈IAφ fAn , x∗〉. (6.3.5)

It follows from (6.3.5) that ∑n≥1

γnIψfn =∑n≥1

γnIAφ f

An , (6.3.6)

with convergence in L2(Ω;E) and hence the result and (6.3.2) follow.”⇐” As before (6.3.4) holds and we may define IAφ ∈ B(L2(R+, A), E) as

IAφ f =

∫R+

φ(t)f(t) dA(t), f ∈ L2(R+, A).

It follows from (6.3.3) that for all x∗ ∈ E∗ and f ∈ L2(R+, A) we have

〈IAφ f, x∗〉 =

∫R+

f(t)〈φ(t), x∗〉 dA(t) =

∫[0,S)

f(τ(s)))〈ψ(s), x∗〉 ds = 〈Iψ(f τ), x∗〉,

and we may conclude that (6.3.6) holds. This proves φ ∈ γ(R+, A;E).


Recall the following results (cf. [63, Theorem 18.4] and [68, Section 3.4.B]).

Theorem 6.3.3 (Dambis, Dubins and Schwartz). Define

τs = inft ≥ 0 : [M ]t > s, Gs := Fτs , s ∈ [0,∞).

Then there exist a probability space (Ω,A,P) and a Brownian motion W with respectto an extension of G := (Gs)s∈R+ such that almost surely

W = M τ on [0, [M ]∞) and M = W [M ].

Moreover, (Ω,A,P) may be taken of the form (Ω× [0, 1],A⊗B[0,1],P⊗ λ, where λis the Lebesgue measure on [0, 1]. The extension of G can be chosen as G = G ⊗H fora certain filtration H on ([0, 1],B[0,1]).

Theorem 6.3.4 (Kazamaki). With the notations of Theorem 6.3.3, we have thefollowing time-change formula for stochastic integrals. If φ : R+ × Ω → R is F-progressively measurable and satisfies∫

R+

|φ(s)|2 d[M ]s <∞, almost surely,

then the process

ψ(s) =

φ(τs) if 0 ≤ s < [M ]∞,

0 if [M ]∞ ≤ s <∞(6.3.7)

is G-adapted and satisfies almost surely∫

R+|ψ(r)|2 dr <∞ and∫ t

0

φ(r) dM(r) =

∫ [M ]t

0

ψ(r) dW (r), t ∈ R+, (6.3.8)

∫ τs

0

φ(r) dM(r) =

∫ s

0

ψ(r) dW (r), s ∈ R+. (6.3.9)

Finally, we need the next lemma for weak limits of processes.

Lemma 6.3.5. Let E be a reflexive Banach space. Let ζ : R+ × Ω → E be a stronglymeasurable process such that almost surely ζ(t) : t ∈ R+ is bounded. If for allx∗ ∈ E∗, limt→∞〈ζ(t), x∗〉 exists almost surely, then ζ∞ := weak- limt→∞ ζ(t) existsalmost surely and is strongly measurable.

Proof. Since ζ is strongly measurable, we may assume that E is separable. By thereflexivity of E, we can find a dense sequence (x∗n)n≥1 in E∗. For each n ≥ 1, let Ωn

be such that P (Ωn) = 1 and for all ω ∈ Ωn, limt→∞〈ζ(t, ω), x∗n〉 exists. Let

Ω0 =⋂n≥1

Ωn ∩ ω ∈ Ω : ζ(·, ω) is bounded.


Then it follows from an easy three-ε-argument that limt→∞〈ζ(t, ω), x∗〉 exists for allx∗ ∈ E∗ and all ω ∈ Ω0. For each ω ∈ Ω0, define x∗∗ω ∈ E∗∗ = E as

〈x∗, x∗∗ω 〉 = limt→∞

〈ζ(t, ω), x∗〉

and x∗∗ω = 0 for ω ∈ Ω0. We may define ζ∞ : Ω → E as ζ∞(ω) = x∗∗ω . The Pettismeasurability theorem ensures that ζ∞ is strongly measurable.

We can now prove our main result.

Proof of Theorem 6.3.1. Since φ is strongly measurable, we may assume that E isseparable. Define ψ : R+ × Ω → E as

ψ(s) =

φ(τs) if 0 ≤ s < [M ]∞,

0 if [M ]∞ ≤ s <∞.(6.3.10)

Notice that against functionals from E∗, (6.3.10) coincides with (6.3.7). By Proposition6.3.4 and the Pettis measurability theorem, ψ is strongly measurable and G-adapted.Moreover, from the substitution rule (6.3.3) it follows that pointwise in Ω for allx∗ ∈ E∗,

‖〈ψ, x∗〉‖L2(R+) = ‖〈φ, x∗〉‖L2(R+,[M ]) (6.3.11)

if one of the expressions is finite. In particular ψ is scalarly in L2(R+) a.s.Let W , (Ω,A,P) and G be as in Theorem 6.3.3. We will prove the result by showing

that (1), (2) and (3) for φ are equivalent with (1), (2) and (3) in Theorem 4.5.9 for ψ.(Notation (k, φ) ⇔ (k, ψ) for k = 1, 2, 3).

(1, φ) ⇒ (1, ψ): Assume that (1) holds for a sequence of elementary processes(φn)n≥1. For all n ≥ 1, define ψn : [0,∞]× Ω → E as

ψn(s) :=

φn(τs) if 0 ≤ s < [M ]∞,

0 if [M ]∞ ≤ s <∞.

Then it follows from the Pettis measurability theorem and Proposition 6.3.4 that eachψn is strongly measurable and strongly adapted and since φn is elementary it followsfrom (6.3.9) that for all n ≥ 1 for all s ∈ R+, almost surely we have

ζψn(s) :=

∫ s

0

ψn(r) dW (r) =

∫ τs

0

φn(r) dM(r).

By the assumption, (ζψn)n≥1 is a Cauchy sequence in L0(Ω;Cb(R+;E)), hence it isconvergent to some ζψ ∈ L0(Ω;Cb(R+;E)). By (6.3.3) and Theorem 6.3.1 (1) (i) itfollows that for all x∗ ∈ E∗ we have lim

n→∞〈ψn, x∗〉 = 〈ψ, x∗〉 in L0(Ω;L2(R+)). Since

each ψn takes values in a finite-dimensional subspace of E, we may approximate it toobtain a sequence of elementary processes (ψn)n≥1 that satisfies Theorem 4.5.9 (1) (i)and (ii).


(1, ψ) ⇒ (1, φ): Let Theorem 4.5.9 (1) be satisfied for ψ on the extended probabil-ity space Ω. Then it follows from Theorem 4.5.9 that ψ ∈ γ(R+;E), P-almost surely.By the special choice of Ω and by Fubini’s theorem we may conclude that ψ ∈ γ(R+;E),P-almost surely. Now Lemma 3.5.2 assures that ψ ∈ L0(Ω; γ(R+;E)). By Lemma 4.5.3we can find elementary processes (ψn)n≥1 in L0(Ω; γ(R+;E)) such that ψ = limn→∞ ψnin L0(Ω; γ(R+;E)). In view of the right-ideal property, ψ = limn→∞ 1[0,[M ]∞)ψn inL0(Ω; γ(R+;E)) and it follows that we may assume that ψn(s) = 0 for s ≥ [M ]∞. Foreach n ≥ 1, define φn : R+ × Ω → E as φn = ψn [M ]. Then (φn)n≥1 is a sequence ofstrongly progressively measurable processes. Moreover, φn τ = ψn. By substitutionit follows that for all x∗ ∈ E∗,

‖〈φ, x∗〉 − 〈φn, x∗〉‖L0(Ω;L2([M ])) = ‖〈ψ, x∗〉 − 〈ψn, x∗〉‖L0(Ω;L2(R+)).

Since the latter converges to 0 we obtain (1) (i). By the Proposition 4.5.7 we have∫ ·

0

ψ(t) dW (t) = limn→∞

∫ ·

0

ψn(t) dW (t) in L0(Ω;Cb(R+;E)).

Since the ψn are elementary processes one easily checks that, almost surely for allt ∈ [0, T ], ∫ [M ]t

0

ψn(t) dW (t) =

∫ t

0

φn(t) dM(t).

It follows that (∫ t

0φn(t) dM(t))n≥1 is a Cauchy sequence in L0(Ω;Cb(R+;E)). Now

as in the proof of (1, φ) ⇒ (1, ψ), we may conclude (1) (ii) via an approximationargument.

(2, φ) ⇒ (2, ψ): Let ζ : [0,∞) × Ω → E be the given integral process. Letζψ : [0,∞)× Ω → E be defined as

ζψ(s) =

ζ(τs). if 0 ≤ s < [M ]∞,

weak− limt→∞ ζ(t) if [M ]∞ ≤ s <∞.

The weak limit exists almost surely and is strongly measurable by Lemma 6.3.5. Theresult would follow immediately if ζψ ∈ L0(Ω;Cb(R+;E)). This is not clear, since thetrajectories of s 7→ τs are not necessarily continuous. Instead, we do the followingargument to show that Theorem 4.5.9 (2) holds for ψ. Afterwards, in Corollary 6.3.6we will show that almost surely ζψ has continuous trajectories.

It follows from Proposition 6.3.4 that ζψ is weakly continuous almost surely. Choose(x∗n)n≥1 dense in the closed unit ball BE∗ . Let Ω0 with P (Ω0) = 1 be such that for allω ∈ Ω0, 〈ζψ(·, ω), x∗n〉 is continuous. For each k ≥ 1 define

Tk = inft > 0 : ‖ζψ(t)‖ ≥ k.

Since ‖ζψ‖ = supn≥1 |〈ζψ, x∗n〉| is progressively measurable, each Tk is a stopping time.

We claim that ‖ζTkψ ‖ ≤ k on Ω0. Indeed, for all n ≥ 1 we have |〈ζTk

ψ , x∗n〉| ≤ k


on Ω0, and taking the supremum over all n gives the claim. In particular, we haveζψ(Tk) ∈ L2(Ω;E).

Now fix k ≥ 1 and x∗ ∈ E∗. By (6.3.9) we have almost surely for all t ∈ R+,

〈ζTkψ (t), x∗〉 =

∫ t

0

1[0,Tk]〈ψ(s), x∗〉 dW (s).

Since 〈ζTkψ , x∗〉 is in L2(Ω;Cb(R+)) it follows from the Burkholder-Davis-Gundy in-

equalities (4.1.3) that 1[0,Tk]〈ψ, x∗〉 ∈ L2(Ω;L2(R+)). Hence 1[0,Tk]ψ is scalarly inL2(Ω;L2(R+)) and for all x∗ ∈ E∗, almost surely,

〈ζψ(Tk), x∗〉 =

∫ ∞

0

1[0,Tk]〈ψ(s), x∗〉 dW (s).

Therefore, we may apply Theorem 4.4.3 to conclude that for all k ≥ 1, 1[0,Tk]ψ ∈L2(Ω; γ(R+;E)). Since for all ω ∈ Ω, Tk(ω) = ∞ for all k large enough, we deduce thatψ ∈ L0(Ω; γ(R+;E)) and ψ = limk→∞ 1[0,Tk]ψ in L0(Ω; γ(R+;E)). It follows from The-

orem 4.5.9 that ψ is stochastically integrable and we may define ζψ ∈ L0(Ω;Cb(R+;E))as

ζψ(t) =

∫ t

0

ψ(s) dW (s).

We conclude that Theorem 4.5.9 (2) holds for ψ and ζψ.(2, ψ) ⇒ (2, φ): Let ζψ be the stochastic integral process of ψ with respect to W .

Let ζ : [0,∞)×Ω → E be defined as ζ = ζψ [M ]. Then ζ ∈ L0(Ω;Cb(R+;E)) and itfollows from (6.3.8) that for all x∗ ∈ E∗, for all t ∈ R+, almost surely we have

〈ζ(t), x∗〉 = 〈ζψ([M ]t), x∗〉 = (〈ζψ, x∗〉)([M ]t)

=

∫ [M ]t

0

〈ψ(r), x∗〉 dW (r) =

∫ t

0

〈φ(r), x∗〉 dM(r).

This proves (2).(3, φ) ⇔ (3, ψ): This follows from Lemma 6.3.2. Moreover, it follows from (6.3.2)

that for almost all ω ∈ Ω, we have

‖φ(·, ω)‖γ(R+,[M ](·,ω);E) = ‖ψ(·, ω)‖γ(R+;E). (6.3.12)

This shows that ω 7→ ‖φ(·, ω)‖γ(R+,[M ](·,ω);E) is measurable and (6.3.1) follows fromζ(t) = ζψ([M ]t), (4.5.8) and (6.3.12).

As is clear from the proof and Theorem 5.3.4, there is again a one-sided stochasticintegration theory for UMD− spaces.

Proposition 6.3.4 has the following extension to E-valued processes.

Corollary 6.3.6. Let E be a UMD space. If φ : R+ × Ω → E is strongly F-progressively measurable and satisfies, φ ∈ γ(R+, [M ];E) almost surely, then theprocess ψ : R+ × Ω → E defined as (6.3.10) is G-adapted and satisfies, ψ ∈ γ(R+;E)almost surely, and the E-valued versions of (6.3.8) and (6.3.9) hold.


Proof. In Theorem 6.3.1 we already showed that ψ is G-adapted and almost surely,ψ ∈ γ(R+;E). Also the E-valued version of (6.3.8) has been obtained there. From(6.3.8) we deduce that∫ τs

0

φ(r) dM(r) =

∫ [M ]τs

0

ψ(r) dW (r) =

∫ s∧[M ]∞

0

ψ(r) dW (r) =

∫ s

0

ψ(r) dW (r)

and the E-valued version of (6.3.9) follows.

To end this section we give the following useful convergence result for the stochasticintegral.

Corollary 6.3.7. Let E be a UMD space. Let (φn)n≥1 and φ be E-valued stronglyprogressive processes which are in γ(R+, [M ];E) a.s. Then we have φ = limn→∞ φn inL0(Ω; γ(R+, [M ];E)) if and only if∫ ·

0

φ(t) dM(t) = limn→∞

∫ ·

0

φn(t) dM(t)

in L0(Ω;Cb(R+;E)).

Proof. This follows from Proposition 4.5.7, Lemma 6.3.2 and Corollary 6.3.6.

6.4 Criteria for stochastic integrability

As a consequence of Theorem 4.6.1, Lemma (6.3.2), Theorem 6.3.1 and the substitutionrule (6.3.3), we obtain the following result for Banach function spaces.

Corollary 6.4.1. Let E be UMD Banach function space over a σ-finite measure space(S,Σ, µ) and let p ∈ (1,∞). Let φ : R+×Ω → E be a strongly progressively measurableprocess. Then φ is stochastically integrable with respect to M if and only if almostsurely ∥∥∥( ∫

R+

|φ(t, ·)|2 d[M ](t)) 1

2∥∥∥E<∞.

In this case, for all p ∈ (1,∞) we have

E supt∈R+

∥∥∥∫ t

0

φ(t) dM(t)∥∥∥p hp,E E

∥∥∥( ∫R+

|φ(t, ·)|2 d[M ](t)) 1

2∥∥∥pE.

The next result follows from Theorem 6.3.1 and Proposition 3.3.1.

Corollary 6.4.2. Let E be a UMD space and let p ∈ (1,∞).

(1) If E has type 2, then every strongly progressively measurable process φ such thatφ ∈ L2(R+, [M ];E) almost surely is stochastically integrable with respect to Mand we have

E supt∈R+

∥∥∥∫ t

0

φ(t) dM(t)∥∥∥p .p,E E‖φ‖pL2(R+,[M ];E).

6.5. The Ito formula 119

(2) If E has cotype 2, then every H-strongly measurable stochastically integrableprocess φ belongs to L2(R+, [M ];E) almost surely and we have

E‖φ‖pL2(R+,[M ];E) .p,E E supt∈R+

∥∥∥∫ t

0

φ(t) dM(t)∥∥∥p.

6.5 The Ito formula

In this section we extend the Ito formula from Section 4.8 to arbitrary continuous localmartingales. As in Section 4.7 we may extend the stochastic integral to processes whichare locally stochastically integrable.

Theorem 6.5.1 (Ito formula). Let E and F be UMD spaces. Let M be a continuouslocal martingale and let A be adapted and almost surely continuous and locally of finitevariation. Assume that f : R+ × E → F is of class C1,2. Let φ : R+ × Ω → E be astrongly progressively measurable process which is locally stochastically integrable withrespect to M and assume that the paths of φ belong to L2

loc(R+, [M ];E) almost surely.Let ψ : R+ × Ω → E be strongly progressively measurable with paths in L1

loc(R+, A;E)almost surely. Let ξ : Ω → E be strongly F0-measurable. Define ζ : R+ × Ω → E as

ζ = ξ +

∫ ·

0

ψ(s) dA(s) +

∫ ·

0

φ(s) dM(s).

Then s 7→ D2f(s, ζ(s))φ(s) is locally stochastically integrable with respect to M andalmost surely we have for all t ∈ [0, T ],

f(t, ζ(t))− f(0, ζ(0)) =

∫ t

0

D1f(s, ζ(s)) ds+

∫ t

0

D2f(s, ζ(s))ψ(s) dA(s)

+

∫ t

0

D2f(s, ζ(s))φ(s) dM(s)

+ 12

∫ t

0

(D2

2f(s, ζ(s)))(φ(s), φ(s)) d[M ](s).

The proof of this result follows the lines of Theorem 4.8.2. The only non-trivialextension of the arguments in Section 4.8 is Lemma 4.8.5. Therefore, we only provethis lemma.

Lemma 6.5.2. Let E be a UMD space. Let φ : R+ ×Ω → E be strongly progressivelymeasurable. Assume that φ is stochastically integrable with respect to M and φ ∈L2(R+, [M ];E) a.s. Then there exists a sequence of elementary processes (Φn)n≥1

such that

φ = limn→∞

φn in L0(Ω;L2(R+;E)) ∩ L0(Ω; γ(R+, [M ];E)).


Proof. Let ψ be as in Corollary 6.3.6. Then it follows from a substitution that ψ ∈γ(R+;E) ∩ L2(R+;E). It follows from Lemma 4.8.5 that ψ may be approximatedalmost surely in γ(R+;E) ∩ L2(R+;E) by a sequence of elementary processes ψn. Asin the proof of Theorem 6.3.1 we may assume that for all n ≥ 1, ψn(s) = 0 forall s ≥ [M ]∞. Let φn(t) = ψn([M ]t). Then φn τ = ψn. It follows from Lemma6.3.2 that φ = limn→∞ φn in γ(R+, [M ];E) almost surely. It follows from (6.3.3) thatφ = limn→∞ φ in L2(R+, [M ];E) almost surely.


In this chapter we showed that many of the results from Chapter 4 extend to the casethat the integrator is a general continuous local martingales. As a direct consequenceit is possible to integrate E-valued processes with respect to real-valued continuoussemi-martingales. It is an interesting problem to find a generalization to cylindricalcontinuous local (semi)-martingales.

The proof Theorem 6.3.1 is based on a time change. The main ingredients areTheorems 4.5.9, 6.3.3 and 6.3.4. A careful look shows that a time change is also usedin the proofs for real-valued stochastic integration theory (cf. [63, Lemma 17.23] or [68,Section 3.2.A]). In Corollary 6.3.6 we proved an E-valued extension of Kazamaki’sresult.

The results in Sections 6.4 and 6.5 are extensions of results in Chapter 4. Manyresults we did not comment on also extend to the above setting.

Part II

Stochastic equations

121

Chapter 7

Basic concepts

7.1 Introduction

In this chapter (Ω,A,P) is a complete probability space andWH is a cylindrical Brown-ian motion with respect to a complete filtration F = (Ft)t≥0 on (Ω,A,P). Let (E, ‖·‖)be a Banach space and let (H, [·, ·]) be a separable Hilbert space.

Below we study several concepts that will be useful in Chapters 8 and 9. InSection 7.2 we consider evolution families (P (t, s))0≤s≤t≤T . Under the Tanabe con-ditions for parabolic evolution equations, we briefly recall some important existence,uniqueness and regularity results for evolution families. In Section 7.3 we deduce reg-ularity results of deterministic convolutions P ∗ φ from this. Here we use the notationP ∗ φ(t) =

∫ t

0P (t, s)φ(s) ds. We will also prove a factorization lemma in the style

of Da Prato, Kwapien and Zabczyk [35]. This will allow us to obtain regularityresults for stochastic convolutions in Chapters 8 and 9.

In Section 7.4 we give conditions under which stochastic integrals of the formt 7→

∫ t

0Φ(t, s) dWH(s) have a strongly progressive version.

In the final Section 7.5 we introduce different solutions concepts for a class ofstochastic evolution equations. Under suitable conditions we prove equivalences ofthese concepts. We only consider “strong solutions” in the sense that the probabilityspace is fixed. This probabilistic notion of “strong solutions” should not be confusedwith the analytic concept of “strong solutions” as defined in Section 7.5.

Some results in this Chapter are related to [98] and [131].

7.2 Evolution families

Let (A(t), D(A(t)))t∈[0,T ] be a family of closed and densely defined linear operators ona Banach space E. Consider the non-autonomous Cauchy problem:

u′(t) = A(t)u(t), t ∈ [s, T ],

u(s) = x.(7.2.1)

123

124 Chapter 7. Basic concepts

We say that u is a classical solution of (7.2.1) if u ∈ C([s, T ];E) ∩ C1((s, T ];E),u(t) ∈ D(A(t)) for all t ∈ (s, T ], u(s) = x, and du

dt(t) = A(t)u(t) for all t ∈ (s, T ]. We

call u a strict solution of (7.2.1) if u ∈ C1([s, T ];E), u(t) ∈ D(A(t)) for all t ∈ [s, T ],u(s) = x, and du

dt(t) = A(t)u(t) for all t ∈ [s, T ].

A family of bounded operators (P (t, s))0≤s≤t≤T on E is called a strongly continuousevolution family if

(1) P (s, s) = I for all s ∈ [0, T ].

(2) P (t, s) = P (t, r)P (r, s) for all 0 ≤ s ≤ r ≤ t ≤ T .

(3) The mapping (τ, σ) ∈ [0, T ]2 : σ ≤ τ 3 (t, s) → P (t, s) is strongly continuous.

We say that such a family (P (t, s))0≤s≤t≤T solves (7.2.1) (on (Ys)s∈[0,T ]) if (Ys)s∈[0,T ] aredense subspaces of E such that for all 0 ≤ s ≤ t ≤ T , we have P (t, s)Ys ⊂ Yt ⊂ D(A(t))and the function t 7→ P (t, s)x is a strict solution of (7.2.1) for every x ∈ Ys.

Well-posedness (i.e. existence, uniqueness, and continuous dependence on initialvalues from (Ys)s∈[0,T ]) of (7.2.1) is equivalent to the existence and uniqueness of astrongly continuous evolution family that solves (7.2.1) on (Ys)s∈[0,T ] (see [103, 104]and the references therein).

We will assume that (A(t))t∈[0,T ] generates a unique strongly continuous evolutionfamily (P (t, s))0≤s≤t≤T that solves (7.2.1). In the literature many sufficient conditionsfor this can be found, both in the hyperbolic and parabolic setting (cf. [4, 9, 82, 83,108, 124, 125] and the references therein).

The next result which will be useful in Section 7.5.

Proposition 7.2.1. Assume that (A(t), D(A(t)))t∈[0,T ] a family of closed and denselydefined linear operators on a Banach space E that generates a strongly continuous evo-lution family. Assume that for each t ∈ [0, T ], (A∗(t− s), D(A∗(t− s)))s∈[0,t] generatesan evolution family that solves (7.2.1) on dense subspaces (Zs)s∈[0,t] of E∗. Then forall t ∈ [0, T ] and all x∗ ∈ Zt, s 7→ P ∗(t, s)x∗ is in C1([0, t];E∗) and for all s ∈ [0, t],P ∗(t, s)x∗ ∈ D(A∗(s)) and

d

dsP ∗(t, s)x∗ = −A∗(s)P ∗(t, s)x∗ s ∈ [0, t].

Proof. For each t0 ∈ [0, T ], let (W (t0; t, s))0≤s≤t≤t0 in B(E∗) be the evolution familygenerated by (A∗(t0 − s), D(A∗(t0 − s)))s∈[0,t0]. For each 0 ≤ s ≤ t ≤ T let V (t, s) =W (t; t− s, 0). Then for all x∗ ∈ Zt, s 7→ V (t, s)x∗ ∈ C1([0, t];E∗) with

d

dsV (t, s)x∗ = −A∗(s)V (t, s)x∗ s ∈ [0, t].

To complete the proof it suffices to show that U∗(t, s) = V (t, s) for 0 ≤ s < t ≤ T .Let Ys be as in the definition of the evolution family. Let x ∈ Ys and x∗ ∈ Zt. Thenfor r ∈ (s, t),

d

dr〈U(r, s)x, V (t, r)x∗〉 = 〈A(r)U(r, s)x, V (t, r)x∗〉 − 〈U(r, s)x,A∗(r)V (t, r)x∗〉 = 0.

7.2. Evolution families 125

Therefore, there is a constant c independent of r such that 〈U(r, s)x, V (t, r)x∗〉 = c.Letting r ↓ s and r ↑ t, this gives 〈x, V (t, s)x∗〉 = 〈U(t, s)x, x∗〉. This implies theresult.

We will briefly recall the parabolic setting for constant domains of Tanabe (cf.[124, Section 5.2]). If E is a real Banach space the assumptions below should beunderstood for the complexification of the objects under consideration.

Tanabe conditions Assume that D(A(t)) does not depend on time, i.e. for allt ∈ [0, T ], D := D(A(0)) = D(A(t)) is dense in E. For w ∈ R, let Aw(t) = A(t) − wand

Σ(ϕ,w) = w ∪ λ ∈ C \ w : | arg(λ− w)| ≤ ϕ.

(T1) There are constants w ∈ R, K ≥ 0, and ϕ ∈ (π2, π) such that for all t ∈ [0, T ],

Σ(ϕ,w) ⊂ %(A(t)) and for all λ ∈ Σ(ϕ,w) and t ∈ [0, T ],

‖R(λ,A(t))‖ ≤ K

1 + |λ− w|.

(T2) There are constants L ≥ 0 and µ ∈ (0, 1] such that for all t, s ∈ [0, T ] we have

‖Aw(t)A−1w (0)− Aw(s)A−1

w (0)‖ ≤ L|t− s|µ.

The first condition may be seen as analyticity uniformly in t ∈ [0, T ]. The secondcondition is equivalent to µ-Holder continuity of A : [0, T ] → B(D,E). In [124] it hasbeen shown that condition (T2) implies that there is a constant L ≥ 0, such that forall t, s, r ∈ [0, T ] we have

‖Aw(t)A−1w (r)− Aw(s)A−1

w (r)‖ ≤ L|t− s|µ. (7.2.2)

Taking r = s, it follows that

‖Aw(t)A−1w (s)− I‖ ≤ L|t− s|µ (7.2.3)

and one can show that this condition implies (T2) again. In particular, the familyAw(s)A−1

w (t) : s, t ∈ [0, T ] is uniformly bounded. Therefore, D(A(t)) = D(A(0))isomorphically with estimates that do not depend on t ∈ [0, T ].

Under the above conditions one can construct a strongly continuous evolutionfamily for (A(t))t∈[0,T ] that solves (7.2.1) on D. Moreover, the following theorem holds(cf. [124, Section 5.2] and [83, Corollary 6.1.8]).

For p ∈ [1,∞] and θ ∈ (0, 1) we write (E,D)θ,p for the real interpolation space and[E,D]θ for the complex interpolation space between E and D (cf. [83]).


Theorem 7.2.2. If conditions (T1) and (T2) hold, then there exists a unique stronglycontinuous evolution family (P (t, s))0≤s≤t≤T that solves (7.2.1) on D and for all x ∈ E,P (t, s)x is the unique classical solution of (7.2.1). For all θ ∈ (0, 1), β ∈ (0, θ) andp ∈ [1,∞] there is a constant C only depending on θ, β, p and the constants in (T1),(T2) and T ∨ 1 such that for all x ∈ (E,D)θ,p

‖P (t, s)x− P (t, r)x‖(E,D)β,p≤ C(t− s)θ−β‖x‖(E,D)θ,p

. (7.2.4)

For all θ ∈ (0, 1), β ∈ [θ, 1) and p ∈ [1,∞] there is a constant C only dependingon θ, β, p and the constants in (T1), (T2) and T ∨ 1 such that for all x ∈ (E,D)θ,p

‖P (t, s)x‖(E,D)β,p≤ C

(t− s)β−θ‖x‖(E,D)θ,p

. (7.2.5)

In particular it follows that P (t, s)0≤s≤t≤T is a strongly continuous evolution familyon (E,D)β,p for all β ∈ (0, 1) and p ∈ [1,∞).

Finally, we recall the result of Yagi (see [136, Theorem 2.1]) that there is a constantC > 0 such that for all θ ∈ (0, µ), 0 ≤ s < t ≤ T and x ∈ D((w − A(s))θ),

‖P (t, s)(w − A(s))θx‖ ≤ C(µ− θ)−1(t− s)−θ‖x‖. (7.2.6)

Actually, in [136] this result has been proved in a more general setting.

7.3 Deterministic convolutions

For α ∈ (0, 1], p ∈ [1,∞] and f ∈ Lp(0, T ;E), define the function Rαf ∈ Lp(0, T ;E)by

(Rαf)(t) =1

Γ(α)

∫ t

0

(t− s)α−1P (t, s)f(s) ds. (7.3.1)

This is well-defined by Young’s inequality and there is a constant C ≥ 0 that onlydepends on α, p and sup0≤s≤t≤T ‖P (t, s)‖ such that

‖Rαf‖Lp(0,T ;E) ≤ CTα‖f‖Lp(0,T ;E).

It is well-known (cf. [35, Lemma 1] and [122, Lemma 2.1(i)]) that for α ∈ (0, 1] andp ∈ (1,∞) such that pα > 1 and f ∈ Lp(0, T ;E), Rαf is continuous and satisfies

‖Rαf‖C([0,T ];E) ≤ CTα−1p‖f‖Lp(0,T ;E),

for some constant C > 0 that only depends on α, p and sup0≤s≤t≤T ‖P (t, s)‖. In thenext lemma we extend this under the conditions (T1) and (T2). It can be seen as ageneralization of [35, Lemma 2] and [122, Lemma 2.1 (ii)]. A more precise comparisonof both results is made in Section 7.6. The lemma will be useful for proving regularityof deterministic and stochastic convolutions.

7.3. Deterministic convolutions 127

Lemma 7.3.1. Assume that (A(t))t∈[0,T ] satisfies (T1) and (T2). Let α ∈ (0, 1],δ ∈ [0, 1), γ ∈ [0, δ] be such that δ − γ < α. The following assertions hold:

(1) Let λ ∈ (0, 1) and p ∈ [1,∞] be such that δ − γ + λ < α − 1p. Then for

every f ∈ Lp(0, T ; (E,D)γ,1) we have (Rαf)(t) ∈ (E,D)δ,1 for all t ∈ [0, T ].Moreover, the mapping t 7→ (Rαf)(t) into (E,D)δ,1 is λ-Holder continuous andthere is a constant C ≥ 0 depending on the constants in (T1) and (T2) and onT ∨ 1, α, δ, γ, λ, p such that for all f ∈ Lp(0, T ; (E,D)γ,1),

‖Rαf‖Cλ([0,T ];(E,D)δ,1) ≤ C‖f‖Lp(0,T ;(E,D)γ,1).

(2) For every p ∈ [1,∞] and f ∈ Lp(0, T ; (E,D)γ,1) the function Rαf is en elementof Lp(0, T ; (E,D)δ,1) and there is a constant C ≥ 0 depending on the constantsin (T1) and (T2) and on T ∨1, α, δ, γ, p such that for all f ∈ Lp(0, T ; (E,D)γ,1),

‖Rαf‖Lp(0,T ;(E,D)δ,1) ≤ C‖f‖Lp(0,T ;(E,D)γ,1).

Proof. (1): We may assume that p < ∞. It is already noted that the integral in(7.3.1) is well defined and that Rαf ∈ C([0, T ];E). Moreover, (Rαf)(0) = 0. So toprove the result it is sufficient to show that there is a constant C such that for allf ∈ Lp(0, T ;E) and for all 0 ≤ s < t ≤ T ,

‖(Rαf)(t)− (Rαf)(s)‖(E,D)δ,1≤ C|t− s|λ‖f‖Lp(0,T ;(E,D)γ,1). (7.3.2)

For the proof of (7.3.2) we split the left-hand side into three parts,

Γ(α)‖(Rαf)(t)− (Rαf)(s)‖(E,D)δ,1

≤ ‖∫ t

s

(t− r)α−1P (t, r)f(r) dr‖(E,D)δ,1

+ ‖∫ s

0

[(t− r)α−1 − (s− r)α−1

]P (t, r)f(r) dr‖(E,D)δ,1

+ ‖∫ s

0

(s− r)α−1(P (t, r)− P (s, r))f(r) dr‖(E,D)δ,1

=: I1 + I2 + I3

The terms I1, I2, and I3 are estimated separately.It follows from (7.2.5) and Holder’s inequality that

I1 ≤∫ t

s

(t− r)α−1‖P (t, r)f(r)‖(E,D)δ,1dr

≤ C1

((α− 1− δ + γ)p′ + 1)1p′

(t− s)α−1p−δ+γ‖f‖Lp(0,T ;(E,D)γ,1),

where p′ > 1 is such that 1p

+ 1p′

= 1.


For I2, note that by (7.2.5) for all x ∈ (E,D)γ,1 and r ∈ [0, s) we have((s− r)α−1 − (t− r)α−1

)‖P (t, r)x‖(E,D)δ,1

≤ C((s− r)α−1 − (t− r)α−1

)(t− r)−δ+γ‖x‖(E,D)γ,1

≤ C((s− r)α−1−δ+γ − (t− r)α−1−δ+γ)‖x‖(E,D)γ,1 .

Therefore, by Holder’s inequality,

I2 ≤ C

∫ s

0

((s− r)α−1−δ+γ − (t− r)α−1−δ+γ) ‖f(r)‖(E,D)γ,1 dr

≤ C

(∫ s

0

((s− r)α−1−δ+γ − (t− r)α−1−δ+γ)p′ dr) 1

p′

‖f‖Lp(0,T ;(E,D)γ,1)

Using the estimate (a− b)q ≤ aq − bq for q ≥ 1, a ≥ b ≥ 0 we proceed by

I2 ≤ C

(∫ s

0

(s− r)(α−1−δ+γ)p′ − (t− r)(α−1−δ+γ)p′ dr

) 1p′

‖f‖Lp(0,T ;(E,D)γ,1))

= CCα,p,δ,γ

(sα−1−δ+γ+ 1

p′ + (t− s)α−1−δ+γ+ 1

p′ − tα−1−δ+γ+ 1

p′)‖f‖Lp(0,T ;(E,D)γ,1)

≤ CCα,p,δ,γ(t− s)α−1p−δ+γ‖f‖Lp(0,T ;(E,D)γ,1).

For I3 we may apply (7.2.4) with θ = λ+ δ and β = δ and (7.2.5) to obtain

‖(P (t, r)− P (s, r))x‖(E,D)δ,1= ‖(P (t, s)− I)P (s, r)x‖(E,D)δ,1

≤ C(t− s)λ‖P (s, r)x‖(E,D)λ+δ,1

≤ C(t− s)λ(s− r)−δ−λ+γ‖x‖(E,D)γ,1

for all x ∈ (E,D)γ,1. Applying this together with Holder’s inequality gives

I3 ≤∫ s

0

(s− r)α−1‖(P (t, r)− P (s, r))f(r)‖(E,D)δ,1dr

≤ C(t− s)λ∫ s

0

(s− r)α−1−δ−λ+γ‖f(r)‖(E,D)γ,1 dr

≤ C ′(t− s)λ‖f‖Lp(0,T ;(E,D)γ,1) dr,

where we used (α− 1− δ − λ+ γ)p′ > −1.(2): We may assume that p <∞. From (7.2.5) and Young’s inequality it follows

that

‖Rαf‖Lp(0,T ;(E,D)δ,1) ≤1

Γ(1− α)

( ∫ T

0

( ∫ t

0

(t− s)α−1‖P (t, s)f(s)‖(E,D)δ,1ds

)pdt

) 1p

≤ C

Γ(1− α)

( ∫ T

0

( ∫ t

0

(t− s)α−δ+γ−1‖f(s)‖(E,D)γ,1 ds)pdt

) 1p

≤ C ′‖f‖Lp(0,T ;(E,D)γ,1).

7.3. Deterministic convolutions 129

The next regularity result is formulated for functions φ such that t 7→ (w −A(t))−θφ(t) is an E-valued function, where θ ∈ [0, µ). In the case that θ = 0 there isa more direct way to obtain it (see [82, Proposition 3.5]). In that case one can eventake λ+ δ = 1− 1

p. We will write

P ∗ φ(t) =

∫ t

0

P (t, s)φ(s) ds.

First we explain some general measurability properties which hold under the Tan-abe conditions. One has that for all 0 ≤ s < t ≤ T , P (t, s)(−Aw(s))θ has an ex-tension to an operator in B(E) (see (7.2.6)). Moreover as a function of (s, t) where0 ≤ s < t ≤ T , this extension is E-strongly measurable. Indeed, by approximation itsuffices to show that for all x ∈ D, P (t, s)(−Aw(s))θx is strongly measurable. How-ever, this function is continuous by the strong continuity of the evolution family, thecontinuity of s 7→ (−Aw(s))θ−1 (cf. [121, (2.10)]) and s 7→ (−Aw(s))x and by writing

P (t, s)(−Aw(s))θx = P (t, s)(−Aw(s))θ−1(−Aw(s))x.

Proposition 7.3.2. Assume that (A(t))t∈[0,T ] satisfies (T1) and (T2). Let θ ∈ [0, µ)

(1) Let p ∈ (1,∞], δ ∈ [0, 1) and λ ∈ (0, 1) be such that λ+δ < 1− 1p−θ. Then there

exists a constant CT with limT↓0CT = 0 and depending on the constants in (T1)and (T2) and T ∨ 1, p, λ, δ and θ such that for all (−Aw)−θφ ∈ Lp(0, T ;E),

‖P ∗ φ‖Cλ([0,T ];(E,D)δ,1) ≤ CT‖(−Aw)−θφ‖Lp(0,T ;E). (7.3.3)

(2) Let p ∈ [1,∞], δ ∈ [0, 1) be such that δ < 1− θ. Then there exists a constant CTwith limT↓0CT = 0 and depending on the above constant, the constants in (T1)and (T2) and T ∨ 1, p, δ and θ such that for all (−Aw)−θφ ∈ Lp(0, T ;E),

‖P ∗ φ‖Lp(0,T ;(E,D)δ,1) ≤ CT‖(−Aw)−θφ‖Lp(0,T ;E). (7.3.4)

To make the above formulation rigorous in the case that θ 6= 0, we interpret φ asa function taking values in appropriate extrapolation spaces of order θ.

Proof. By the above condirations it follows that

(t, s) : 0 ≤ s < t ≤ T 3 (t, s) 7→ P (t, s)φ(s) = P (t, s)(−Aw(s))θ(−Aw(s))−θφ(s)

a strongly measurable E-valued function.(1): Let α ∈ (0, 1) be such that λ+ δ < α < 1− 1

p− θ. Define ζ : [0, T ] → E as

ζα(t) =1

Γ(1− α)

∫ t

0

(t− s)−αP (t, s)φ(s) ds.


This is well defined and by (7.2.6) and Holder’s inequality for each t ∈ [0, T ],

‖ζα(t)‖ ≤1

Γ(1− α)

∫ t

0

(t− s)−α‖P (t, s)φ(s)‖ ds

≤ C

Γ(1− α)

∫ t

0

(t− s)−α−θ‖(−Aw(s))−θφ(s)‖ ds

≤ CT‖(−Aw)−θφ‖Lp(0,T ;E).

This shows that t 7→ ζα(t) is well-defined and ζα ∈ Lp(0, T ;E). Define ζ : [0, T ] → Eas

ζ(t) =

∫ t

0

P (t, s)φ(s) ds.

We claim that ζ = Rα(ζα). This would complete the proof by Lemma 7.3.1 and

‖ζ‖Cλ([0,T ];(E,D)δ,1) = ‖Rα(ζα)‖Cλ([0,T ];(E,D)δ,1)

≤ C‖ζα‖Lp(0,T ;E) ≤ T1pCT‖(−Aw)−θφ‖Lp(0,T ;E).

To prove the claim notice that by Fubini’s theorem for all t ∈ [0, T ],

Rα(ζα) =1

Γ(α)

∫ t

0

(t− s)α−1P (t, s)ζα(s) ds

=1

Γ(1− α)Γ(α)

∫ t

0

∫ s

0

(t− s)α−1(s− r)−αP (t, r)φ(r) dr ds

=1

Γ(1− α)Γ(α)

∫ t

0

∫ t

r

(t− s)α−1(s− r)−αP (t, r)φ(r) ds dr

=

∫ t

0

P (t, r)φ(r) dr = ζ(t).

(2): This may be proved in the same way as (1). Let α ∈ (δ, 1− θ). This time onecan use (7.2.6) and Young’s inequality to estimate

‖ζα‖Lp(0,T ;E) ≤1

Γ(1− α)

( ∫ T

0

( ∫ t

0

(t− s)−α‖P (t, s)φ(s)‖ ds)pdt

) 1p

≤ C

Γ(1− α)

( ∫ T

0

( ∫ t

0

(t− s)−α−θ‖(−Aw(s))−θφ(s)‖ ds)pdt

) 1p

≤ CT‖(−Aw)−θφ‖Lp(0,T ;E).

Therefore, by Lemma 7.3.1 (2),

‖P ∗ φ‖Lp(0,T ;(E,D)δ,1) = ‖Rα(ζα)‖Lp(0,T ;(E,D)δ,1)

≤ C‖ζα‖Lp(0,T ;E) ≤ CT‖(−Aw)−θφ‖Lp(0,T ;E).

7.4. Measurability of stochastic convolutions 131

7.4 Measurability of stochastic convolutions

Below we study processes as (t, ω) 7→( ∫ t

0P (t, s)Φ(s) dWH(s)

)(ω). We will need that

such processes are strongly measurable or even strongly progressively measurable.The following lemma establishes strong progressive measurability of a general class ofprocesses.

Proposition 7.4.1. Let E be a UMD− space. Assume that Φ : R+×R+×Ω → B(H,E)is H-strongly measurable and for each t ∈ R+, Φ(t, ·) is adapted and for all t ∈ R+,Φ(t, ·) is in γ(R+;H,E) a.s., then the process ζ : R+ × Ω → E defined by

ζ(t) =

∫ t

0

Φ(t, s) dWH(s)

has a strongly progressive version.

If only for almost all t ∈ R+, Φ(t, ·) is in γ(R+;H,E) a.s., say on a set A ∈ BR+ offull λ-measure, where λ is the Lebesgue measure on R+, then the above result may beapplied to Φ1A, to obtain a strongly progressive process ζ such that ζ = ζ, λ⊗ P-a.e.

The proof is based on techniques of [39, Section IV.30] and we will first partlyextend the result therein.

Lemma 7.4.2. Let E be a complete separable metric space. If ζ : R+ × Ω → Eis adapted and measurable, then ζ has a progressive version (in the sense of inverseimages).

Proof. It follows from [63, Theorem A1.2] that E is Borel isomorphic to a Borel subsetof [0, 1], i.e. there exists a Borel set A ⊂ [0, 1] and a bijective measurable map f : E →A such that f−1 is also measurable. Let η = f(ζ). Then η is adapted and measurable,so it follows from [39, Section IV.30] that η has a progressively measurable version η.Let C = η−1(A). Then C is progressively measurable and since η is a version of ηwe deduce that for all t ∈ R+, P (Ct) = 1, where Ct = ω ∈ Ω : (t, ω) ∈ C. Defineη : R+ × Ω → A as η = η1C . Then η is progressive measurable and is a version of η;hence it is a version of η. Now define ζ = f−1(η). This is the required version.

Proof of Proposition 7.4.1. If we show that ζ has a strongly measurable version ζ,then the result will follow. Indeed, ζ and hence ζ are adapted, so the existenceof a progressively measurable version will follow from Lemma 7.4.2. By the Pettismeasurability theorem this version is strongly progressively measurable as well.

Next we prove the existence of a strongly measurable version. Let G ⊂ R+ × Ωbe the set of elements (t, ω) such that Φ(t, ·, ω) ∈ γ(R+;H,E). Since Φ is H-stronglymeasurable, we have G ∈ BR+ ⊗ A. Moreover, letting Gt = ω ∈ Ω : (t, ω) ∈ G fort ∈ R+, we have P(Gt) = 1 and therefore Gt ∈ F0. Define the H-strongly measurablefunction Ψ : R+ × R+ × Ω → B(H,E) as Ψ(t, s, ω) := Φ(t, s, ω)1[0,t](s)1G(t, ω). Itfollows from Lemma 3.5.2 that the map R+ × Ω 3 (t, ω) → Ψ(t, ·, ω) ∈ γ(R+;H,E)


is strongly measurable. Hence, the map R+ 3 t → Ψ(t, ·) ∈ L0(Ω; γ(R+;H,E))is strongly measurable. Since it takes values in L0

F(Ω; γ(R+;H,E)) it follows froman approximation argument that it is strongly measurable as an L0

F(Ω; γ(R+;H,E))-valued map. Since the elementary elements in L0

F(Ω; γ(L2(R+;H), E)) are dense itfollows from the Pettis measurability theorem that we can find a sequence of functions(Φn)n≥1 such that Φn : R+ → L0

F(Ω; γ(R+;H,E)) is a countably valued simple function

Φn =∑k≥1

1BnkΦnk , with Bn

k ∈ BR+ and Φnk ∈ L0

F(Ω; γ(R+;H,E)),

and for all t ∈ R+ we have ‖Ψ(t) − Φn(t)‖L0(Ω;γ(R+;H,E)) ≤ 2−n. Notice that by theMarkov inequality, for a random variable ξ : Ω → F , where F is a normed space, andε ∈ (0, 1], we have

P(‖ξ‖F > ε) = P(‖ξ‖F ∧ 1 > ε) ≤ ε−1‖ξ‖L0(Ω;F ).

It follows from Proposition 5.3.3 and Theorem 5.3.4 that for all t ∈ R+, for all n ≥ 1and for all ε, δ ∈ (0, 1],

P(∥∥∥∫

R+

Ψ(t, s)− Φn(t, s) dWH(s)∥∥∥ > ε

)≤ C2,Eδ

2

ε2+

1

δ2n.

Taking ε ∈ (0, 1] arbitrary and δ = 1n, it follows from the Borel-Cantelli lemma that

for all t ∈ R+,

P( ⋂N≥1

⋃n≥N

∥∥∥∫R+

Ψ(t, s)− Φn(t, s) dWH(s)∥∥∥ > ε

)= 0.

Since ε ∈ (0, 1], was arbitrary, we may conclude that for all t ∈ R+, almost surely,

ζ(t, ·) =

∫R+

Ψ(t, s) dWH(s) = limn→∞

∫R+

Φn(t, s) dWH(s).

Clearly, ∫R+

Φn(·, s) dWH(s) =∑k≥1

1Bnk(·)

∫R+

Φnk(s) dWH(s)

has a strongly BR+⊗F∞-measurable version, say ζn : R+×Ω → E. Let C ⊂ R+×Ω bethe set of all points (t, ω) such that (ζn(t, ω))n≥1 converges in E. Then C ∈ BR+ ⊗F∞and we may define the process ζ as ζ = limn→∞ ζn1C . It follows that ζ is stronglyBR+ ⊗F∞-measurable and for all t ∈ R+, almost surely, ζ(t, ·) = ζ(t, ·).

7.5 Solution concepts

On a Banach space E we consider the problem

(SE)


U(0) = u0.

7.5. Solution concepts 133

Here (A(t))t∈[0,T ] is a family of densely defined closed operators, which generates astrongly continuous evolution family (P (t, s))0≤s≤t≤T as defined in Section 7.2.

Assume that there exists a Banach space DF such that DF → E and F : [0, T ]×Ω × DF → E is strongly measurable and adapted. Similarly, we assume that thereexists a Banach space DB → E such that B : [0, T ]×Ω×DB → B(H,E) is H-stronglymeasurable and adapted. Finally, we assume that the initial value u0 : Ω → E is F0

measurable.Let E be a UMD− space. Below several different type of solutions are defined:

strong, variational, mild and weak solutions.

Definition 7.5.1. A DF ∩ DB-valued process (U(t))t∈[0,T ] is called a strong solutionof (SE), if

(i) U : [0, T ] × Ω → DF and U : [0, T ] × Ω → DB are strongly measurable andadapted,

(ii) t 7→ A(t)U(t) is in L0(Ω;L1(0, T ;E)) and F (·, U) is in L0(Ω;L1(0, T ;E)),

(iii) t 7→ B(t, U(t)) ∈ L0F(Ω; γ(0, T ;H,E)),

(iv) almost surely, for all t ∈ [0, T ],

U(t)− u0 =

∫ t

0

A(s)U(s) + F (s, U(s)) ds+

∫ t

0

B(s, U(s)) dWH(s). (7.5.1)

In particular, strong solutions always have continuous paths a.s.In the case of unbounded operators, strong solutions are rare and not easy to

construct. Therefore, we introduce several additional solution concepts. To motivatethe next definition let

Υt =ϕ ∈ C1([0, t];E∗) : for all s ∈ [0, t] ϕ(s) ∈ D(A∗(s))

and s 7→ A∗(s)ϕ(s) ∈ C([0, t];E∗)

for t ∈ [0, T ]. Fix some t ∈ [0, T ] and ϕ ∈ Υt. Formally, applying Ito formula to(7.5.1) yields

〈U(t),ϕ(t)〉 − 〈u0, ϕ(0)〉

=

∫ t

0

〈U(s), ϕ′(s)〉 ds+

∫ t

0

〈U(s), A∗(s)ϕ(s)〉+ 〈F (s, U(s)), ϕ(s)〉 ds

+

∫ t

0

B∗(s, U(s))ϕ(s) dWH(s).

(7.5.2)

Definition 7.5.2. A DF ∩DB-valued process (U(t))t∈[0,T ] is called a variational solu-tion of (SE), if



(ii) U and F (·, U) are in L0(Ω;L1(0, T ;E)),

(iii) B(·, U) is in L0(Ω;L2(0, T ; γ(H,E))) and is H-strongly adapted,

(iv) for all t ∈ [0, T ] and all ϕ ∈ Υt, almost surely, (7.5.2) holds.

Notice that the stochastic integral in (7.5.2) is well-defined. Indeed, since s 7→B∗(s, U(s))ϕ(s) is strongly measurable and adapted by the Pettis measurability the-orem and the H-strong measurability and adaptedness of s 7→ B(s, U(s)) this followsfrom (iii).

The argument to obtain (7.5.2) can be made rigorous with Theorems 5.3.5 or 5.3.6.This is the contents of the next result.

Proposition 7.5.3. Let E be a UMD− space. If U ∈ L0(Ω;L1(0, T ;E)) is a strongsolution of (SE) such that B(·, U) is in L0(Ω;L2(0, T ; γ(H,E))) and is H-stronglyadapted, then U is a variational solution of (SE).

To define a mild solution we require the evolution system P (t, s) : 0 ≤ s ≤ t ≤ Tthat is generated by (A(t))t∈[0,T ]. Let E be a UMD− space. We will use the short handnotations

P ∗ F (U)(t) :=

∫ t

0

P (t, s)F (s, U(s)) ds

P B(U)(t) :=

∫ t

0

P (t, s)B(s, U(s)) dWH(s).

Definition 7.5.4. We call a DF ∩ DB-valued process (U(t))t∈[0,T ] a mild solution of(SE), if


(ii) for all t ∈ [0, T ], s 7→ P (t, s)F (s, U(s)) is in L0(Ω;L1(0, t;E)),

(iii) for all t ∈ [0, T ], s 7→ P (t, s)B(s, U(s)) is in L0F(Ω; γ(0, t;H,E)),

(iv) for all t ∈ [0, T ], a.s.

U(t) = P (t, 0)u0 + P ∗ F (U)(t) + P B(U)(t).

To prove equivalences between variational and mild solutions, we need the followingcondition.

(C1) Assume that for all t ∈ [0, T ], there is a σ(E∗, E)-sequentially dense subspace Γtof E∗ such that for all x∗ ∈ Γt, we have ϕ(s) := P ∗(t, s)x∗ is in C1([0, t];E∗) andϕ(s) ∈ D(A∗(s)) for all s ∈ [0, t] and

d

dsϕ(s) = −A∗(s)ϕ(s). (7.5.3)


If the family (A∗(t))t∈[0,T ] generates an evolution family, then (C1) is usually fulfilled(cf. Proposition 7.2.1). If A = A(s) is independent of s and generates a stronglycontinuous semigroup, then (C1) is fulfilled with Γ = Γt = D(A), where A denotesthe sun adjoint of A (cf. [92]).

Proposition 7.5.5. Let E be a UMD− space. The following assertions hold:

(1) If U ∈ L0(Ω;L1(0, T ;E)) is a mild solution of (SE) with

F (·, U) ∈ L0(Ω;L1(0, T ;E)) and B(·, U) ∈ L0(Ω;L2(0, T ; γ(H,E))),

then U is a variational solution of (SE).

(2) If condition (C1) holds and U is a variational solution of (SE) and for all t ∈[0, T ], s 7→ P (t, s)B(s, U(s)) is in L0

F(Ω; γ(0, t;H,E)), then U is a mild solutionof (SE).

Proof. (1): Let t ∈ [0, T ] be arbitrary and ϕ ∈ Υt. Since U is a.s. in L1(0, T ;E)we have that s 7→ 〈U(s), A∗(s)ϕ(s)〉 is integrable and from the definition of a mildsolution we obtain that a.s.,∫ t

0

〈U(s), A∗(s)ϕ(s)〉 ds

=

∫ t

0

〈P (s, 0)u0, A∗(s)ϕ(s)〉 ds

+

∫ t

0

∫ s

0

〈P (s, r)F (r, U(r)), A∗(s)ϕ(s)〉 dr)ds

+

∫ t

0

∫ s

0

B∗(r, U(r))P ∗(s, r)A∗(s)ϕ(s) dWH(r))ds.

(7.5.4)

Since (P (t, s))0≤s≤t≤T is an evolution family that solves (7.2.1), it follows from anapproximation argument that for all x ∈ E, 0 ≤ r ≤ t ≤ T ,

〈P (t, r)x, ϕ(t)〉−〈x, ϕ(r)〉

=

∫ t

r

〈P (s, r)x,A∗(s)ϕ(s)〉 ds+

∫ t

r

〈P (s, r)x, ϕ′(s)〉 ds.(7.5.5)

As a consequence one obtains that for all R ∈ B(H,E), 0 ≤ r ≤ t ≤ T ,

R∗P ∗(t, r)ϕ(t)−R∗ϕ(r)

=

∫ t

r

R∗P ∗(s, r)A∗(s)ϕ(s) ds+

∫ t

r

R∗P ∗(s, r)ϕ′(s) ds.(7.5.6)

Indeed, this follows from (7.5.5) by applying h ∈ H on both sides.


By the Fubini theorem and (7.5.5) we obtain a.s.,∫ t

0

∫ s

0

〈P (s, r)F (r, U(r)), A∗(s)ϕ(s)〉 dr ds

=

∫ t

0

〈P (t, r)F (r, U(r)), ϕ(t)〉 dr −∫ t

0

〈F (r, U(r)), ϕ(r)〉 dr

−∫ t

0

∫ s

0

〈P (s, r)F (r, U(r)), ϕ′(s)〉 dr ds.

By the stochastic Fubini theorem and (7.5.6) we obtain a.s.,∫ t

0

∫ s

0

B∗(r, U(r))P ∗(s, r)A∗(s)ϕ(s) dWH(r) ds =

=

∫ t

0

B∗(r, U(r))P ∗(t, r)ϕ(t) dWH(r)−∫ t

0

B∗(r, U(r))ϕ(r) dWH(r)

−∫ t

0

∫ t

r

B∗(r, U(r))P ∗(s, r)ϕ′(s) dWH(r) ds.

Therefore, it follows from (7.5.4), (7.5.5) and the definition of a mild solution that∫ t

0

〈U(s), A∗(s)ϕ(s)〉 ds = 〈U(t), ϕ(t)〉 −∫ t

0

〈U(s), ϕ′(s)〉 ds− 〈u0, ϕ(0)〉

−∫ t

0

〈F (r, U(r)), ϕ(r)〉 dr −∫ t

0

B∗(r, U(r))ϕ(r) dWH(r)

and we obtain that U is a variational solution.(2): Let t ∈ [0, T ] be arbitrary. We show that for all x∗ ∈ Γt, a.s.

〈U(t), x∗〉 = 〈P (t, 0)u0, x∗〉+

∫ t

0

〈P (t, s)F (s, U(s)), x∗〉 ds

+

∫ t

0

B∗(s, U(s))P ∗(t, s)x∗ dWH(s).

(7.5.7)

By the existence of the integral, the existence of the stochastic integral, the weak∗-sequential density of Γt (see (C1)) and the Hahn-Banach theorem this suffices. Forx∗ ∈ Γt, let ϕ(s) = P ∗(t, s)x∗. Then it follows from (7.5.2) and (7.5.3) that

〈U(t), x∗〉−〈P (t, 0)u0, x∗〉+

∫ t

0

〈U(s), A∗(s)P ∗(t, s)x∗〉 ds

=

∫ t

0

〈U(s), A∗(s)P ∗(t, s)x∗〉 ds+

∫ t

0

〈P (t, s)F (s, U(s)), x∗〉 ds

+

∫ t

0

B∗(s, U(s))P ∗(t, s)x∗ dWH(s)

and we may conclude (7.5.7).


The assumption in Proposition 7.5.5 (2) on s 7→ P (t, s)B(s, U(s)) can be omittedin the following two situations.

Corollary 7.5.6. Assume that U is a variational solution of (SE) and that condition(C1) holds.

(1) If E is a UMD space with type 2, then U is a mild solution of (SE).

(2) If E is a UMD space and for all t ∈ [0, T ], for some p ∈ (1,∞), U ∈ Lp(Ω;E),then U is a mild solution of (SE).

Proof. (1): One can argue as in the proof of Proposition 7.5.5 (2). The fact thats 7→ P (t, s)B(s, U(s)) is in L0

F(Ω; γ(0, t;H,E)) follows from Proposition 3.3.1, thestrong continuity of (P (t, s))0≤s≤t≤T and Definition 7.5.2 (iii).

(2): Apply Theorem 4.4.3 in the proof of Proposition 7.5.5 (2) to obtain that forall t ∈ [0, T ], s 7→ P (t, s)B(s, U(s)) is in L0

F(Ω; γ(0, t;H,E)).

Remark 7.5.7. In some situations it is useful to consider more general functionsF and B. Namely functions F and B such that only (−Aw)−θFF and (−Aw)−θBBare mappings with values in E and B(H,E). In this case under the assumption that(A(t))t∈[0,T ] satisfies the Tanabe conditions with µ > maxθF , θB the result in Propo-sition 7.5.5 still holds. However, one has to replace Definition 7.5.2 (ii) and (iii)by

(ii)′ U and (−Aw)−θFF (·, U) are in L0(Ω;L1(0, T ;E)) for some θF ∈ (0, 1).

(iii)′ (−Aw)−θBB(·, U) is in L0(Ω;L2(0, T ; γ(H,E))) and is H-strongly adapted forsome θB ∈ (0, 1

2).

Moreover, to obtain a version Proposition 7.5.5 (1) one has to assume (ii)′ and (iii)′

instead and to prove a version Proposition 7.5.5 (2) one needs that for all t ∈ [0, T ],s 7→ P (t, s)F (s, U(s)) is in L0

F(Ω;L1(0, t;E)) as well.

Sketch of the proof: Observe that (7.2.6) holds and therefore, P (t, s)F (s, U(s)) ∈ Eand P (t, s)B(s, U(s)) ∈ B(H,E) are well-defined. Moreover, by approximation (7.5.5)and (7.5.6) hold for x and R in Er

−θ and B(H,Er−θ) for all 0 < θ < µ. Here Er

−θ isdefined as the completion of E with respect to the norm ‖ · ‖Er

−θ= ‖(−Aw(r))−θ · ‖E.

Also note that one should use (P (t, s)B(s, U(s)))∗ instead of B(s, U(s)∗P ∗(t, s) as anadjoint operator everywhere. One may then follows the same line of proof as before.

Next we consider variational solutions where ϕ in Definition 7.5.2 is constant intime. Such solutions will be called weak solutions. Of course for such a definition weneed some conditions on the adjoint operators (A∗(t))t∈[0,T ].

(C2) Assume that D(A∗(t)) = D(A∗(0)) is time independent and⋂t∈[0,T ] ρ(A

∗(t))

is non-empty and for some w ∈⋂t∈[0,T ] ρ(A

∗(t)), t 7→ A∗(t)R(w,A∗(0)) and

t 7→ A∗(0)R(w,A∗(t)) are strongly continuous.


Notice that (C2) is fulfilled if (A∗(t))t∈[0,T ] satisfies the Tanabe conditions.In particular (C2) implies that for x∗ ∈ D(A∗(0)), t 7→ A∗(t)x∗ is in C([0, T ];E∗).

Definition 7.5.8. Assume that (C2) holds. An DF ∩DB-valued process (U(t))t∈[0,T ]

is called a weak solution of (SE), if


(ii) U and F (·, U) are in L0(Ω;L1(0, T ;E)),

(iii) B(·, U) is in L0(Ω;L2(0, T ; γ(H,E))) and is H-strongly adapted,

(iv) for all t ∈ [0, T ] and x∗ ∈ D(A∗(0)), a.s. we have

〈U(t), x∗〉 − 〈u0, x∗〉 =

∫ t

0

〈U(s), A∗(s)x∗〉+ 〈F (s, U(s)), x∗〉 ds

+

∫ t

0

B∗(s, U(s))x∗ dWH(s).

(7.5.8)

Proposition 7.5.9. Let E be a reflexive Banach space. Assume that (C2) holds andA(0) generates a strongly continuous semigroup. Then a process (U(t))t∈[0,T ] is a weaksolution if and only if it is a variational solution.

For the proof we need the following elementary lemma.

Lemma 7.5.10. Let E be a Banach space and let A be the generator of a stronglycontinuous semigroup on E. Then

spanf ⊗ x : f ∈ C1([0, T ]), x ∈ D(A)

is dense in C1([0, T ];E) ∩ C([0, T ];D(A)).

Proof of Proposition 7.5.9. Since condition (C2) implies that each x∗ ∈ D(A∗(0)) is aconstant element of Υt, we obtain that every variational solution is a weak solution.

For the converse let U be a weak solution. We follow the argument in [37, Lemma5.5]. Let t ∈ [0, T ], and let f ∈ C1([0, t]) and x∗ ∈ D(A∗(0)) be arbitrary. By the Itoformula we have

〈U(t), f(t)⊗ x∗〉 = 〈U(t), x∗〉f(t)

= 〈u0, x∗〉f(0) +

∫ t

0

〈U(s), A∗(s)x∗〉f(s) + 〈F (s, U(s)), x∗〉f(s) ds

+

∫ t

0

〈U(s), x∗〉f ′(s) ds+

∫ t

0

B∗(s, U(s))x∗f(s) dWH(s).

Writing ϕ = f ⊗ x∗, this becomes (7.5.2). Furthermore, by linearity one can ex-tend (7.5.2) to ϕ : [0, t] → D(A∗(0)) of the form ϕ =

∑Nn=1 fn ⊗ x∗n, with fn ∈


C1([0, t]) and x∗n ∈ D(A∗(0)) for all n = 1, . . . , N . Notice that by condition (C2)Υt = C1([0, t];E∗) ∩ C([0, t];D(A∗(0)) and by Lemma 7.5.10 applied to A∗(0), everyϕ ∈ Υt can be approximated in C1([0, t];E∗) ∩ C([0, t];D(A∗(0))) by such elements.Now one easily obtains that U is a variational solution.

Remark 7.5.11. The equivalence of variational and weak solutions from Proposition7.5.9 has an extension to the setting of Remark 7.5.7.

In Propositions 7.5.3 and 7.5.9, conditions are given under which a strong solutionis a variational solution and hence a weak solution of (SE). In the converse directionthe following result holds.

Proposition 7.5.12. Let E be a UMD− space and assume (C2). If a weak solutionU of (SE) satisfies AU ∈ L1(0, T ;E) a.s. and

s 7→ B(s, U(s)) ∈ L0F(Ω; γ(0, T ;H,E)), (7.5.9)

then U is a strong solution of (SE).

If E is a UMD space of type 2 then (7.5.9) follows from Proposition 3.3.1 andDefinition 7.5.8 (iii).

Proof. Fix t ∈ [0, T ]. It follows from (7.5.8) that for all x∗ ∈ D(A∗(0)),

〈U(t)− u0, x∗〉 =

⟨ ∫ t

0

A(s)U(s) + F (s, U(s)) ds+

∫ t

0

B(s, U(s)) dWH(s), x∗⟩.

This extends to all x∗ ∈ E∗ by weak∗-sequentially density of D(A∗(0)). The resultnow follows from the Hahn-Banach theorem.


Proposition 7.2.1 is based on [2, Proposition 2.9] by Acquistapace, Flandoli andTerreni (also see [5, Remark 6.2]). The authors assume that (A∗(t))t∈[0,T ] satisfiesthe conditions of Acquistapace and Terreni (see Chapter 10 and references giventhere), which are more general than the Tanabe conditions (cf. [4, Section 7]). Theproof of Proposition 7.2.1 uses classical solutions instead of strict solutions in thatcase. In this way one does not need that Zs is norm dense. However, our formulationworks for more general families (A∗(t))t∈[0,T ] (e.g. the hyperbolic case).

The Tanabe conditions have been studied by many authors (cf. [9, 83, 108, 124] andthe references therein). We mention that Lunardi in [83] also considers the case whereD = D(A(t)) is not necessarily dense. The estimate (7.2.6) has been proved in [136,Theorem 2.1] by Yagi. It is even valid under the Acquistapace-Terreni conditions.

Lemma 7.3.1 has been proved in [35] by Da Prato, Kwapien and Zabczyk inthe autonomous case. They formulate the result in terms of fractional domains spaces.


In [122] Seidler observed that their argument extends to the non-autonomous settingif one assumes that the fractional domain spaces do not vary in time and there existsa constant C such that for all α ∈ (0, 1), for all t ∈ [0, T ] and for all x ∈ D,

‖(−Aw(0))αx‖ ≤ ‖(−Aw(t))αx‖ ≤ ‖(−Aw(0))αx‖.

This holds for instance if the fractional domain spaces coincides with the complexinterpolation spaces. We do not need such an assumption, since we have formulatedour result in terms of real interpolation spaces. In [131] a version of Lemma 7.3.1 fortime-dependent domains has been proved under the Acquistapace-Terreni conditions.

The solution concepts we study in Section 7.5 are generally known (cf. [37] and thereferences therein). Many authors only study the relations in the autonomous case. Inthat situation variational solutions are often used as a step to go from a weak solutionto a mild solution (cf. [37, Lemma 5.5]). In the non-autonomous setting one couldargue that variational solutions are more natural than weak solutions. This is becauseD(A∗(t)) may depend on time even if D(A(t)) is constant. Variational solutions fordeterministic equations have been studied for instance in [124, Section 5.3]. Variationalsolutions in the case of additive noise have also been considered in [131].

Condition (C2) and Definition 7.5.8 can be weakened in the sense that one onlyneeds that

⋂t∈[0,T ]D(A∗(t)) is large enough. In some cases a version of Proposition

7.5.9 holds for UMD− spaces as well. In that case one cannot use Lemma 7.5.10,because E is not necessarily reflexive. However, sometimes one can show that (7.5.2)holds for a class of functions ϕ which is large enough to deduce that the solutionis a mild solution and therefore a variational solution by Proposition 7.5.5. In thesemigroup case one could take for instance C1([0, t];E∗) ∩ C([0, t];D(A∗)) instead ofthe space Υt in Definition 7.5.2.

Chapter 8

Equations in type 2 spaces

8.1 Introduction

Let (E, ‖ · ‖) be a UMD Banach space with type 2 and let (H, [·, ·]) be a separableHilbert space. Let (Ω,A,P) be a complete probability space and WH be a cylindricalBrownian motion with respect to a complete filtration F = (Ft)t≥0 on (Ω,A,P).

In Section 8.2 we will obtain regularity results for stochastic convolutions of theform

P Φ(t) :=

∫ t

0

P (t, s)Φ(s) dWH(s).

Here (P (t, s))0≤s≤t≤T is the evolution family generated by a family (A(t))t∈[0,T ], whichsatisfies the Tanabe conditions. We extend the factorization method of Da Prato,Kwapien and Zabczyk from [35], and use it to prove combined space-time regularityresults. For Hilbert spaces E, we give conditions on (A(t))t∈[0,T ] under which thestochastic convolution has maximal L2-regularity.

In Sections 8.3-8.5 we study the following evolution equation on E:dU(t) = (A(t)U(t) + F (t, U(t))) dt+B(t, U(t)) dWH(t), t ∈ [0, T0],

U(0) = u0.(8.1.1)

Here (A(t))t∈[0,T ] satisfies the Tanabe conditions again. The functions F and B aremeasurable and adapted in an appropriate way and u0 is an F0-measurable initialvalue.

In Section 8.3 the functions F and B satisfy certain Lipschitz and linear growthconditions and u0 is an integrable initial value. In this case we construct a mild so-lution of (8.1.1) using a fixed point argument in the subspace of adapted processesin Lr(Ω;Lp(0, T ; (E,D)a,1)). The results in Section 8.2 allow us to deduce regularityresults for the obtained mild solution. In Section 8.4 existence, uniqueness and regu-larity results are extended. This is done by localization and cut off arguments. Thisis further extended in Section 8.5 to the case where F and B are locally Lipschitzfunctions. There the solution is only constructed up to an explosion time. Underlinear growth assumptions we show that the solution is a global solution.

141

142 Chapter 8. Equations in type 2 spaces

In the last part of this chapter we apply the abstract results to stochastic partialdifferential equations. The equations are modelled on Lp-spaces for p ∈ [2,∞). InSection 8.6 we consider a second order equation with colored noise. In Section 8.7 westudy a general elliptic equation with space-time white noise.

8.2 Stochastic convolutions

For UMD spaces E with type 2 we have the following regularity result.

Theorem 8.2.1. Let E be a UMD space with type 2. Assume that (A(t))t∈[0,T ] satisfies(T1) and (T2). Let θ ∈ [0, µ). Let (−Aw)−θΦ : [0, T ] × Ω → γ(H,E) be H-stronglymeasurable and adapted. Then for all t ∈ [0, T ], s 7→ P (t, s)Φ(s) ∈ γ(H,E) is H-strongly measurable and adapted and the following results hold:

(1) Let δ ∈ [0, 12), λ ∈ (0, 1

2) and p ∈ (2,∞) be such that δ + θ + λ < 1

2− 1

p. There

exists a constant CT ≥ 0 with limT↓0CT = 0, which depends only on the constantsin (T1), (T2), T ∨ 1, E, p, δ, λ, θ and such that

E∥∥∥t 7→∫ t

0

P (t, s)Φ(s) dWH(s)∥∥∥pCλ([0,T ];(E,D)δ,1)

≤ CTE‖(−Aw)−θΦ‖pLp(0,T ;γ(H,E)).

(8.2.1)

(2) Let δ ∈ [0, 12−θ) and p ∈ [2,∞). There exists a constant CT ≥ 0 with limT↓0CT =

0, which only depends on the constants in (T1), (T2), T ∨ 1, E, p, δ, θ and suchthat

E∥∥∥t 7→∫ t

0

P (t, s)Φ(s) dWH(s)∥∥∥pLp(0,T ;(E,D)δ,1)

≤ CTE‖(−Aw)−θΦ‖pLp(0,T ;γ(H,E)).

(8.2.2)

Proof. In the same way as above Proposition 7.3.2 one can see that

(t, s) : 0 ≤ s < t ≤ T 3 (t, s) 7→ P (t, s)Φ(s) ∈ γ(H,E)

is H-strongly measurable and for all t ∈ [0, T ],

(0, t) 3 s 7→ P (t, s)Φ(s) ∈ γ(H,E)

is H-strongly adapted.

(1): Take α ∈ (0, 12− θ) such that δ + λ < α− 1

p. Define ζα : [0, T ]× Ω → E as

ζα(t) =1

Γ(1− α)

∫ t

0

(t− s)−αP (t, s)Φ(s) dWH(s).

8.2. Stochastic convolutions 143

Then ζα is well-defined and from Theorem 4.6.2, (7.2.6) and Holder’s inequality itfollows that for all t ∈ [0, T ],

‖ζα(t)‖Lp(Ω;E) ≤ C(E

( ∫ t

0

‖(t− s)−αP (t, s)Φ(s)‖2γ(H,E) ds

) p2) 1

p

≤ C ′(E

( ∫ t

0

(t− s)−2α−2θ‖(−Aw(s))−θΦ(s)‖2γ(H,E) ds

) p2) 1

p

≤ C ′( ∫ t

0

(t− s)(−2α−2θ)q ds) 1

q(E

∫ t

0

‖(−Aw(s))−θΦ(s)‖pγ(H,E) ds) 1

p

≤ CT

(E‖(−Aw)−θΦ‖pLp(0,T ;γ(H,E))

) 1p.

Here q ∈ (1,∞) satisfies 2p+ 1q

= 1. By Fubini’s theorem and Proposition 7.4.1 it follows

that ζα ∈ Lp(Ω× (0, T );E). Let Ω0 with P (Ω0) = 1 be such that ζα(·, ω) ∈ Lp(0, T ;E)for all ω ∈ Ω0. We may apply Lemma 7.3.1 to obtain that for all ω ∈ Ω0,

t 7→ (Rαζα(·, ω))(t) ∈ Cλ([0, T ]; (E,D)δ,1)

and‖Rαζα(·, ω)‖Cλ([0,T ];(E,D)δ,1) ≤ Cα,λ,δ,p,T∨1‖ζα(·, ω)‖Lp(0,T ;E). (8.2.3)

Define ζ : [0, T ]× Ω → E as

ζ(t) =

∫ t

0

P (t, s)Φ(s) dWH(s).

We claim that for all t ∈ [0, T ], for almost all ω ∈ Ω, we have

ζ(t, ω) = (Rαζα(·, ω))(t). (8.2.4)

It suffices to check that for all t ∈ [0, T ] and x∗ ∈ E∗, almost surely we have

〈ζ(t), x∗〉 =1

Γ(α)

∫ t

0

(t− s)α−1〈P (t, s)ζα(s), x∗〉 ds.

This follows as in Proposition 7.3.2, now using the stochastic Fubini theorem (see [35]).Therefore, the above estimates imply (8.2.1).

(2): One may proceed as the first part, but now using Lemma 7.3.1 (2). By Theorem4.6.2, (7.2.6) and Young’s inequality, we have

‖ζα‖Lp(Ω×(0,T );E) ≤ C( ∫ T

0

E( ∫ t

0

‖(t− s)−αP (t, s)Φ(s)‖2γ(H,E) ds

) p2dt

) 1p

≤ C ′(E

∫ T

0

( ∫ t

0

(t− s)−2α−2θ‖(−Aw)−θΦ(s)‖2γ(H,E) ds

) p2dt

) 1p

≤ CT

(E‖(−Aw)−θΦ‖pLp(0,T ;γ(H,E))

) 1p.


Notice that it is possible that ζ(t) is only defined for almost all t ∈ [0, T ], and (8.2.4)holds for almost all t ∈ [0, T ]. From Lemma 7.3.1 (2) we conclude that

(E‖P Φ‖pLp(0,T ;(E,D)δ,1))1p = (E‖Rα(ζα)‖pLp(0,T ;(E,D)δ,1))

1p

≤ C‖ζα‖Lp(Ω×(0,T );E) ≤ C ′T (E‖(−Aw)−θΦ‖pLp(0,T ;γ(H,E)))

1p .

For Hilbert spaces E we obtain a maximal regularity result. Assume (T1) and (T2)and the following condition on the operators (A(t))t∈[0,T ].

(H∞) There exists constant w ∈ R, C > 0 and ϕ ∈ [0, 12π) such that for all t ∈ [0, T ],

−Aw(t) admits a bounded H∞-calculus on Σϕ and

supt∈[0,T ]

(‖f(−Aw(t))‖ : ‖f‖H∞(Σϕ) ≤ 1

)<∞.

Equivalently, we could assume that −Aw(t) has bounded imaginary powers uniformlyin t ∈ [0, T ] (see [89, Section 8]). As sufficient condition is that there exists a w, suchthat for all t ∈ [0, T ], Aw(t) is dissipative, i.e. 〈Aw(t)x, x〉 ≤ 0 for all x ∈ D (cf. [7]).

Recall that for θ ∈ (0, 1), [E,D]θ stands for the complex interpolation space be-tween D and E.

Theorem 8.2.2. Let E be a Hilbert space. Assume that (A(t))t∈[0,T ] satisfies (T1),(T2) and (H∞). If Φ : [0, T ] × Ω → γ(H,E) is H-strongly measurable and adaptedand Φ ∈ L2(0, T ; γ(H,E)) a.s., then P Φ ∈ L2(0, T ; [E,D] 1

2) a.s. Moreover there is

a constant C independent of Φ such that

E‖P Φ‖2L2(0,T ;[E,D] 1

2) ≤ CE‖Φ‖2

L2(0,T ;γ(H,E)). (8.2.5)

Proof. It follows from (H∞) and [89, Section 8] that there exists a constant C suchthat for all t ∈ [0, T ] and x ∈ E,∫

R+

‖(−Aw(t))12 e−sAw(t)x‖2 ds ≤ C‖x‖2. (8.2.6)

Moreover, it follows from (H∞) and [126, Theorem 1.15.3] that for all t ∈ [0, T ],

D(−Aw(t))12 = [E,D] 1

2and there exists constants c1, C1 ≥ 0 such that for all t ∈ [0, T ]

and x ∈ [E,D] 12,

c1‖(−Aw(t))12x‖ ≤ ‖x‖[E,D] 1

2

≤ C1‖(−Aw(t))12x‖. (8.2.7)

First assume that Φ ∈ L2(Ω;L2(0, T ; γ(H,E))). Notice that γ(H,E) = C2(H,E) isthe space of Hilbert-Schmidt operators from H into E. Let (hn)n≥1 be an orthonormal

8.3. Lipschitz coefficients and integrable initial values 145

basis for H. By (8.2.7), the Ito isometry (or Theorem 4.6.2 with p = 2 and constants1) and the Fubini theorem, we have

E‖P Φ‖2L2(0,T ;[E,D] 1

2) h E‖(−Aw(0))

12P Φ‖2

L2(0,T ;E)

= E∫ T

0

∫ t

0

‖(−Aw(0))12P (t, s)Φ(s)‖2

γ(H,E) ds dt

= E∫ T

0

∑n≥1

∫ T

s

‖(−Aw(0))12P (t, s)Φ(s)hn‖2 dt ds.

Recall from [83, Section 6.1] that P (t, s) = W (t, s) + e(t−s)A(s) where W (t, s) satisfies(see [83, the proof of Corollary 6.1.8]),

‖(−Aw(0))12W (t, s)x‖ ≤ C2‖W (t, s)x‖

12D ‖W (t, s)x‖

12 ≤ C3

(t− s)12−α

2

. (8.2.8)

Therefore, by (8.2.6) (8.2.7) and (8.2.8) it follows that( ∫ T

s

‖(−Aw(0))12P (t, s)x‖2 dt

) 12

≤( ∫ T

s

‖(−Aw(0))12 e(t−s)A(s)x‖2 dt

) 12

+( ∫ T

s

‖(−Aw(0))12W (t, s)x‖2 dt

) 12

≤ C4

( ∫ T

s

‖(−Aw(s))12 e(t−s)Aw(s)x‖2 dt

) 12

+ C5

( ∫ T

s

(t− s)α−1‖x‖2 dt) 1

2

≤ C6‖x‖.

We may conclude that

E‖P Φ‖2L2(0,T ;[E,D] 1

2) ≤ C7E

∫ T

0

∑n≥1

‖Φ(s)hn‖2 ds = C7E‖Φ‖2L2(0,T ;γ(H,E)).

This proves (8.2.5).The general result now follows from a localization argument.

8.3 Lipschitz coefficients and integrable initial val-

ues

We recall the equation from Section 7.5. On a Banach space E we consider the problemdU(t) = (A(t)U(t) + F (t, U(t))) dt+B(t, U(t)) dWH(t), t ∈ [0, T0],

U(0) = u0.(SE)

Here (A(t))t∈[0,T ] is a family of closed unbounded operators on E with constant domainD. Consider the follows assumptions.


(A1) The family (A(t))t∈[0,T ] satisfies the Tanabe conditions (T1) and (T2).

Fix a ∈ [0, 1). Let DF = DB = (D,E)a,1 if a ∈ (0, 1) and DF = DB = E if a = 0.Assume that there is a θF ∈ [0, µ) such that θF + a < 1 and (−Aw)−θFF is a stronglymeasurable adapted mapping from [0, T ] × Ω × DF to E. Similarly, we assume thatthere is a θB ∈ [0, µ) such that θB+a < 1

2and (−Aw)−θBB is an H-strongly measurable

and adapted mapping from [0, T ]× Ω×DB to γ(H,E).

(A2) Let a ∈ [0, 1) and θF ∈ [0, µ) be such that a+ θF < 1. Let DF be as above. Forall x ∈ DF , (t, ω) 7→ (−Aw(t))−θFF (t, ω, x) is strongly measurable and adapted.The function (−Aw)−θFF has linear growth and is Lipschitz continuous in spaceuniformly in [0, T ]× Ω, that is there are constants LF and CF such that for allt ∈ [0, T ], ω ∈ Ω, x, y ∈ DF ,

‖(−Aw(t))−θF (F (t, ω, x)− F (t, ω, y))‖E ≤ LF‖x− y‖DF, (8.3.1)

‖(−Aw(t))−θFF (t, ω, x)‖E ≤ CF (1 + ‖x‖DF). (8.3.2)

(A3) Let a ∈ [0, 1) and θB ∈ [0, µ) be such that a+ θB <12. Let DB be as above. For

all x ∈ DB, (t, ω) 7→ (−Aw(t))−θBB(t, ω, x) is strongly measurable and adapted.The function (−Aw)−θBB has linear growth and is Lipschitz continuous in spaceuniformly in [0, T ]× Ω, that is there are constants LB and CB such that for allt ∈ [0, T ], ω ∈ Ω, x, y ∈ DB,

‖(−Aw(t))−θB(B(t, ω, x)−B(t, ω, y))‖γ(H,E) ≤ LB‖x− y‖DB, (8.3.3)

‖(−Aw(t))−θBB(t, ω, x)‖γ(H,E) ≤ CB(1 + ‖x‖DB). (8.3.4)

For r ∈ [1,∞) and p ∈ [1,∞] let V r,pa (0, T ;E) be the space of all strongly mea-

surable (a.s. pathwise continuous if p = ∞) and adapted processes φ : [0, T ] × Ω →(E,D)a,1 such that ‖φ‖V r,p

a (0,T ;E) <∞, where

‖φ‖V r,pa (0,T ;E) := ‖φ‖Lr(Ω;Lp(0,T ;(E,D)a,1)).

if p <∞ and‖φ‖V r,p

a (0,T ;E) := ‖φ‖Lr(Ω;C([0,T ];(E,D)a,1)).

if p = ∞.Identifying processes which are equal a.e. on Ω × [0, T ], this defines a norm on

V r,pa (0, T ;E) for p < ∞ which turns V r,p

a (0, T ;E) into a Banach space. Identifyingindistinguishable processes this defines a norm on V r,p

a (0, T ;E) for p = ∞ and againV r,pa (0, T ;E) is a Banach space. Our main interest concerns the cases p = ∞ andp = r <∞.

Define the fixed point operator LT : V r,pa (0, T ;E) → V r,p

a (0, T ;E) as

LT (φ) = t 7→ P (t, 0)u0 + P ∗ F (φ)(t) + P B(φ)(t).

In the next lemma we show that LT is well-defined and that it is a strict contractionfor T small enough. Recall that P ∗ F (φ) and P B(φ) stand for the convolution andstochastic convolution respectively.


Lemma 8.3.1. Let E be a UMD space with type 2 and let H be a separable Hilbertspace. Assume (A1) - (A3). Let r ∈ (2,∞), p ∈ [r,∞] be such that θB < 1

2− 1

rand

θF < 1− 1p

and let u0 ∈ Lr(Ω,F0; (E,D)a,1). Then the operator LT is well-defined and

there exist a constant CT with limT↓0CT = 0 such that for all φ1, φ2 ∈ V r,pa (0, T ;E),

‖LT (φ1)− LT (φ2)‖V r,pa (0,T ;E) ≤ CT‖φ1 − φ2‖V r,p

a (0,T ;E). (8.3.5)

Moreover, there is a constant C independent of u0 such that for all φ ∈ V r,pa (0, T ;E),

‖LT (φ)‖V r,pa (0,T ;E) ≤ C(1 + (E‖u0‖r(E,D)a,1

)1r . (8.3.6)

Remark 8.3.2. In the case that r ∈ [2,∞) and p ∈ [r,∞), we do not need anyrestrictions in terms of θB and θF in Lemma 8.3.1.

Proof. Initial value part –By (7.2.5) we may estimate

‖P (t, 0)u0‖(E,D)a,1 ≤ C‖u0‖(E,D)a,1 .

This clearly implies

‖t 7→ P (t, 0)u0‖V r,pa (0,T ;E) ≤ CT

1p‖u0‖Lr(Ω;(E,D)a,1). (8.3.7)

Deterministic convolution –(a): Let (−Aw)−θFφ ∈ Lp(0, T ;E). Then it follows from Proposition 7.3.2 that

there exists a constant CT independent of φ with limT↓0CT = 0 and such that

‖P ∗ φ‖Lp(0,T ;(E,D)δ,1) ≤ CT‖(−Aw)−θFφ‖Lp(0,T ;E), (8.3.8)

and if p = ∞, we note that P ∗ φ is continuous.(b): Let φ1, φ2 ∈ V r,p

a (0, T ;E). Then (−Aw))−θFF (·, φ1) and (−Aw(t))−θFF (·, φ2)are adapted and in Lr(Ω;Lp(0, T ;E)) and by (8.3.8) applied pointwise in Ω, P ∗F (·, φ1)and P ∗ F (·, φ2) define an element of V r,p

a (0, T ;E) and

‖P ∗ F (·, φ1)− P ∗ F (·, φ2)‖V r,pa (0,T ;E)

≤ CT‖(−Aw)−θF (F (·, φ1)− F (·, φ2))‖Lr(Ω;Lp(0,T ;E))

≤ CTLF‖φ1 − φ2‖V r,pa (0,T ;E).

(8.3.9)

Stochastic convolution –(a): Let (−Aw)−θBΦ ∈ Lr(Ω;Lp(0, T ; γ(H,E))). Then it follows from Theorem

8.2.1 that there exists a constant CT independent of φ with limT↓0CT = 0 and suchthat

‖P Φ‖V r,pa (0,T ;E) ≤ CT‖(−Aw)−θBΦ‖Lr(Ω;Lr(0,T ;γ(H,E))

≤ CTT1r− 1

p‖(−Aw)−θBΦ‖Lr(Ω;Lp(0,T ;γ(H,E)).(8.3.10)


(b): Let φ1, φ2 ∈ V r,pa (0, T ;E). Then (−Aw)−θBB(·, φ1), (−Aw)−θBB(·, φ2) are

adapted and in Lr(Ω;Lp(0, T ; γ(H,E))) and by (8.3.10) we obtain P B(·, φ1), P B(·, φ2) ∈ V r,p

a (0, T ;E) and

‖PB(·, φ1)− P B(·, φ2)‖V r,pa (0,T ;E)

≤ CTT1r− 1

p‖(−Aw)−θB(B(·, φ1)−B(·, φ2))‖Lr(Ω;Lp(0,T ;γ(H,E)))

≤ CTT1r− 1

pLB‖φ1 − φ2‖V r,pa (0,T ;E).

(8.3.11)

Conclusions –It follows from the above considerations that LT is well-defined and (8.3.5) follows

from (8.3.9) and (8.3.11). Moreover, the estimate (8.3.6) follows from (8.3.5) and

‖LT (0)‖V r,pa (0,T ;E) ≤ C(1 + ‖u0‖Lr(Ω;(E,D)a,1)).

We can now obtain a first existence and uniqueness result for (SE).

Theorem 8.3.3. Let E be a UMD space with type 2 and let H be a separable Hilbertspace. Assume (A1) - (A3). Let r ∈ (2,∞) be such that θB < 1

2− 1

rand let u0 ∈

Lr(Ω,F0; (E,D)a,1). Then there exists a unique mild solution

U ∈ V r,∞a (0, T0;E) = Lr(Ω;C([0, T0]; (E,D)a,1))

of (SE). Moreover, there exists a constant C := CE,q,S,T0,λ,δ,a,a0,a1,r,F,B such that

‖U‖V r,∞a (0,T0;E) ≤ C(1 + (E‖u0‖r(E,D)a,1

)1r ). (8.3.12)

Proof. It follows from Lemma 8.3.1 that we can find a T ∈ (0, T0] such that CT <12

and LT has a unique fixed point U ∈ V r,∞a (0, T ;E). This gives an almost surely

pathwise continuous and adapted process U : [0, T ]×Ω → (E,D)a,1 such that almostsurely for all t ∈ [0, T ],

U(t) = P (t, 0)u0 + P ∗ F (·, U)(t) + P B(·, U)(t). (8.3.13)

Construction for arbitrary finite T – Using a standard induction argument one mayconstruct a mild solution on each of the intervals [T, 2T ], . . . , [(n − 1)T, nT ], [nT, T0]for an appropriate integer n. The induced U solution on [0, T0] is the mild solution of(SE). Moreover, from (8.3.6) we deduce that

‖U‖V r,∞a (0,T ;E) = ‖LT0(U)‖V r,∞

a (0,T ;E) ≤ C(1 + ‖u0‖Lr(Ω;(E,D)a,1))

and (8.3.12) follows.Uniqueness: For small T ∈ (0, T0] this follows from the uniqueness of the fixed

point. For larger T the result follows from an induction argument as in Step 2.

8.4. Lipschitz coefficients and general initial values 149

The solution has some further regularity properties.

Corollary 8.3.4. In the situation of Theorem 8.3.3 for every δ, λ > 0 such thatδ + λ < min1− (a+ θF ), 1

2− (a+ θB)− 1

r, there exists a constant C independent of

u0 such that(E‖U − P (·, 0)u0‖rCλ([0,T0];(E,D)a+δ,1)

) 1r ≤ C(1 + (E‖u0‖r(E,D)a,1

)1r . (8.3.14)

Proof. Let U ∈ V r,∞a (0, T0;E) be the solution of (SE) constructed in Theorem 9.5.3.

By Proposition 7.3.2

E‖P ∗ F (·, U)‖rCλ([0,T0];(E,D)a+δ ,1)≤ CE‖(−Aw)−θFF (·, U)‖rL∞(0,T0;E).

By Theorem 8.2.1

E‖P B(·, U)‖rCλ([0,T0];(E,D)a+δ,1) ≤ CE‖(−Aw)−θBB(·, U(s))‖rL∞(0,T0;E).

Define U : [0, T0]× Ω → (E,D)a,1 as

U(t) = P (t, 0)u0 + P ∗ F (·, U)(t) + P B(·, U)(t),

where we take the versions of the convolutions as above. Clearly, U ≡ U a.s. andtherefore U is the required mild solution. Since F and B are of linear growth as inLemma 8.3.1 we may deduce that there is a constant C such that

E‖U − P (·, 0)u0)‖rCλ([0,T0];(E,D)a+δ,1) ≤ C(1 + ‖U‖rV r,∞a

(0, T0;E)).

Now (8.3.14) can be obtained using (8.3.12).

8.4 Lipschitz coefficients and general initial values

For p ∈ [1,∞] define V p,loca (0, T ;E) as the space of all processes φ : [0, T ] × Ω →

(E,D)a,1 such that φ is strongly measurable (a.s. pathwise continuous if p = ∞) andadapted and almost surely satisfies

‖φ‖Lp(0,T ;(E,D)a,1) <∞.

Notice that for all r ∈ [1,∞), V r,pa (0, T ;E) ⊂ V p,loc

a (0, T ;E).

Theorem 8.4.1. Let E be a UMD space with type 2 and let H be a separable Hilbertspace. Assume (A1)-(A3). Let u0 : Ω → (E,D)a,1 be F0-measurable. Then there existsa unique mild solution U : [0, T0] × Ω → (E,D)a,1 of (SE) in V 2,loc

a (0, T0;E) and ithas a version with continuous paths.

For the proof we need the following uniqueness result.


Lemma 8.4.2. Under the conditions of Theorem 8.3.3 let U and V in V r,∞a (0, T0;E)

be the mild solutions of (SE) with initial values u0 and v0 in Lr(Ω,F0; (E,D)a,1).Then almost surely on the set u0 = v0 we have U ≡ V .

Proof. Let Γ = u0 = v0. First consider small T ∈ (0, T0] as in Step 1 in the proof ofTheorem 8.3.3. Since Γ is F0-measurable we have

‖U1Γ − V 1Γ‖V r,∞a (0,T ;E) = ‖LT (U)1Γ − LT (V )1Γ‖V r,∞

a (0,T ;E)

= ‖(LT (U1Γ)− LT (V 1Γ))1Γ‖V r,∞a (0,T ;E)

≤ 1

2‖U1Γ − V 1Γ‖V r,∞

a (0,T ;E),

hence almost surely U |[0,T ]×Γ ≡ V |[0,T ]×Γ.For [0, T0] one may proceed as in the proof of Theorem 8.3.3.

Proof of Theorem 8.4.1. Let r ∈ (2,∞) be as in Theorem 8.3.3. Define (un)n≥1 inLr(Ω,F0; (E,D)a,1) as un = 1‖u0‖≤nu0. By Theorem 8.3.3, for each n ≥ 1, thereis a solution Un ∈ V r,∞

a (0, T0;E) of (SE) with initial value un. Lemma 8.4.2 impliesthat for 1 ≤ m ≤ n almost surely on the set ‖u0‖ ≤ m, for all t ∈ [0, T0], Un(t) =Um(t). It follows that almost surely, for all t ∈ [0, T0], limn→∞ Un(t) exists in (E,D)a,1.Define U : [0, T0] × Ω → (E,D)a,1 as U(t) = limn→∞ Un(t) if this limit exists and 0otherwise. Clearly, U is strongly measurable and adapted. Moreover, almost surelyon ‖u0‖ ≤ n, for all t ∈ [0, T0], U(t) = Un(t) and hence U a.s. has continuous paths.Furthermore, for all t ∈ [0, T0], s 7→ P (t, s)F (·, U(s)) is in L1(0, t; (E,D)a,1) almostsurely, and s 7→ P (t, s)B(·, U(s)) is in L2(0, t; γ(H, (E,D)a,1)) ⊂ γ(0, t;H, (E,D)a,1)almost surely.

We show that U is a solution of (SE). Since the deterministic convolution isdefined pathwise, it follows that almost surely on ‖u0‖ ≤ n, for all t ∈ [0, T0],P ∗ F (·, U)(t) = P ∗ F (·, Un)(t). In particular, almost surely for all t ∈ [0, T0], P ∗F (·, U)(t) = limn→∞ P ∗ F (·, Un)(t).

From Proposition 4.2.3, we deduce that for all m ≤ n, for all t ∈ [0, T0], almostsurely on ‖u0‖ ≤ n, P B(·, Um)(t) = P B(·, Un)(t) and by path continuity weobtain that almost surely on ‖u0‖ ≤ n for all m ≤ n, P B(·, Um) ≡ P B(·, Un). Itfollows that we may define ζ : [0, T0]×Ω → (E,D)a,1 pathwise as limn→∞ P B(·, Un)if this limit exists in C([0, T0]; (E,D)a,1) and 0 otherwise. Then ζ has almost all pathsin C([0, T0]; (E,D)a,1), and again by Proposition 4.2.3 for all t ∈ [0, T0], almost surelyin ‖u0‖ ≤ n,

P B(·, U)(t) = P B(·, Un)(t) = ζ(t).

Taking ζ as a version of P B(·, U), we obtain that almost surely, P B(·, U) =limn→∞ P B(·, Un) in C([0, T0]; (E,D)a,1).

We may conclude that almost surely for all t ∈ [0, T0],

U(t) = limn→∞

Un(t) = limn→∞

(P (t, 0)un + P ∗ F (·, Un)(t) + P B(·, Un)(t)

)= P (t, 0)u0 + P ∗ F (·, U)(t) + P B(·, U)(t).


This proves that U is a mild solution of (SE).

Uniqueness: Let T ∈ [0, T0]. Let U1, U2 ∈ V 2,loca (0, T ;E) be mild solutions of (SE).

For each n ≥ 1 let the stopping times ν1n for U1 be defined as

ν1n = inf

r ∈ [0, T ] :

∫ T

0

‖U1(s)1[0,r](s)‖2(E,D)a,1

ds ≥ n.

Similarly, we define the stopping times ν2n for U2. For each n ≥ 1 let

τn = ν1n ∧ ν2

n,

and let U1n = U11[0,τn] and U2

n = U21[0,τn]. Then for all n ≥ 1, U1n and U2

n are inV 2,2a (0, T ;E). One easily checks that

U1n = 1[0,τn](LT (U1

n))τn and U2

n = 1[0,τn](LT (U2n))

τn ,

where LT is the mapping introduced before Lemma 8.3.1 and

(LT (U1n))

τn(t) := (LT (U1n))(t ∧ τn).

By Remark 8.3.2 we can find T ∈ (0, T0] such that CT ≤ 12

and therefore

‖U1n − U2

n‖V 2,2a (0,T ;E) = ‖1[0,τn](LT (U1

n)− LT (U2n))

τn‖V 2,2a (0,T ;E)

≤ ‖LT (U1n)− LT (U2

n)‖V 2,2a (0,T ;E)

≤ 1

2‖U1

n − U2n‖V 2,2

a (0,T ;E).

We obtain that U1n = U2

n in V 2,2a (0, T ;E), hence U1

n(t, ω) = U2n(t, ω) for almost all

(t, ω) ∈ [0, T ]× Ω. Choosing n ≥ 1 arbitrary large, we may conclude that U1(t, ω) =U2(t, ω) for almost all (t, ω) ∈ [0, T ] × Ω. This proves the result for small T . Theresult can now be obtained as in Theorem 8.3.3.

Corollary 8.4.3. In the situation of Theorem 8.4.1 the mild solution U of (SE) has aversion such that almost all paths satisfy U−P (·, 0)u0 ∈ Cλ([0, T0]; (E,D)a+δ,1), whereone may take λ > 0 and δ ≥ 0 according to λ+ δ < min1− (a+ θF ), 1

2− (a+ θB).

Proof. Let r ∈ (2,∞) be so large that λ + δ < 12− (a + θB) − 1

r. Let U be the

solution constructed in Theorem 8.4.1. Define (un)n≥1 in Lr(Ω,F0; (E,D)a,1) as un =1‖u0‖≤nu0. By Corollary 8.3.4, for each n ≥ 1, we can find a solution Un of (SE) withinitial value un such that Un−P (·, 0)un has paths in Cλ([0, T0]; (E,D)a+δ,1). Since bythe proof of Theorem 8.4.1, almost surely on u0 = un we have U ≡ Un, the resultfollows.


8.5 Locally Lipschitz coefficients

In this section we study (SE) in the case that F and B are locally Lipschitz functions.In this case a mild solution does not always exists. To solve this problem we studylocal mild solutions.

Let T > 0 and let τ be a stopping time with values in [0, T ]. For t ∈ [0, T ] let

Ωt(τ) = ω ∈ Ω : t < τ(ω),

[0, τ)× Ω = (t, ω) ∈ [0, T ]× Ω : 0 ≤ t < τ(ω).A process ζ : [0, τ) × Ω → E (or (ζ(t))t∈[0,τ)) is called admissible if for all t ∈ [0, T ],Ωt(τ) 3 ω → ζ(t, ω) is Ft-measurable and for almost all ω ∈ Ω, [0, τ(ω)) 3 t 7→ ζ(t, ω)is continuous.

Let E be a UMD− space.

Definition 8.5.1. We call an admissible (E,D)a,1-valued process (U(t))t∈[0,τ ] a localmild solution of (SE), if τ ∈ (0, T0] almost surely and there exists an increasingsequence of stopping times (τn)n≥1 with τ = limn→∞ τn such that for all t ∈ [0, T0] andall n ≥ 1 almost surely, the following conditions hold:

(i) for all t ∈ [0, T0], s 7→ P (t ∧ τn, s)F (s, U(s)) is in L0(Ω;L1(0, t;E)),

(ii) for all t ∈ [0, T0], s 7→ P (t ∧ τn, s)B(s, U(s)) is in L0F(Ω; γ(0, t;H,E)),

(iii) for all t ∈ [0, T0], a.s.

U(t ∧ τn) = P (t ∧ τn, 0)u0 + P ∗ F (·, U)(t ∧ τn) + P B(·, U)(t ∧ τn).

Here the deterministic convolution is defined pathwise as a Bochner integral. ViaTheorem 4.5.9 (1), the stochastic convolution is defined as

P B(·, U)(t ∧ τn) =

∫ t∧τn

0

Pt∧τn,sB(s, U(s))1[0,τn](s) dWH(s).

A local mild solution (U(t))t∈[0,τ) is called maximal for a certain space V consisting

of (E,D)a,1-valued admissible processes, if for any other local mild solution (U(t))t∈[0,τ)

in V , almost surely we have τ ≤ τ and U ≡ U |[0,τ). Clearly, a maximal local mildsolution in such a space V is always unique in V . We say that a local mild solution(U(t))t∈[0,τ) of (SE) is a global mild solution of (SE) if τ = T0 almost surely and U

has an extension to a mild solution U : [0, T0]× Ω → (E,D)a,1 of (SE). In particular,almost surely “no blow” up occurs at t = T0.

We say that τ is an explosion time if for almost all ω ∈ Ω with τ(ω) < T0,

lim supt↑τ(ω)

‖U(t, ω)‖(E,D)a,1 = ∞.

Notice that if τ = T0 almost surely, then τ is always an explosion time in this definition.However, there does not have to be any “blow up” in this case.

Consider the following assumptions on F and B.

8.5. Locally Lipschitz coefficients 153

(A2)′ Let a ∈ [0, 1) and θF ∈ [0, µ) be such that a + θF < 1. Let DF be as before(A2). For all x ∈ DF , (t, ω) 7→ (−Aw(t))−θFF (t, ω, x) is strongly measurableand adapted. The function (−Aw)−θFF is locally Lipschitz continuous in spaceuniformly in [0, T ]×Ω, that is for each n ≥ 1 there is a constant LF,n such thatfor all t ∈ [0, T ], ω ∈ Ω, x, y ∈ DF with ‖x‖DF

, ‖y‖DF≤ n,

‖(−Aw(t))−θF (F (t, ω, x)− F (t, ω, y))‖E ≤ LF,n‖x− y‖DF. (8.5.1)

There is a constant CF,0 such that for all t ∈ [0, T0] and ω ∈ Ω,

‖(−Aw(t))−θFF (t, ω, 0)‖E ≤ CF,0.

(A3)′ Let a ∈ [0, 1) and θB ∈ [0, µ) be such that a + θB < 12. Let DB be as before

(A2). For all x ∈ DB, (t, ω) 7→ (−Aw(t))−θBB(t, ω, x) is strongly measurableand adapted. The function (−Aw)−θBB is locally Lipschitz continuous in spaceuniformly in [0, T ]×Ω, that is for each n ≥ 1 there is a constant LB,n such thatfor all t ∈ [0, T ], ω ∈ Ω, x, y ∈ DB with ‖x‖DB

, ‖y‖DB≤ n,

‖(−Aw(t))−θB(B(t, ω, x)−B(t, ω, y))‖γ(H,E) ≤ LB,n‖x− y‖DB, (8.5.2)

There is a constant CB,0 such that for all t ∈ [0, T0] and ω ∈ Ω,

‖(−Aw(t))−θBB(t, ω, 0)‖γ(H,E) ≤ CB,0.

Fix a stopping time τ with values in [0, T ]. Define V adma (0, τ ;E) as the space of all

(E,D)a,1-valued admissible processes (φ(t))t∈[0,τ).

Theorem 8.5.2. Let E be a UMD space with type 2 and let H be a separable Hilbertspace. Assume (A1), (A2)′ and (A3)′. Then there exists a unique maximal local mildsolution (U(t))[0,τ) in V adm

a (0, τ ;E) of (SE). Moreover, U has a version such that foralmost all ω ∈ Ω,

t 7→ U(t, ω)− P (t, 0)u0(ω) ∈ Cλloc([0, τ(ω)); (E,D)a+δ,1),

where one may take λ > 0 and δ ≥ 0 according to λ+δ < min1−(a+θF ), 12−(a+θB).

If, additionally F and B are of linear growth, i.e. (8.3.2) and (8.3.4) hold, then theabove function U is the unique global mild solution of (SE) in V adm

a (0, T0;E) and thefollowing assertions hold:

(1) The solution U satisfies the statements of Theorem 8.4.1 and Corollary 8.4.3.

(2) If r ∈ (2,∞) is such that θB < 12− 1

rand u0 ∈ Lr(Ω,F0; (L

p(S), Dp)a,1), thenthe solution U is in V r,∞

a (0, T0;E) and (8.3.12) and the statements of Corollary8.3.4 hold.

Before we proceed, we prove the following local uniqueness result.


Lemma 8.5.3. Assume that the conditions of Theorem 8.5.2 are satisfied. Assumethat (U1(t))t∈[0,τ1) in V adm

a (0, τ1;E) and (U2(t))t∈[0,τ2) in V adma (0, τ2;E) are local mild

solution of (SE) with initial values u10 and u2

0. Let Γ = u10 = u2

0. Then almost surelyon Γ, U1|[0,τ1∧τ2) ≡ U2|[0,τ1∧τ2). Moreover, if τ1 is an explosion time for U1, then almostsurely on Γ, τ1 ≥ τ2. If τ1 and τ2 are explosion times for U1 and U2, then almost surelyon Γ, τ1 = τ2 and U1 ≡ U2.

Proof. Let τ = τ1 ∧ τ2. For n ≥ 1, let

σ1n = inft ∈ [0, τ1) : ‖U1(t)‖(E,D)a,1 ≥ n and σ2

n = inft ∈ [0, τ2) : ‖U2(t)‖(E,D)a,1 ≥ n

and let σn = σ1n ∧ σ2

n.Fix an integer n ≥ 1. Let (−Aw)−θFFn : [0, T0]× Ω×DF → E be defined by

Fn(·, x) = F (·, x) for ‖x‖(E,D)a,1 ≤ n,

and Fn(·, x) = F(·, nx‖x‖(E,D)a,1

)otherwise. Define (−Aw)−θBBn : [0, T0] × Ω × DB →

γ(H,E) in a similar way. Then Fn and Bn satisfy (A2) and (A3). As in the proof ofTheorem 8.4.1 it follows that

‖Uσn1 1[0,σn]×Γ − Uσn

2 1[0,σn]×Γ‖V 2,2a (0,T ;E)

= ‖(LT (Uσn1 1[0,σn]×Γ)− LT (Uσn

2 1[0,σn]×Γ))1[0,σn]×Γ‖V 2,2a (0,T ;E)

≤ ‖LT (Uσn1 1[0,σn]×Γ)− LT (Uσn

2 1[0,σn]×Γ)‖V 2,2a (0,T ;E)

≤ CT‖Uσn1 1[0,σn]×Γ − Uσn

2 1[0,σn]×Γ‖V 2,2a (0,T ;E),

where CT satisfies limT↓0CT = 0. For T small enough it follows that Uσn1 1[0,σn]×Γ =

Uσn2 1[0,σn]×Γ in V 2,2

a (0, T ;E). By an induction argument this holds on [0, T0] as well.By path continuity it follows that almost surely, U1 ≡ U2 on [0, σn] × Γ. Since τ =limn→∞ σn we may conclude that almost surely, U1 ≡ U2 on [0, τ)× Γ.

If τ1 is an explosion time, then this yields τ1 ≥ τ2 on Γ almost surely. Indeed, if forsome ω ∈ Γ, τ1(ω) < τ2(ω), then we can find an n such that τ1(ω) < σ2

n(ω). We haveU1(t, ω) = U2(t, ω) for all 0 ≤ t ≤ σ1

n+1(ω) < τ1(ω). If we combine both assertions weobtain

n+ 1 = ‖U1(σ1n+1(ω), ω)‖(E,D)a,1 = ‖U2(σ

1n+1(ω), ω)‖(E,D)a,1 ≤ n.

This is a contradiction. The final assertion is now obvious.

Proof of Theorem 8.5.2. Let r ∈ (2,∞) be as in Theorem 8.3.3. For n ≥ 1 letΓn = ‖u0‖ ≤ n

2 and un = u01Γn . Let (Fn)n≥1 and (Bn)n≥1 be as in the proof

of Lemma 8.5.3. It follows from Theorem 8.3.3 that there exists a mild solutionUn ∈ V r,∞

a (0, T0;E) of (SE) with u0, F and B replaced by un, Fn and Bn. In par-ticular, Un has a version with continuous paths. Let τn be the stopping time definedby

τn(ω) = inft ∈ [0, T0] : ‖Un(t, ω)‖(E,D)a,1 ≥ n.


It follows from Lemma 8.5.3 that for all 1 ≤ m ≤ n, almost surely, Um ≡ Un on[0, τm ∧ τn] × Γm. By path continuity this implies τm ≤ τn. Therefore, we can defineτ = limn→∞ τn and on Γn, U(t) = Un(t) for t ≤ τn. By approximation and Proposition4.2.3 it is clear that U ∈ V adm

a (0, τ ;E) is a local mild solution of (SE). Moreover, τ isan explosion time. Indeed, if ω ∈ Ω is such that τ(ω) < T0, then

lim supt↑τ(ω)

‖U(t, ω)‖(E,D)a,1 ≥ lim supn→∞

‖U(τn(ω), ω)‖(E,D)a,1 = lim supn→∞

n = ∞.

Maximality: If (U(t))t∈[0,τ) in V adma (0, τ ;E) is a local mild solution of (SE), then

it follows from Lemma 8.5.3 that τ ≤ τ and U |[0,τ ] ≡ U almost surely. Therefore,(U(t))t∈[0,τ) is a maximal local mild solution.

Holder regularity: By Corollary 8.3.4, each Un satisfies the regularity statement.Therefore, the construction yields the required regularity properties of the paths of U .

Global mild solution in the case that F and B are of linear growth. First we show(2). Let (Un)n≥1 be as before. As in the proof of Lemma 8.3.1 one can check that bythe linear growth assumption,

‖Un‖V r,∞a (0,T ;E) = ‖LT (Un)‖V r,∞

a (0,T ;E) ≤ CT‖Un‖V r,∞a (0,T ;E) + C + C‖un‖Lr(Ω;(E,D)a,1),

where the constants do not depend on n and u0 and we have limT↓0CT = 0. Since‖un‖Lr(Ω;(E,D)a,1) ≤ ‖u0‖Lr(Ω;(E,D)a,1), it follows that for T small we have

‖Un‖V r,∞a (0,T ;E) ≤ C(1 + ‖u0‖Lr(Ω;(E,D)a,1)),

where C is a constant independent of n and u0. Repeating this inductively, we ob-tain a constant CT0 independent of n and u0 such that ‖Un‖V r,∞

a (0,T0;E) ≤ CT0(1 +‖u0‖Lr(Ω;(E,D)a,1)). In particular,

E sups∈[0,T0]

‖Un(s)‖r(E,D)a,1≤ Cr

T0(1 + ‖u0‖Lr(Ω;(E,D)a,1))

r.

It follows thatP( sup

s∈[0,T0]

‖Un(s)‖(E,D)a,1 ≥ n) ≤ CrT0n−r.

Since∑

n≥1 n−r <∞, the Borel-Cantelli Lemma implies that

P( ⋂k≥1

⋃n≥k

sup

s∈[0,T0]

‖Un(s)‖(E,D)a,1 ≥ n)

= 0.

This gives that almost surely, for all n large enough τn = T0, where τn is as before. Inparticular, τ = T0 and by Fatou’s lemma

‖U‖V r,∞a (0,T ;E) ≤ lim inf

n→∞‖Un‖V r,∞

a (0,T ;E) ≤ CT0(1 + ‖u0‖Lr(Ω;(E,D)a,1)).

By an approximation argument one can check that U is a global mild solution. Thefinal statement in (2) can be obtained as in Corollary 8.3.4.

For the proof of (1) one may repeat the construction from Theorem 8.4.1, usingLemma 8.5.3 instead of Lemma 8.4.2.


8.6 Second order equation with colored noise

Consider the following stochastic partial differential equation.

du(t, s) = (A(t, s,D)u(t, s) + f(u(t, s))) dt

+ b(u(t, s)) dW (t, s), t ∈ (0, T ], s ∈ S,u(t, s) = 0, t ∈ (0, T ], s ∈ ∂Su(0, s) = u0(s), s ∈ S.

(8.6.1)

Here S ⊂ Rn is a bounded C∞-domain with boundary being locally on one side of S,and

A(t, x,D) =n∑

i,j=1

aij(t, x)DiDj +n∑i=1

ai(t, x)Di + a0(t, x).

We assume that the coefficients are real and satisfy

aij, ai, a0 ∈ Cµ([0, T ];C(S)),

for i, j = 1, . . . , n and a constant µ ∈ (0, 1]. Furthermore, let (aij) be symmetric anduniformly elliptic, i.e. there is a constant κ > 0 such that

n∑i,j=1

aij(t, x)ξiξj ≥ κ|ξ|2, x ∈ S, t ∈ [0, T ], ξ ∈ Rn. (8.6.2)

The functions f, b : R → R are locally Lipschitz and of linear growth. The noise termis assumed to be an Lp(S)-valued Brownian motion, where p ∈ [2,∞). This is forinstance the case if

W (t) =∑k≥1

√λkwk(t)ek, (8.6.3)

where λk ≥ 0 and ek ∈ Lp(S) are such that∫S

( ∑k≥1

λk|ek(s)|2) p

2ds <∞ (8.6.4)

and (wk)k≥1 is a sequence of independent standard Brownian motions. Notice that bythe triangle inequality (8.6.4) is fulfilled if∑

k≥1

λk‖ek‖2Lp(S) <∞. (8.6.5)

Finally u0 : S ⊗ Ω → R is a F0 ⊗ BS-measurable function.We model (8.6.1) as a stochastic evolution equation of the form (SE) on E = Lp(S).

The realization of A on Lp(S) will be denoted by Ap and

D(Ap(t)) = Dp = g ∈ W 2,p(S) : g = 0 on ∂S.

8.6. Second order equation with colored noise 157

Notice that by [126, Theorem 4.3.3] if θ ∈ (0, 1) satisfies θ 6= 14p

, then

[Lp(S), Dp]θ = H2mθ,p0 (S) := g ∈ H2mθ,p(S) : g = 0 on ∂S if θ − 1

2p> 0,

and

(Lp(S), Dp)θ,q = B2mθp,q,0(S) := g ∈ B2mθ

p,q (S) : g = 0 on ∂S if θ − 1

2p> 0,

where q ∈ [1,∞] and Hs,p(S) denotes the Bessel potential space and Bsp,q(S) denotes

the Besov space.The problem (8.6.1) is said to have a mild solution if the corresponding (SE) has

a mild solution.

Theorem 8.6.1. Let p ∈ [2,∞) be such that n < p and let a ∈ [ n2p, 1

2). Under the

above conditions for each u0 which is in (Lp(S), Dp)a,1 a.s. there exists a unique mildsolution u of (8.6.1) which has paths in C([0, T ]; (Lp(S), Dp)a,1).

(1) If u0 ∈ (Lp(S), Dp) 12,1 a.s. then for all λ, δ ≥ 0 such that δ + a + λ < 1

2, u has

a version with paths in Cλ([0, T ]; (Lp(S), Dp)a+δ,1). Moreover, for all r ∈ (2,∞)such that δ + a+ λ < 1

2− 1

rthere exists a constant C not depending on u0, such

that

E‖u‖rCλ([0,T ];(Lp(S),Dp)a+δ,1) ≤ C(1 + E‖u0‖r(Lp(S),Dp) 12 ,1

).

(2) If there exists an ε > 0 such that aij ∈ C([0, T ];Cε(S)), then for all u0 ∈(Lp(S), Dp)a,1 the solution u is in L2(0, T ;H1,2

0 (S)) almost surely.

By [126, Theorems 4.3.1.1, 4.6.1f], for all δ ≥ 0,

(Lp(S), Dp)a+δ,1 → C2(a+δ)−n

p

0 (S) → C2δ0 (S),

where the subscript 0 stands for being zero at ∂S. In particular, the solution u is inC([0, T ]× S). If u0 ∈ (Lp(S), Dp) 1

2,1 the solution u is in

Cη([0, T ];C0(S;E)) ∩ C([0, T ];C2η0 (S;E))

for all η ∈ [0, 12− n

p).

Proof. One may check that the family (Ap(t))t∈[0,T ] satisfies the Tanabe conditionswith parameter µ (cf. [121, Example 2.9] and the references therein) and thereforegenerates a unique strongly continuous evolution family P (t, s)0≤s≤t≤T . Notice thatby (7.2.4) for all θ ∈ (0, 1), β ∈ (0, θ),

‖P (t, 0)u0‖Cθ−β([0,T ];(Lp(S),Dp)β,1) ≤ C‖u0‖(Lp(S),Dp)θ,1.


As above we have (Lp(S), Dp)a,1 → C(S). Let F : (Lp(S), Dp)a,1 → Lp(S) bedefined by F (x)(s) = f(x(s)). Then F is locally Lipschitz and of linear growth. LetH be a separable Hilbert space, i ∈ γ(H,Lp(S)) and WH be a cylindrical Brownianmotion such that WH(t) : i∗x∗ 7→ 〈W (t), x∗〉 for x∗ ∈ E∗ (cf. Example 3.4.2). If(8.6.3) holds, then we can take H = L2(S) and iek = λkek. Let B : (Lp(S), Dp)a,1 →γ(H,Lp(S)) be defined as (B(x)h)(s) = b(x(s))(ih)(s). This is well-defined since

ess sups∈S

|b(x(s))| ≤ C(1 + ess sups∈S

|x(s)|) ≤ C(1 + ‖x‖(Lp(S),Dp)a,1).

Furthermore, B is locally Lipschitz and of linear growth. Now the first statement and(1) follow from Theorem 8.5.2.

To prove (2) we use Theorem 8.2.2. First note that A2(t) satisfies (H∞) (cf.[40, 64]). We already showed that u is in C([0, T ]; (Lp(S), Dp)a,1) a.s. and thereforea.s.

‖B(u)‖L2(0,T ;γ(H,L2(S))) ≤ CS‖B(u)‖L2(0,T ;γ(H,Lp(S))) ≤ CSC(1 + ‖u‖L2(0,T ;(Lp(S),Dp)a,1))

≤ CSC(1 +√T‖u‖C([0,T ];(Lp(S),Dp)a,1)) <∞.

An application of Theorem 8.2.2 shows that P B(u) ∈ L2(0, T ;H1,20 (S)) a.s. Since

F (u) ∈ L∞(0, T ;Lp(S)) ⊂ L∞(0, T ;L2(S)) a.s. by Proposition 7.3.2 we obtain thatP ∗ F (u) is in C([0, T ]; (L2(S), H2,2

0 (S)) 12,1 a.s. Since

(L2(S), H2,20 (S)) 1

2,1 → [L2(S), H2,2

0 (S)] 12

= H1,20 (S),

we obtain P ∗ F (u) ∈ L2(0, T ;H1,20 (S)) a.s. For the initial value part by (7.2.5) a.s.

for all t ∈ (0, T ] we have

‖P (t, 0)u0‖H1,20 (S) ≤ C1‖P (t, 0)u0‖H1,p

0 (S) h C2‖P (t, 0)u0‖[Lp(S),Dp] 12

≤ C2‖P (t, 0)u0‖(Lp(S),Dp) 12 ,1≤ C4t

− 12+a‖u0‖(Lp(S),Dp)a,1 .

It follows that t 7→ P (t, 0)u0 is in L2(0, T ;H1,20 (S)) a.s. The result can now be deduced

from the definition of a mild solution.

Remark 8.6.2.

(1) The regularity conditions on S can be weakened. The only thing we need is in(1) that (Ap(t))t∈[0,T ] satisfies (T1) and (Lp(S), Dp)a,1 → C(S). For (2) we needthat (A(t))t∈[0,T ] satisfies (H∞).

(2) By Proposition 7.5.5 (1) the mild solution u is also a variational solution. If aij ∈Cµ([0, T ];C2(S)) and ai ∈ Cµ([0, T ];C1(S)), then one can show that (A∗

p(t))t∈[0,T ]

satisfies the Tanabe conditions. In that case by Corollaries 7.5.6 (1) and 7.5.9,u is the unique variational and weak solution as well.

8.7. Elliptic equations with space-time white noise 159

8.7 Elliptic equations with space-time white noise

Let S ⊂ Rd be a bounded C∞-domain and consider

∂u

∂t(t, s) = A(t, s,D)u(t, s) + f(t, s, u(t, s))

+g(t, s, u(t, s))∂w

∂t(t, s), s ∈ S, t ∈ (0, T ],

Bj(s,D)u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], j = 1, . . . ,m (8.7.1)

u(0, s) = u0(s), s ∈ S.

Here A is of the formA(t, s,D) =

∑|α|≤2m

aα(t, s)Dα

where D = (∂1, . . . , ∂d), and for j = 1, . . . ,m,

Bj(s,D) =∑|β|≤mj

bjβ(s)Dβ

where 1 ≤ mj < 2m is an integer. The principal part Aπ(t, s,D) =∑

|α|=2m aα(t, s)Dα

of A is assumed to be uniformly elliptic, i.e. there is a κ > 0 such that

(−1)m+1∑

|α|=2m

aα(t, s)ξα ≥ κ|ξ|2m, t ∈ [0, T ], s ∈ S, ξ ∈ Rd.

For |α| ≤ 2m the coefficients aα are in Cµ([0, T ];C(S)). For the coefficients of theboundary value operator assume that for j = 1, . . . ,m and |β| ≤ mj, bjβ ∈ C2m−mj(S).and (Bj)

mj=1 is a normal system of Dirichlet type, i.e. 0 ≤ mj < m (cf. [124, Section

3.7]).The functions f, g : [0, T ]×Ω×S×R → R are jointly measurable, and adapted in

the sense that for each t ∈ [0, T ], f(t, ·), g(t, ·) is Ft ⊗BS ⊗BR-measurable. Finally, wis a spatio-temporal white noise and u0 : S × Ω → R is a BS ⊗ F0-measurable initialvalue condition. We say that u : [0, T ]×Ω× S → R is a mild solution of (8.7.1) if thecorresponding functional analytic model (SE) has a mild solution.

Consider the following conditions:

(C1) The functions f and g are locally Lipschitz in the fourth variable uniformly in[0, T ]× Ω× S, i.e. for each R > 0 the exists constants LRf and LRg such that

|f(t, ω, s, x)−f(t, ω, s, y)| ≤ LRf |x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R,

|g(t, ω, s, x)−g(t, ω, s, y)| ≤ LRg |x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and |x|, |y| < R.

The functions f and g satisfy the following boundedness condition:

sup |f(t, ω, s, 0)| <∞ and sup |g(t, ω, s, 0)| <∞,

where the suprema are taken over all t ∈ [0, T ], ω ∈ Ω and s ∈ S.


(C2) The functions f and g are of linear growth in the fourth variable uniformly in[0, T ]× Ω× S, i.e. the exists constants Cf and Cg such that

|f(t, ω, s, x)| ≤ Cf (1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R,

|g(t, ω, s, x)| ≤ Cg(1 + |x|), t ∈ [0, T ], ω ∈ Ω, s ∈ S, x ∈ R.

Obviously, if f and g are Lipschitz and f(·, 0) and g(·, 0) are bounded, i.e. (C1)holds with constants Lf and Lg not depending on R, then (C2) is automaticallyfulfilled.

The main theorem of this section will be formulated in the terms of the spacesBsp,1,Bj(S). For some details we refer to [126, Section 4.3.3] and the references therein.

For p ∈ (1,∞) and s > 0, let

Hs,pBj(S) := φ ∈ Hs,p(S) : Bjφ = 0, for mj < s− 1

p, j = 1, . . . ,m,

Bsp,1,Bj(S) := φ ∈ Bs

p,1(S) : Dβφ = 0, for mj < s− 1

p, j = 1, . . . ,m.

For all p ∈ (1,∞) let Ap(t) be the realization of A(t, ·) on the space E = Lp(S) withdomain Dp := H2m,p

Bj(S). In this way Ap(t) is the generator of an analytic semigroup

(etAp)t≥0 and it satisfies (T1) and (T2) (cf. [124, Section 3.7]. Since we may replace Apand f in (8.7.1) by w+Ap and w+ f , we will assume that w = 0 and that (etAp)t≥0 isexponentially stable. From [126, Theorem 4.3.3] we may deduce that if θ ∈ (0, 1) andp ∈ (1,∞) are such that

2mθ − 1

p6= mj, for all j = 1, . . . ,m, (8.7.2)

then(Lp(S), Dp)θ,1 = (Lp(S), H2m,p

Bj(S))θ,1 = B2mθp,1,Bj(S)

isomorphically. For the other cases of θ ∈ (0, 1) one can still obtain a formula for theinterpolation space (Lp(S), H2m,p

Bj(S))θ,1. We will not need this result.

Theorem 8.7.1. Assume dm< 2. Assume (C1) and let p ∈ [2,∞) be such that

d2mp

< 12− d

4m.

(1) If a ∈ [ d2mp

, 12− d

4m) is such that (8.7.2) holds for (a, p), and for almost all

ω ∈ Ω, u0(·, ω) ∈ B2map,1,Bj(S), then there exists a unique maximal mild solution

(u(t))t∈[0,τ) of (8.7.1) such that almost surely t 7→ u(t, ·) is continuous as anB2map,1,Bj(S)-valued process.

(2) Moreover, if for almost all ω ∈ Ω, u0(·, ω) ∈ Bm− d

2


Cλloc([0, τ);B

2mδp,1,Bj(S)) for all δ > d

2mpand λ > 0 that satisfy δ + λ < 1

2− d

4m

and (8.7.2) for (δ, p).

8.7. Elliptic equations with space-time white noise 161

Furthermore, if condition (C2) holds as well, then

(3) If a ∈ [ d2mp

, 12− d

4m) is such that (8.7.2) holds (a, p) and for almost all ω ∈ Ω,

u0(·, ω) ∈ B2map,1,Bj(S), then there exists a unique global mild solution process

(u(t))t∈[0,T ] of (8.7.1) such that almost surely t 7→ u(t, ·) is continuous as anB2map,1,Bj(S)-valued process.


2


Cλ([0, T ];B2mδp,1,Bj(S)) for all δ > d

2mpand λ > 0 that satisfy δ+λ < 1

2− d

4mand

(8.7.2) for (δ, p).

By Sobolev embedding one obtains Holder continuous solutions in time and space.For instance assume in (4) that

u0 ∈ Cm− d

2

Bj (S) := φ ∈ Cm− d2 (S) : Bjφ = 0 if mj < m− d

2.

Then it follows from Cm− d

2

Bj (S) → Bap,1,Bj(S) for all p ∈ (1,∞) and a < m − d

2

and [126, Theorem 4.6.1] that the solution u has paths in Cλ([0, T ];C2mδBj(S)) for all

δ, λ > 0 that satisfy δ + λ < 12− d

4m.

For the proof we need the following lemma.

Lemma 8.7.2. Assume that S ⊂ Rd is a bounded C∞-domain. If λ > d2, then for all

p ∈ [1,∞), the embedding I : Hλ,2(S) → Lp(S) is γ-radonifying.

Proof. For λ > d2

one has Hλ,2(S) → C(S) (cf. [126, Theorem 4.6.1]). Let C bethe norm of this embedding. Let g : S → R be defined as g = C1S. Then for allh ∈ Hλ,2(S) and almost all s ∈ S,

|h(s)| ≤ ‖h‖C(S) ≤ ‖h‖Hλ,2(S)g(s)

and the result follows from Proposition 3.2.3 (3).

Proof of Theorem 8.7.1. By rewriting f and A we may assume that (etAp(r)(t))t≥0 and(etA2(r))t≥0 are exponentially stable uniformly in r ∈ [0, T ].

If β ∈ [ d2mp

, 1), then by [126, Theorem 4.6.1] we have

(Lp(S), Dp)β,1 → C(S).

In particular, (Lp(S), Dp)a,1 → C(S).Let F,G : [0, T ]× Ω× (Lp(S), Dp)a,1 → L∞(S) be defined as

(F (t, ω, x))(s) = f(t, ω, s, x(s)) and G(t, ω, x))(s) = g(t, ω, s, x(s)).

We show that this is well-defined and locally Lipschitz. Fix x, y ∈ (Lp(S), Dp)a,1 andlet

R := maxess sups∈S

|x(s)|, ess sups∈S

|y(s)| <∞.


From the measurability of x, y and f it is clear that s 7→ (F (t, ω, x))(s) and s 7→(F (t, ω, y))(s) are measurable functions. By (C1) it follows that for almost all s ∈ Sfor all t ∈ [0, T ] and ω ∈ Ω,

|(F (t, ω, x))(s)− (F (t, ω, y))(s)| = |f(t, ω, s, x(s))− f(t, ω, s, y(s))|≤ LRf |x(s)− y(s)|≤ LRf ‖x− y‖L∞(S) .a,p,S L

Rf ‖x− y‖(Lp(S),Dp)a,1 .

Also by the second part of (C1), for almost all s ∈ S for all t ∈ [0, T ] and ω ∈ Ω,

|(F (t, ω, 0))(s)| = |f(t, ω, s, 0)| < supt,s,ω

|f(t, ω, s, 0)| <∞.

Combing the above results yields that F is well-defined and locally Lipschitz. Similarly,one can show that F has linear growth (see (8.3.2)) if (C2) holds. The same argumentworks for G.

Since L∞(S) → Lp(S) we may consider F as an Lp(S)-valued mapping. It fol-lows from the Pettis measurability theorem that for all x ∈ (Lp(S), Dp)a,1, (t, ω) 7→F (t, ω, x) ∈ Lp(S) is strongly measurable and adapted.

To model g(t, x, u(t, s)) ∂w(t,s)∂t

, let H = L2(S) and let WH be a cylindrical Brownianmotion. Define the multiplication operator function GM : [0, T ]×Ω×(Lp(S), Dp)a,1 →B(H) as

(GM(t, ω, x)h)(s) = (G(t, ω, x))(s)h(s), s ∈ S.This is well-defined, because for all t ∈ [0, T ], ω ∈ Ω, G(t, ω, x) ∈ L∞(S).

Now let θ > θ′ > d4m

be such that (8.7.2) holds for (θ, 2) and (θ′, 2). Notice that(−Ap)−θ coincides with (−A2)

−θ on Lp(S), and there exists constants (Ci)4i=1 such

that for all t ∈ [0, T ] and φ ∈ Lp(S),

‖(− Ap(t))−θφ‖H2mθ′,2(S) = ‖(−A2(t))

−θφ‖H2mθ′,2(S)

(i)

≤ C1‖(−A2(t))−θφ‖(L2(S),D2)θ′,2

(ii)

≤ C2‖(−A2(t))−θφ‖(L2(S),D2)θ,∞

(iii)

≤ C3‖(−A2(t))−θφ‖D((−A2(t))θ) ≤ C4‖φ‖L2(S).

Here (i) follows from [126, Theorem 4.3.1]. The estimate (ii) can be deduced from[83, Proposition 1.2.3] and (iii) has been proved in [83, Proposition 2.2.15]. Therefore,each (−Ap(t))−θ has a bounded extension to an operator from L2(S) into H2mθ′,2(S)with norm less than or equal to C4. This extension coincides with (−A2(t))

−θ.By Lemma 8.7.2 for the inclusion operator we have i ∈ γ(H2mθ′,2(S), Lp(S)) and

we may define (−Ap(t))−θB : [0, T ]× Ω× (Lp(S), Dp)a,1 → γ(H,Lp(S)) as

(−Ap(t))−θB(t, ω, x)h = i (−Ap(t))−θ G(t, ω, x)h

using the bounded extension of (−Ap(t))−θ. We show that this is well-defined. By theright-ideal property we obtain

‖i (−Ap(t))−θ‖γ(L2(S),Lp(S)) ≤ ‖(−Ap(t))−θ‖B(L2(S),H2mθ′,2(S))‖i‖γ(H2mθ′,2(S),Lp(S))

≤ C4‖i‖γ(H2mθ,2(S),Lp(S)).


Fix x, y ∈ (Lp(S), Dp)a,1 and let

R := maxess sups∈S

|x(s)|, ess sups∈S

|y(s)| <∞.

It follows that from the right-ideal property

‖(−Ap(t))−θB(t, ω, x)− (−Ap(t))−θB(t, ω, y)‖γ(L2(S),Lp(S))

≤ C4‖i‖γ(H−2ma1,2(S),Lp(S))‖GM(t, ω, x)−GM(t, ω, y)‖B(L2(S))

≤ C4‖i‖γ(H2mθ,2(S),Lp(S))‖G(t, ω, x)−G(t, ω, y)‖L∞(S)

≤ C4‖i‖γ(H2mθ,2(S),Lp(S))LRg ‖x− y‖L∞(S)

.a,p C4‖i‖γ(H2mθ,2(S),Lp(S))LRg ‖x− y‖(Lp(S),Dp)a,1 .

In the same way one obtains

‖B(t, ω, 0)‖γ(L2(S),Lp(S)) ≤ C4‖i‖γ(H2mθ,2(S),Lp(S)) sup |g(t, ω, s, 0)|,

where the supremum is taken over all t ∈ [0, T ], ω ∈ Ω and s ∈ S. Similarly, one canshow that B has linear growth if (C2) holds. Notice that (−Ap(t))−θB is H-stronglymeasurable and adapted by the Pettis measurability theorem.

If for almost all ω ∈ Ω, s 7→ u0(s, ω) is in B2mβp,1,Bj(S), where β ∈ [ d

2mp, 1

2− d

4m] is

such that (8.7.2) holds for (β, p), then ω 7→ u0(·, ω) ∈ B2mβp,1,Bj(S) = (Lp(S), Dp)β,1 is

strongly F0-measurable. This follows from the Pettis measurability theorem.(1): It follows from Theorem 8.5.2 with a, θB = θ as above and with a+ θ < 1

2and

θF = 0, that there is a unique maximal local mild solution (U(t))t∈[0,τ) in V adma (0, τ, E).

In particular U has almost all paths in C([0, τ), (Lp(S), Dp)a,1)). Now take u(t, ω, s) =U(t, ω)(s). This proves (1).

(2): Let δ = a > d2mp

and λ > 0 be such that λ + δ < 12− d

4m. Choose θ >

d4m

such that λ + δ < 12− θ. It follows from Theorem 8.5.2 that almost surely,

U −Su0 ∈ Cλloc([0, τ(ω));B2mδ

p,1,Bj(S)). First consider the case that (12− d

4m, p) satisfies

(8.7.2). Since u0 ∈ Bm− d

2

p,1,Bj(S) = (Lp(S), Dp) 12− d

4m,1 ⊂ (Lp(S), Dp)δ,1 almost surely

and λ + δ < 12− d

4m, Su0 ∈ Cλ([0, T ];B2mδ

p,1,Bj(S)). almost surely. Therefore, almost

all paths of U are in Cλloc([0, τ(ω));B2mδ

p,1,Bj(S)). In the case that (12− d

4m, p) does not

satisfy (8.7.2) one can do above argument with 12− d

4m− ε for ε > 0 small. This proves

(2).(3), (4): This follows from Theorem 8.5.2 (1) and (2).


In case E is a Hilbert space a result similar to Theorem 8.2.1 can be found in [122]by Seidler. For type 2 space in the autonomous setting it has been proved by


Brzezniak in [20]. However, both authors formulate their results in terms of frac-tional domain spaces and in [122] the author needs an assumption on the fractionaldomains space (cf. Section 7.6).

Theorem 8.2.2 is an extension of [37, Theorem 6.14] by Da Prato and Zabczyk,where the autonomous case has been considered. Recall that for a Hilbert space Eand an invertible sectorial operator A, the fractional domain spaces coincides with thecomplex interpolation spaces if and only if A has a bounded H∞-calculus (cf. [56, page162]). The H∞-calculus has been studied by McIntosh in [89] and by many otherauthors. For an overview we refer to [56, 73].

In Sections 8.3-8.5 we studied a class of parabolic stochastic evolution equations.We only considered operators (A(t))t∈[0,T ] with constant domains. The methods areinspired by [20, 122]. Many results existence and uniqueness results for (SE) stillhold if one only assumes that (A(t))t∈[0,T ] generates a strongly continuous evolutionfamily (P (t, s))0≤s≤t≤T . Only the space-time regularity results no longer extend to thissetting. However, it is still possible to obtain continuity in time (see [122]).

In [119] Sanz-Sole and Vuillermot have considered the problem (8.6.1) of Sec-tion 8.6 in divergence form with time-dependent boundary conditions. They assumedthat the noise term is of the form (8.6.3) and that (ek)k≥1 is an orthonormal basis forL2(S) that satisfies (8.6.5). The authors have related several type of solution conceptswhich are strongly related to the variational, weak and mild solutions from Section7.5. They have proved existence and regularity properties of mild solutions usingcalculations with the associated Green’s function.

The example in Section 8.7 is a general elliptic equation with space-time whitenoise. Summarizing, one could say that it is possible to solve such an equation ifthe dimension d is less than the order of the elliptic operator 2m. The second orderautonomous case has been considered by Brzezniak in [20]. A related result forunbounded domains has been studied in [25] by Brzezniak and Peszat. Here thenoise term is a so-called homogenous Wiener process.

In Sections 8.6 and 8.7 we considered boundary conditions of Dirichlet type. It isalso possible to consider more general boundary conditions (Bj)

mj=1, i.e. mj < 2m, as

long as (A, (Bj)mj=1) satisfies the Lopatinskii-Shapiro condition.

Chapter 9

Equations in UMD spaces

9.1 Introduction

Let (E, ‖ · ‖) be a UMD− Banach space and let (H, [·, ·]) be a separable Hilbert space.Let (Ω,A,P) be a complete probability space and WH be a cylindrical Brownianmotion with respect to a complete filtration F = (Ft)t≥0 on (Ω,A,P). Recall fromChapter 5 that in case E is a UMD− space without the UMD property, then we needan assumption on the filtration in order to have a stochastic integration theory.

In this chapter we extend results from Chapter 8 to UMD− spaces. We againconsider

dU(t) = (AU(t) + F (t, U(t))) dt+B(t, U(t)) dWH(t), t ∈ [0, T0],

U(0) = u0.(9.1.1)

To solve this equations with fixed point arguments many difficulties occur. Therefore,we only consider the autonomous case and we assume that A generates an analyticC0-semigroup (S(t))t∈[0,T ].

The main difficulty is to find “the right” fixed point space. The space has to bein such a way that both the deterministic and stochastic convolution behave well. Wewill use an intersection of certain Bochner spaces and γ-spaces. This implies that wehave to estimate both the deterministic and stochastic convolution in a Bochner normand in a γ-norm. Also the fixed point space has to be small enough to be able toderive good regularity properties.

The γ-space will have a weight of the form s 7→ (t − s)−α. This turns out to beconvenient for the stochastic convolution. In Section 9.2 we prove several results onconvolutions in Besov and weighted γ-spaces. Most os these results are based on theembeddings of Besov and γ-spaces from Section 3.3.

In Section 9.3 the stochastic convolution will be considered. Once again we willapply the factorization method of Da Prato, Kwapien and Zabczyk from [35] toobtain space-time regularity results for stochastic convolutions S Φ with values in aUMD− space.

165

166 Chapter 9. Equations in UMD spaces

We will need that the function B from our equation (SE) is a Lipschitz function inour fixed point space in an appropriate way. Therefore, we will introduce a randomizedLipschitz notion in Section 9.4. The functions which fulfill this property will be calledL2γ-Lipschitz functions.

In Section 9.5 we explain the fixed point argument for our equation and deduceexistence, uniqueness and regularity results. From this point we follow the schemefrom Chapter 8.

In Section 9.8 we explain a functional analytic example where A is bounded orequivalently A = 0. In Section 9.9 we consider a perturbed heat equation in Lp-spaceswith p ∈ [1,∞). The last two sections are concerned with generalizations of theexamples from Sections 8.6 and 8.7 to Lp-spaces with p ∈ (1, 2).


In this chapter we consider abstract interpolation spaces of the following form. Fora ∈ (0, 1) we will denote Ea for an arbitrary Banach space of type Ja between E andD(A), that is D(A) ⊂ Ea ⊂ E and there exists a constant C ≥ 0 such that

‖x‖Ea ≤ C‖x‖1−a‖x‖aD(A), x ∈ D(A).

Recall that for a ∈ (0, 1) and a Banach space F with D(A) ⊂ F ⊂ E one has that Fis of type Ja if and only if (E,D(A))a,1 → F (cf. [83, Section 2.2]). Here (E,D(A))a,1denotes the real interpolation space. Moreover, we set E0 = E. Examples of spacesof type Ja are the real interpolation spaces (E,D(A))a,p for p ∈ [1,∞], the complexinterpolation spaces [E,D(A)]a and the fractional domain spaces D((−Aw)a) (cf. [83,Section 2.2] and [84]).

9.2 Convolutions in Besov spaces and γ-spaces

Let E be a Banach space and let I = (a, b) with −∞ ≤ a < b ≤ ∞. Below we givea definition of the vector-valued space Λs

p,q(I;E), where s ∈ (0, 1) and 1 ≤ p, q ≤ ∞.This space is related to the vector-valued Besov spaces Bs

p,q(Rd;E) defined in Section2.7. We follow the presentation in [69, Section 3.b] by Konig. It is also possible todefine Λs

p,q(S;E), where s > 0 and S is a regular domain, but we will not need this.For h ∈ R and a function f : I → E we define the function T (h)f : I → E as the

translate of f by h, i.e.

(T (h)f)(t) :=

f(t+ h) if t+ h ∈ I,0 otherwise.

For h ∈ R put

I[h] :=s ∈ I : s+ h ∈ I

.

For f ∈ Lp(I;E) and t > 0 let

%p(f, t) := sup|h|≤t

‖T (h)f − f‖Lp(I[h];E).

9.2. Convolutions in Besov spaces and γ-spaces 167

Now define

Λsp,q(I;E) := f ∈ Lp(I;E) : ‖f‖Λs

p,q(I;E) <∞,

where

‖f‖Λsp,q(I;E) = ‖f‖Lp(I;E) +

( ∫ 1

0

(t−s%p(f, t)

)q dtt

) 1q

(9.2.1)

with the obvious modification for q = ∞. Endowed with the norm ‖ · ‖Λsp,q(I;E),

Λsp,q(I;E) is a Banach space. The following continuous inclusions hold:

Λsp,q1

(I;E) → Λsp,q2

(I;E), Λs1p,q(I;E) → Λs2

p,q(I;E)

for all s, s1, s2 ∈ (0, 1), p, q, q1, q2 ∈ [1,∞] with q1 ≤ q2, s2 ≤ s1. If I is bounded, thenalso

Λsp1,q

(I;E) → Λsp2,q

(I;E)

for 1 ≤ p2 ≤ p1 ≤ ∞.

In the case that I = R and 1 ≤ p, q < ∞, then Λsp,q(I;E) = Bs

p,q(I;E) (see [109,Proposition 3.1] and [120, Theorem 4.3.3]).

The following multiplication lemma will be useful.

Lemma 9.2.1. Let p ∈ (1,∞), r ∈ [1,∞) and let s ∈ (0, 1) and q ∈ [1, p) be such thats < 1

q− 1

p. Let 0 < α < 1

q− 1

p−s. For all φ ∈ Λs

p,r(0, T ;E), we have t 7→ t−αφ(t)1(0,T )(t)

is in Λsq,r(R;E) and there exists a constant C depending on p, q, r, s, α and T ∨ 1 such

that

‖t 7→ t−αφ(t)1(0,T )(t)‖Λsq,r(R;E) ≤ CT

1q− 1

p−α−s‖φ‖Λs

p,r(0,T ;E).

Proof. By Holder’s inequality we have

‖t 7→ t−αφ(t)1(0,T )(t)‖Lq(R;E) ≤ T1q− 1

p−αCα,p,q‖φ‖Lp(0,T ;E).

Fix u ∈ [0, T ] and |h| < u. First assume that h > 0. We may estimate

( ∫R

∥∥∥φ(t+ h)1(0,T )(t+ h)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥q dt) 1q

≤( ∫

R

∥∥∥φ(t+ h)1(0,T )(t+ h)− φ(t)1(0,T )(t)

(t+ h)α

∥∥∥q dt) 1q

+( ∫

R

∥∥∥φ(t)1(0,T )(t)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥q dt) 1q.

(9.2.2)

The first term on the right-hand side in (9.2.2) may be estimated using Holder’s


inequality( ∫R

∥∥∥φ(t+ h)1(0,T )(t+ h)− φ(t)1(0,T )(t)

(t+ h)α

∥∥∥q dt) 1q

≤( ∫ 0

−h

∥∥∥φ(t+ h)

(t+ h)α

∥∥∥q dt) 1q

+( ∫ T−h

0

∥∥∥φ(t+ h)− φ(t)

(t+ h)α

∥∥∥q dt) 1q

+( ∫ T

T−h

∥∥∥ φ(t)

(t+ h)α

∥∥∥q dt) 1q

≤ ‖φ‖Lp(0,T ;E)Cp,q,αu1q− 1

p−α + T

1q− 1

p−αCp,q,α

( ∫I[h]

‖φ(t+ h)− φ(t)‖p dt) 1

p,

where we used (T + h)a − T a ≤ ha for a ∈ (0, 1) in estimating the last integral. Forthe second term on the right-hand side in (9.2.2) again by Holder’s inequality( ∫

R

∥∥∥φ(t)1(0,T )(t)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥q dt) 1q

≤ ‖φ‖Lp(0,T ;E)

( ∫ T

0

∣∣∣(t+ h)−α − t−α∣∣∣ pq

p−qdt

) p−qpq.

We have∫ T

0

∣∣∣(t+ h)−α− t−α∣∣∣ pq

p−qdt ≤

∫ T

0

t−αpqp−q − (t+ h)−

αpqp−q dt ≤ Cα,p,qh

1− αpqp−q ≤ Cα,p,qu

1− αpqp−q .

It follows that( ∫R

∥∥∥φ(t)1(0,T )(t)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥q dt) 1q ≤ ‖φ‖Lp(0,T ;E)Cα,p,qu

p−qpq

−α.

Similar estimates hold for h < 0.We may write [0, 1] = [0, T ∧ 1]∪ [T ∧ 1, 1] in the definition of the Besov norm and

estimate the first part as( ∫ T∧1

0

u−sr sup|h|≤u

∥∥∥φ(t+ h)1(0,T )(t+ h)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥rLq(R;E)

du

u

) 1r

≤ Cα,p,q

( ∫ T∧1

0

u−sr[T

1q− 1

p−α sup

|h|≤u‖φ(t+ h)− φ(t)‖Lp(I[h];E)

+ ‖φ‖Lp(0,T ;E)up−qpq

−α]r du

u

) 1r

(i)

≤ Cα,p,qT1q− 1

p−α

( ∫ 1

0

u−sr[

sup|h|≤u

‖φ(t+ h)− φ(t)‖Lp(I[h];E)

]r duu

) 1r

+ Cα,p,q‖φ‖Lp(0,T ;E)

( ∫ T

0

u−sru(p−q)r

pq−αr du

u

) 1r

(ii)

≤ T1q− 1

p−αCp,q,r,s,α‖φ‖Λs

p,r(0,T ;E) + Cp,q,r,s,αT1q− 1

p−α−s‖φ‖Lp(0,T ;E).


In (i) we used the triangle inequality in Lr(0, T ∧ 1, duu

). In (ii) we used α < 1q− 1

p− s.

Now fix T < |h|. Then( ∫R

∥∥∥φ(t+ h)1(0,T )(t+ h)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥q dt) 1q

≤ 2( ∫ T

0

∥∥∥φ(t)

tα

∥∥∥q dt) 1q ≤ T

1q− 1

p−αCα,p,q‖φ‖Lp(0,T ;E).

So the other part on [T ∧ 1, 1] can be estimated as( ∫ 1

T∧1

u−sr sup|h|≤u

∥∥∥φ(t+ h)1(0,T )(t+ h)

(t+ h)α−φ(t)1(0,T )(t)

tα

∥∥∥rLq(R;E)

du

u

) 1r

≤ T1q− 1

p−αCα,p,q‖φ‖Lp(0,T ;E)

( ∫ 1

T∧1

u−srdu

u

) 1r

≤ T1q− 1

p−α−sCα,p,q,r,s,T∨1‖φ‖Lp(0,T ;E).

Putting everything together we obtain that

‖t 7→ t−αφ(t)‖Λsq,r(0,T ;E)

= ‖t 7→ t−αφ(t)‖Lq(0,T ;E) +( ∫ 1

0

u−sr sup|h|≤u

∥∥∥φ(t+ h)

(t+ h)α− φ(t)

tα

∥∥∥rLq(0,T ;E)

du

u

) 1r

≤ T1q− 1

p−αCα,p,q‖φ‖Lp(0,T ;E)

+ T1q− 1

p−αCp,q,r,s,α‖φ‖Λs

p,r(0,T ;E) + T1q− 1

p−α−sCp,q,r,s,α,T∨1‖φ‖Lp(0,T ;E).

The result clearly follows from this.

As a consequence of this we have the following lemma.

Lemma 9.2.2. Let E be a Banach space with type q ∈ [1, 2). Let α ∈ [0, 12) and p > 2

be such that α < 12− 1

p. Then there exists a constant C depending on p, q, α, E and

T ∨ 1 such that for all φ ∈ Λ1q− 1

2p,q (0, T ;E), we have

supt∈(0,T )

‖s 7→ (t− s)−αφ(s)‖γ(0,t;E) ≤ CT12− 1

p−α‖φ‖

Λ1q−

12

p,q (0,T ;E). (9.2.3)

Proof. Fix t ∈ (0, T ). We may estimate

‖s 7→ (t− s)−αφ(s)‖γ(0,t;E) = ‖s 7→ (t− s)−αφ(s)1(0,t)(s)‖γ(R;E)

(i)

≤ CE,q‖s 7→ (t− s)−αφ(s)1(0,t)(s)‖Λ

1q−

12

q,q (R;E)

= CE,q‖s 7→ s−αφ(t− s)1(0,t)(s)‖Λ

1q−

12

q,q (R;E)

(ii)

≤ Cp,q,α,E,T∨1T12− 1

p−α‖s 7→ φ(t− s)‖

Λ1q−

12

p,q (0,t;E)

= Cp,q,α,E,T∨T12− 1

p−α‖φ‖

Λ1q−

12

p,q (0,T ;E).


In (i) we used Theorem 3.3.2 and the fact that Λ1q− 1

2q,q (R;E) = B

1q− 1

2q,q (R;E) isomorphi-

cally and (ii) follows from Lemma 9.2.1.

Next we have two lemmas for convolutions. The first result is a maximal regularityresult for convolutions in Besov spaces.

Lemma 9.2.3. Assume that the analytic C0-semigroup (S(t))t≥0 generated by A isexponentially stable. For all p, q ∈ [1,∞), s ∈ R, δ ∈ (0, 1], θ ∈ [0, 1) such thatδ + θ < 1 or δ = 1 and θ = 0 there exists a constant C depending on S, p, q, δ, θ and ssuch that for all (−A)−θf ∈ Bs+δ−1

p,q (R;E),

‖S ∗ f‖Bsp,q(R;Eδ) ≤ C‖(−A)−θf‖Bs+δ+θ−1

p,q (R;E), (9.2.4)

here S ∗ f := 0 on (−∞, 0) and E1 := D(A).

Proof. Let f ∈ S(R;E)). Notice that for all k ∈ N we have

ϕk ∗ f =1∑

l=−1

ψk+l ∗ ϕk ∗ f,

where ψk = ϕk for all k 6= 0 and ψ0 = φ. Fix k ∈ N, and denote fk = ϕk ∗ f . Forn = −1, 0, 1, 2, . . . we may write

ψn ∗ ϕk ∗ S ∗ f = ψn ∗ S ∗ fk.

We estimate ‖ϕn ∗S ∗ fk‖Lp(R;Eδ). For each n ≥ −1, we may use Young’s inequality toestimate

‖ψn ∗ S ∗ fk‖Lp(R;Eδ) = ‖F−1(ψn (−A)θR(·i, A)) ∗ (−A)−θfk‖Lp(R;Eδ)

≤ ‖F−1(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ))‖(−A)−θfk‖Lp(R;E).

Let t > 0 be fixed. First consider n ≥ 1. Clearly, it holds that∫r>t

‖F−1(ψn (−A)θR(·i, A))(r)‖B(E,Eδ) dr

=

∫r>t

r−2‖F−1D2(ψn (−A)θR(·i, A))(r)‖B(E,Eδ) dr

≤ supr∈R

‖F−1D2(ψn (−A)θR(·i, A))(r)‖B(E,Eδ)

∫r>t

r−2 dr

≤ ‖(D2ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ))

1

t,

where D stands derivation. One also has that∫0≤r<t

‖F−1(ψn (−A)θR(·i, A))(r)‖B(E,Eδ) dr ≤ t‖ψn (−A)θR(·i, A)‖L1(R;B(E,Eδ)).


Therefore, we deduce that

‖F−1(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ))

≤ t−1‖D2(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ)) + t‖ψn (−A)θR(·i, A)‖L1(R;B(E,Eδ)).

Minimization over t > 0 gives

‖F−1(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ))

≤ ‖(D2ψn (−A)θR(·i, A))‖12

L1(R;B(E,Eδ))‖ψn (−A)θR(·i, A)‖12

L1(R;B(E,Eδ)).(9.2.5)

Since ψn has support in In := [−2n+1,−2n−1] ∪ [−2n−1,−2n+1] it follows that

‖D2(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ))

≤ ‖D2ψn‖∞‖(−A)θR(·i, A)‖L1(In;B(E,Eδ))

+ 2‖Dψn‖∞‖(−A)θR(·i, A)2‖L1(In;B(E,Eδ)) + ‖ψn‖∞‖(−A)θR(·i, A)3‖L1(In;B(E,Eδ))

≤ C12−2n2(n+1)(δ+θ) + C22

−n2(n+1)(δ+θ−1) + C32(n+1)(δ+θ−2) ≤ C42

−n(2−δ−θ)

where we used‖(−A)θR(λi, A)‖B(E,Eδ) ≤ C|λ|δ+θ−1. (9.2.6)

Indeed,‖(−A)θx‖Eδ

≤ C‖x‖(E,D(A))δ+θ,1, for x ∈ ‖x‖(E,D(A))δ+θ,1

,

where we take (E,D(A))δ+θ,1 = D(A) if δ + θ = 1. Therefore,

‖(−A)θR(λi, A)x‖Eδ≤ C‖R(λi, A)x‖(E,D(A))δ+θ

≤ C ′‖AR(λi, A)x‖δ+θE ‖R(λi, A)x‖1−δ+θE

and (9.2.6) follows from standard resolvent estimates. Similarly one can show that

‖ψn (−A)θR(·i, A)‖L1(R;B(E,Eδ)) ≤ CS,δ,θ,ψ2n(δ+θ).

Combining these estimates with (9.2.5) we arrive at

‖F−1(ψn (−A)θR(·i, A))‖L1(R;B(E,Eδ)) ≤ CS,δ,θ,ψ2−n(1−δ−θ).

We may conclude that

‖S ∗ f‖Bsp,q(R;Eδ) ≤

1∑l=−1

(∑k≥0

(2sk‖ψk+l ∗ S ∗ fk‖Lp(R;Eδ))q)

1q

≤ CS,δ,θ,ψ

1∑l=−1

(∑k≥0

(2sk2−(k+l)(1−δ−θ)‖fk‖Lp(R;E))q)

1q

= CS,δ,θ,ψ

1∑l=−1

2−l(1−δ−θ)(∑k≥0

(2(s+δ+θ−1)k‖fk‖Lp(R;E))q)

1q

= CS,δ,θ,ψ‖f‖Bs+δ+θ−1p,q (R;E)


and (9.2.4) follows.The general case follows since S(R; R) is dense in Bs

p,q(R;E).

As a consequence we obtain the following estimate for the weighted γ-norm of aconvolution.

Proposition 9.2.4. Let E be a Banach space with type q ∈ [1, 2]. Assume that S isexponentially stable. Let α ≥ 0 and p ∈ [2,∞) be such that α < 1

2− 1

p. Let a, θ ≥ 0

be such that a + θ < 32− 1

q. Let (−A)−θφ ∈ Lp(0, T ;E). Then there is a constant C

which depends on S, q, a, θ, α and T ∨ 1 such that for all t ∈ [0, T ],

‖s 7→ (t− s)−αS ∗ φ‖γ(0,t;Ea) ≤ CT12−α− 1

p‖(−A)−θφ‖Lp(0,T ;E).

Proof. We may extend S as zero outside R+ and φ as zero outside (0, T ). We mayassume that a > 0. First assume that q ∈ [1, 2). There exist constants (Ci)

4i=1 which

depend only on q, p, α, E, a, S, θ and T ∨ 1 such that

‖s 7→(t− s)−αS ∗ φ(s)‖γ(0,t;Ea)

(i)

≤ C1T12− 1

p−α‖S ∗ φ‖

Λ1q−

12

p,q (0,T ;Ea)

≤C1T12− 1


Λ1q−

12

p,q (R;Ea)

(ii)

≤ C2T12− 1


B1q−

12

p,q (R;Ea)

(iii)

≤ C3T12− 1

p−α‖(−A)−θφ‖

B1q−

12−(1−a−θ)

p,q (R;E)

(iv)

≤ C4T12− 1

p−α‖(−A)−θφ‖Lp(R;E)

= C4T12− 1

p−α‖(−A)−θφ‖Lp(0,T ;E).

In (i) we used Lemma 9.2.2. Since Λ1q− 1

2−(1−a−θ)

p,q (R;E) = B1q− 1

2−(1−a−θ)

p,q (R;E) areisomorphic (ii) holds. To obtain (iii) one may apply Lemma 9.2.3. For the estimatein (iv) we note that 1

q− 1

2− (1− a− θ) > 0 so that general results apply (see Section

2.7).If q = 2, then by Proposition 3.3.1 and Young’s inequality with 1

p+ 1

p′= 1 + 1

2it

follows that there exist constants (Ci)3i=1 which depend only on the same parameters

as before, and such that

‖s 7→ (t− s)−αS ∗ φ(s)‖γ(0,t;Ea) ≤ C1‖s 7→ (t− s)−αS ∗ φ(s)‖L2(0,t;Ea)

≤ C2‖s 7→ s−α−a−θ‖Lp′ (0,t;E)‖(−A)−θφ‖Lp(0,T ;E)

≤ C3T12−α− 1

p+1−a−θ‖(−A)−θφ‖Lp(0,T ;E).

9.3 Stochastic convolutions

The following lemma gives some R-boundedness properties of analytic semigroups andwill often be combined with Proposition 3.2.4. Recall that A is the generator of ananalytic semigroup S on E.


Lemma 9.3.1. Let a ∈ [0, 1) and let ε > 0. Then for all T > 0, the family ta+εS(t) ∈B(E,Ea) : t ∈ [0, T ] is R-bounded and there is a constant C depending on S,Ea, εand T ∨ 1 such that

R(ta+εS(t) ∈ B(E,Ea) : t ∈ [0, T ]) ≤ CT ε.

Proof. It suffices consider the case that Ea = (E,D(A))a,1. Let N : [0, T ] → B(E,Ea)be defined as N(t) = ta+εS(t). Then N is continuously differentiable on (0, T ) andN ′(t) = (a+ ε)ta+ε−1S(t) + ta+εAS(t). Moreover, by [83, Proposition 2.2.9]

‖N ′(t)‖B(E,Ea) ≤ Ctε−1 for t ∈ (0, T ].

Therefore, the result follows as in [73, Example 2.18].

Theorem 9.3.2. Let E be a UMD− space. Let θ ∈ [0, 12) and let (−Aw)−θΦ : [0, T ]×

Ω → B(H,E) be H-strongly measurable and adapted. Let α ∈ (0, 12), λ, δ ≥ 0 and

p ∈ (2,∞) be such that λ+ δ+ θ < α− 1p. Then there exists a constant C, which only

depends on E, S, p, α, λ, δ and T ∨ 1, and such that

E∥∥∥t 7→∫ t

0

S(t− s)Φ(s) dWH(s)∥∥∥pCλ([0,T ];Eδ)

≤ CpT supt∈[0,T ]

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E).(9.3.1)

Remark 9.3.3. It will be clear from the proof below there exists ε > 0 such that

E∥∥∥t 7→∫ t

0

S(t− s)Φ(s) dWH(s)∥∥∥pCλ([0,T ];Eδ)

≤ CpT εp∫ T

0

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E) dt.

However, if the right-hand side is finite, the stochastic integral is possibly only definedfor almost all t ∈ [0, T ].

Proof. Let β ∈ (0, 12) be such that λ+ δ < β − 1

p< α− θ− 1

p. It follows from Lemma

9.3.1 and Proposition 3.2.4 that for all t ∈ [0, T ] almost surely,

‖s 7→ (t−s)−βS(t−s)Φ(s)‖γ(0,t;H,E) ≤ Tα−β−θC1‖s 7→ (t−s)−α(−Aw)−θΦ(s)‖γ(0,t;H,E).

By Theorem 4.5.9, we may define ζβ : [0, T ]× Ω → E as

ζβ(t) =1

Γ(1− β)

∫ t

0

(t− s)−βS(t− s)Φ(s) dWH(s)

and for all t ∈ [0, T ],E‖ζβ(t)‖p ≤ Tα−β−θC2M,


whereM := sup

t∈[0,T ]

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E).

So, by Proposition 7.4.1 and Fubini’s theorem we have

‖ζβ‖Lp(Ω;Lp(0,T ;E)) ≤ C3T1pTα−β−θM

1p .

Let Ω0 with P (Ω0) = 1 be such that ζβ(·, ω) ∈ Lp(0, T ;E) for all ω ∈ Ω0. We mayapply Lemma 7.3.1 to obtain that for all ω ∈ Ω0,

t 7→ (Rβζβ(·, ω))(t) ∈ Cλ([0, T ];Eδ)

and‖Rβζβ(·, ω)‖Cλ([0,T ];Eδ) ≤ Cβ,λ,δ,p,T∨1‖ζβ(·, ω)‖Lp(0,T ;E). (9.3.2)

By the right-ideal property we have

E‖s 7→ S(t− s)Φ(s)‖pγ(0,t;H,E) ≤ T βpM.

It follows from Theorem 4.5.9 that s 7→ S(t− s)Φ(s) is stochastically integrable for allt ∈ [0, T ] and we may define ζ : [0, T ]× Ω → E as

ζ(t) =

∫ t

0

S(t− s)Φ(s) dWH(s).

We claim that for all t ∈ [0, T ], for almost all ω ∈ Ω,

ζ(t, ω) = (Rβζβ(·, ω))(t). (9.3.3)

It suffices to check that for all t ∈ [0, T ] and x∗ ∈ E∗, almost surely we have

〈ζ(t), x∗〉 =1

Γ(β)

∫ t

0

(t− s)β−1〈S(t− s)ζβ(s), x∗〉 ds.

This follows from a standard argument based on the stochastic Fubini theorem (see[35]). Now the results and (9.3.1) follow from (9.3.3) and taking Lp moments in(9.3.2).

As a consequence we have the following regularity result of stochastic convolutionsin spaces with type q ∈ [1, 2).

Corollary 9.3.4. Let E be a UMD− space with type q ∈ [1, 2). Let p, r ∈ (2,∞)and θ ∈ (0, 1

2). If (−Aw)−θΦ is strongly measurable and adapted and (−Aw)−θΦ is in

Lr(Ω; Λ1q− 1

2p,q (0, T ; γ(H,E))), then there exists a constant C = CS,E,p,q,r,λ,δ,θ,T such that

E∥∥∥t 7→ ∫ t

0

S(t− s)Φ(s) dWH(s)∥∥∥rCλ([0,T ];Eδ)

≤ CrE‖(−Aw)−θΦ‖rΛ

1q−

12

p,q (0,T ;γ(H,E)),

(9.3.4)

where one may take λ > 0 and δ ≥ 0 according to λ+ δ < min12− 1

p, 1

2− 1

r− θ.


Proof. Let λ + δ < α < min12− 1

p, 1

2− 1

r− θ. For t ∈ (0, T ) it follows from Lemma

9.2.2 that

‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖γ(0,t;H,E) ≤ C1‖(−Aw)−θΦ‖Λ

1q−

12

p,q (0,T ;γ(H,E)).

The result now follows from Theorem 9.3.2 and

E∥∥∥t 7→∫ t

0

Φ(s) dWH(s)∥∥∥rCλ([0,T ];Eδ)

≤ Cr2 supt∈[0,T ]

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖rγ(0,t;H,E)

≤ Cr3E‖(−Aw)−θΦ‖r

Λ1q−

12

p,q (0,T ;γ(H,E)).

The next lemma is crucial for estimating stochastic convolutions in the γ-norm.We first recall the definition of Pisier’s property (α).

Let (rmn)m,n≥1 be a double indexed Rademacher sequence on Ω and let (r′n)n≥1 and(r′′m)m≥1 be Rademacher sequences on Ω′ and Ω′′. A Banach space E has property (α)if there are constants c, C > 0 such that for all finite sequences (xmn)

Nm,n=1 in E we

have

cE′E′′∥∥∥ N∑m,n=1

r′mr′′nxmn

∥∥∥2

≤ E∥∥∥ N∑m,n=1

rmnxmn

∥∥∥2

≤ C2E′E′′∥∥∥ N∑m,n=1

r′mr′′nxmn

∥∥∥2

.

A space E has property (α−) if the left-hand side inequality holds for all finite sequences(xmn)

Nm,n=1 in E. A space E has property (α+) if the right-hand side inequality holds for

all finite sequences (xmn)Nm,n=1 in E. One obtains equivalent notions if the Rademacher

sequence is replaced with a Gaussian sequence. Every Banach function space withfinite cotype has property (α). For more information we refer to [99, 113].

Lemma 9.3.5. Let E be a UMD− Banach space with property (α−). Let θ ∈ [0, 12).

Let b ∈ [0, 12− θ), α ∈ (b+ θ, 1

2) and p ∈ (1,∞) be fixed. Assume that

supt∈[0,T ]

‖s 7→ (t− s)−α(−Aw)−θΦ(s))‖Lp(Ω;γ(0,t;H,E)) <∞.

Then for all t ∈ [0, T ], s 7→ S(t − s)Φ(s) is stochastically integrable as an Eb-valuedprocess and ζ := S Φ is progressively measurable and satisfies

supt∈[0,T ]

E‖s 7→ (t− s)−αζ(s)‖pγ(0,t;H,Eb)

≤ CT ( 12−b)p sup

t∈[0,T ]

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E),(9.3.5)

where C depends on E, S, α, b, p and T ∨ 1.


Remark 9.3.6. With the same proof one can show that Lemma 9.3.5 has the followingLr-version for r ∈ [1,∞).( ∫ T

0

(E‖s 7→ (t− s)−αζ(s)‖pγ(0,t;H,Eb)

) rp dt

) 1r

≤ CT ( 12−b)p

( ∫ T

0

(E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E)

) rp dt

) 1r.

(9.3.6)

Only in this case the stochastic convolution exists possibly only for almost all t ∈ [0, T ].

Proof. First of all since Eb is of class Jb between E and D(A) we may assume thatEb = (E,D(A))b,1. Therefore, noting that D((−Aw)b

′) → (E,D(A))b,1 if b′ > b, and

by replacing b we may assume that Eb = D((−Aw)b).We claim that s 7→ (−Aw)bS(t−s)Φ(s) is stochastically integrable on [0, t]. Indeed,

let ε > 0 be such that b+θ+ε < α. Then it follows from Lemma 9.3.1 and Proposition3.2.4 that

E‖s 7→(−Aw)bS(t− s)Φ(s)‖pγ(0,t;H,E)

= E‖s 7→ (t− s)b+θ+ε(−Aw)b+θS(t− s)(t− s)−b−θ−ε(−Aw)−θΦ(s)‖pγ(0,t;H,E)

≤ C1TεE‖s 7→ (t− s)−b−ε(−Aw)−θΦ(s)‖pγ(0,t;H,E).

and the latter is finite by the assumption. The stochastic integrability now followsfrom Theorem 4.5.9. Define ζ2 : [0, T ]× Ω → E as

ζ2(t) =

∫ t

0

(−Aw)bS(t− s)Φ(s) dWH(s).

Then by Proposition 7.4.1, ζ2 is strongly progressively measurable, and for all t ∈ [0, T ],almost surely (−Aw)−bζ2(t) = ζ(t). It follows that ζ has a version with almost all pathsin Eb and such that (−Aw)bζ is progressively measurable.

We may estimate

‖s 7→ (t− s)−α(−Aw)bζ(s)‖Lp(Ω;γ(0,t;H,E))

= ‖s 7→ (t− s)−α(−Aw)bIWHu (S(s− u)Φ(u)1[0,s](u))‖Lp(Ω;γ(0,t;H,E))

(i)hp ‖s 7→ (t− s)−α(−Aw)bIWH

u (S(s− u)Φ(u)1[0,s](u))‖γ(0,t;H,Lp(Ω;E))

= ‖s 7→ (t− s)−αIWHu (u 7→ (−Aw)bS(s− u)Φ(u)1[0,s](u))‖γ(0,t;H,Lp(Ω;E))

(ii)

.p,E ‖s 7→ (u 7→ (t− s)−α(−Aw)bS(s− u)Φ(u)1[0,s](u))‖γ(0,t;Lp(Ω;γ(0,t;H,E)))

(i)hp ‖s 7→ (u 7→ (t− s)−α(−Aw)bS(s− u)Φ(u)1[0,s](u))‖Lp(Ω;γ(0,t;γ(0,t;H,E)))

(iii)

. E ‖(u, s) 7→ (t− s)−α(−Aw)bS(s− u)Φ(u)1[0,s](u)‖Lp(Ω;γ((0,t)2;H,E)).

In (i) we used the γ-Fubini isomorphism from Proposition 3.5.3. In (ii) we used the leftideal property of γ-radonifying operators and Theorem 4.5.9. In (iii) we used property(α−) and [99].


Let ε > 0 be such that b + θ + ε < 12. Pointwise in Ω and with t ∈ [0, T ] fixed we

may estimate

‖(u, s) 7→ (t− s)−α(−Aw)bS(s− u)Φ(u)1[0,s](u)‖γ((0,t)2;H,E)

(i)

≤ C2Tε‖(u, s) 7→ (t− s)−α(s− u)−b−θ−ε(−Aw)−θΦ(u)1[0,s](u)‖γ((0,t)2;H,E)

(ii)

≤ C3T12−b‖u 7→ (t− u)−α(−Aw)−θΦ(u)‖γ(0,t;H,E).

In (i) we used Proposition 3.2.4 and Lemma 9.3.1. For (ii) define P : L2(0, t;H) →L2((0, t)2;H) as

(Pf)(u, s) = (t− u)α(t− s)−α(s− u)−b−θ−εf(u)1[0,s](u).

Then P is bounded with ‖P‖ ≤ C4T12−b−θ−ε. Indeed, using Fubini’s theorem we

deduce that

‖Pf‖2L2((0,t)2;H) =

∫ t

0

∫ s

0

(t− u)2α(t− s)−2α(s− u)−2b−2θ−2ε‖f(u)‖2H1[0,s](u) du ds

=

∫ t

0

‖f(u)‖2H(t− u)2α

∫ t

u

(t− s)−2α(s− u)−2b−2θ−2ε ds du

=

∫ t

0

‖f(u)‖2H(t− u)−2b−2θ−2ε+1

∫ 1

0

(1− s)−2αs−2b−2ε ds du

=

∫ t

0

‖f(u)‖2H(t− u)−2b−2θ−2ε+1

∫ 1

0

(1− s)−2αs−2b−2ε ds du

= C5

∫ t

0

‖f(u)‖2H(t− u)−2b−2θ−2ε+1 du

≤ C6T−2b−2θ−2ε+1‖f‖2

L2(0,t;H),

where we used −2b − 2θ − 2ε + 1 > 0 in the last step. Now (ii) follows from theright-ideal property and

‖(u, s) 7→(t− s)−α(s− u)−b−θ−ε(−Aw)−θΦ(u)1[0,s](u)‖γ((0,t)2;H,E)

= ‖u 7→ (t− u)−α((−Aw)−θΦ P )(u)‖γ(0,t;H,E)

≤ C4T12−b−θ−ε‖u 7→ (t− u)−α(−Aw)−θΦ(u)‖γ(0,t;H,E).

If we combine the estimates we obtain (9.3.5).

We do not know whether property (α−) is really needed in Lemma 9.3.5. However,if E has non-trivial type, then the following result can be useful. In particular, it canbe useful in the case that b = θ = 0 or when E has type close to 2.


Lemma 9.3.7. Let E be a UMD− Banach space with type q ∈ (1, 2]. Let b, θ ≥ 0 besuch that b + θ < 1

2. Let α ∈ (0, 1

2) and p ∈ (2,∞) be such that b + 1

q− 1

2< α − 1

p.

Assume that

supt∈[0,T ]

(E‖s 7→ (t− s)−α(−Aw)−θΦ(s))‖pγ(0,t;H,E))

) 1p<∞.

Then for almost all t ∈ [0, T ], s 7→ S(t − s)Φ(s) is stochastically integrable and theprocess ζ := S Φ has a version which satisfies, almost surely, for all s ∈ [0, T ],ζ(s) ∈ Eb. Moreover (−Aw)bζ is progressively measurable and for all t0 ∈ [0, T ] itsatisfies

supt∈[0,T ]

(E‖s 7→ (t− s)−αζ(s)‖pγ(0,t;H,Eb)

dt) 1

p

≤ CT1p supt∈[0,T ]

(E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E)

) 1p,

(9.3.7)

where C depends on E, S, α, b, θ, p, q and T ∨ 1.

Remark 9.3.8. In the proof below one can also show that the above result holds with

supt∈[0,T ]

(E‖s 7→ (t− s)−αζ(s)‖pγ(0,t;H,Eb)

dt) 1

p

≤ CT ε( ∫ T

0

E‖s 7→ (t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E) dt) 1

p

(9.3.8)

assuming only that the right-hand side is finite. This uses Remark 9.3.3 and in thatcase the stochastic convolution exists possibly only for almost all t ∈ [0, T ].

Proof. The first part may be proved in the same way as in Lemma 9.3.5. We mayassume that 1 < q < 2. To obtain (9.3.7) fix t0 ∈ [0, T ]. Let r ∈ (2,∞) be such thatα < 1

2− 1

r. Let ε > 0 be such that b + θ + 1

q− 1

2+ ε < α − 1

pThen it follows from

Lemma 9.2.2 and Theorem 9.3.2 that(E‖s 7→(t0 − s)−αζ(s)‖pγ(0,t0;Eb)

) 1p ≤ C1T

12− 1

r−α(E‖ζ‖p

Λ1q−

12

r,q (0,T ;Eb)

) 1p

≤ C2T12− 1

r−α(E‖ζ‖p

C1q−

12+ε

(0,T ;Eb)

) 1p

≤ C3T12− 1

r−αT

1p supt∈[0,T ]

(E‖(t− s)−α(−Aw)−θΦ(s)‖pγ(0,t;H,E)

) 1p .

9.4 L2γ-Lipschitz functions

In this section we will introduce a randomized Lipschitz condition. Let E1 and E2 beBanach spaces and let H be a separable Hilbert space with dim(H) ≥ 1. Let (S,Σ, µ)be a separable finite measure space.

9.4. L2γ-Lipschitz functions 179

One easily checks that the simple functions are dense in L2γ(S;E1) := γ(S;E1) ∩

L2(S;E1). Here γ(S;E1) ∩ L2(S;E1) denotes the Banach space of all functions forwhich

‖φ‖L2γ(S;E1) := ‖φ‖γ(S;E1)∩L2(S;E1) := ‖φ‖γ(S;E1) + ‖φ‖L2(S;E1) <∞.

Let f : S × E1 → B(H,E2) be a function such that for all x ∈ E1 we havef(·, x) ∈ γ(S;H,E2). Then for simple functions φ : S → E1 one can check thats 7→ f(s, φ(s)) ∈ γ(S;H,E2) and we say that f L2

γ-Lipschitz function for µ if f isstrongly continuous in the second variable and for all simple functions φ1, φ2 : S → E1,

‖f(·, φ1)− f(·, φ2)‖γ(S;H,E2) ≤ C‖φ1 − φ2‖L2γ(S;E1). (9.4.1)

In this case the mapping f(φ)(s) = f(s, φ(s)) extends uniquely to a Lipschitz mappingfrom L2

γ(S;E1) into γ(L2(S;H), E2). The Lipschitz constant of f will be denoted byLγµ,f and one has Lγµ,f = infC : (9.4.1) holds for C. Since f is assumed to be strongly

continuous in its second variable one can check that f(φ) is represented by f(·, φ) forφ ∈ L2

γ.If f is L2

γ-Lipschitz for all finite measures µ on (S,Σ) and

Lγf := supLγµ,f : µ is a finite measure on (S,Σ) <∞,

we just say that f is a L2γ-Lipschitz function.

In type 2 spaces there is an easy sufficient condition which implies that a functionis L2

γ-Lipschitz function.

Lemma 9.4.1. Let E2 be a type 2 space. Let f : S × E1 → γ(H,E2) be such that forall x ∈ E1, f(·, x) is strongly measurable. If there is a constant C such that

‖f(s, x)‖γ(H,E2) ≤ C(1 + ‖x‖), s ∈ S, x ∈ E1, (9.4.2)

‖f(s, x)− f(s, y)‖γ(H,E2) ≤ C‖x− y‖, s ∈ S, x, y ∈ E1, (9.4.3)

then f is a L2γ-Lipschitz function and Lγf ≤ T2(E2)C, where T2(E2) is the type 2

constant of E2. Moreover, it satisfies the following linear growth condition

‖f(s, φ)‖γ(S;H,E2) ≤ T2(E2)C(1 + ‖φ‖L2(S;E1)).

If f does not depend on S, one can check that (9.4.1) implies (9.4.2) and (9.4.3)and in that case L2

γ-Lipschitz and Lipschitz functions coincides.

Proof. Let φ1, φ2 ∈ L2(S;E1). Using an approximation argument and (9.4.3) oneeasily checks that f(·, φ1) and f(·, φ2) are strongly measurable. It follows from (9.4.2)that f(·, φ1) and f(·, φ2) are in L2(S; γ(H,E2)) and from (9.4.3) we obtain

‖f(·, φ1)− f(·, φ2)‖L2(S;γ(H,E2)) ≤ C‖φ1 − φ2‖L2(S;E1). (9.4.4)

From Proposition 3.3.1 and (9.4.4) we may conclude that

‖f(·, φ1)− f(·, φ2)‖γ(S;H,E2) ≤ T2(E2)C‖φ1 − φ2‖L2(S;E1).

This clearly implies the result. The second statement follows in the same way.


If f : E1 → B(H,E2) (i.e f depends only on the Banach space E1), then f is L2γ-

Lipschitz for all measure spaces (S,Σ, µ) if and only if there exists a constant C ≥ 0such that for some (for all) orthonormal basis (hm)m≥1 and all (xn)

Nn=1 and (yn)

Nn=1 in

E,

E∥∥∥ N∑n=1

∑m≥1

γnm(f(xn)hm − f(yn)hm)∥∥∥2

≤ C2E∥∥∥ N∑n=1

γn(xn − yn)∥∥∥2

+ C2

N∑n=1

‖xn − yn‖2.

(9.4.5)

Moreover, Lγf = infC : (9.4.5) holds. This follows from a well-known scaling ar-gument for independent Gaussian random variables (see [61, Proposition 1]) and theexact expression of the γ-norm for simple functions (see (3.2.5))

Clearly, every L2γ-Lipschitz function f : E1 → γ(H,E2) is a Lipschitz function. The

other direction is not true. Nevertheless, many functions arising in stochastic partialdifferential equation turn out to be L2

γ-Lipschitz.

Example 9.4.2. Let b : R → R be a Lipschitz function. Let p ∈ [1,∞) and let(S,Σ, µ) be a σ-finite measure space. Define the Nemitskii operator B : Lp(S) → Lp(S)as B(x)(s) = b(x(s)). Then B is L2

γ-Lipschitz. Indeed, it follows from the Kahane-Khinchine inequalities that

(E

∥∥∥ N∑n=1

γn(B(xn)−B(yn))∥∥∥2) 1

2 hp

( ∫S

( N∑n=1

|b(xn(s))− b(yn(s))|2) p

2dµ(s)

) 1p

≤ Lb

( ∫S

( N∑n=1

|xn(s)− yn(s)|2) p

2dµ(s)

) 1p

h(E

∥∥∥ N∑n=1

γn(xn − yn)∥∥∥2) 1

2.

9.5 Lipschitz coefficients and integrable initial val-

ues

On the space E we consider the stochastic equation:dU(t) = (AU(t) + F (t, U(t))) dt+B(t, U(t)) dWH(t), t ∈ (0, T0],

U(0) = u0,(SE)

here the initial value u0 : Ω → E is F0-measurable and T0 > 0 is some finite time. Inthe case of infinity time horizons one can apply the results below to every finite timeinterval (0, n], to obtain a solution (U(t))t∈R+ .

Consider the following assumption on A.


(A1) The operator A is the generator of an analytic C0-semigroup S on E and thereexists an M ≥ 0 and ω > 0 such that for all t ∈ R+, ‖S(t)‖ ≤Me−ωt.

The uniform exponential stability assumption is not restrictive, because A + F =(A− cI) + (F + cI) for c ∈ R.


(A2) Let q ∈ [1, 2] be the type of E. Let a, θF ≥ 0 be such that 0 ≤ a + θF <32− 1

q.

The function (−A)−θFF : [0, T0]× Ω× Ea → E is a Lipschitz function of lineargrowth uniformly in [0, T0]×Ω, i.e. there are constants LF and CF such that forall t ∈ [0, T0], ω ∈ Ω and x, y ∈ Ea,

‖(−A)−θF (F (t, ω, x)− F (t, ω, y))‖E ≤ LF‖x− y‖Ea ,

‖(−A)−θFF (t, ω, x)‖E ≤ CF (1 + ‖x‖Ea). (9.5.1)

Moreover, for all x ∈ Ea, (t, ω) 7→ (−A)−θFF (t, ω, x) is strongly measurable andadapted.

(A3) Let a, θB > 0 be such that 0 ≤ a + θB < 12. The function (−A)−θBB : [0, T0] ×

Ω×Ea → B(H,E) is a L2γ-Lipschitz function of linear growth uniformly in Ω, i.e.

there are constants LγB and CγB such that for all measures µ on ([0, T0],B[0,T0]),

for all ω ∈ Ω and φ1, φ2 ∈ L2γ(0, T0, µ;Ea),

‖(−A)−θB(B(·, ω, φ1)−B(·, ω, φ2))‖γ(0,T0,µ;H,E) ≤ LγB‖φ1 − φ2‖L2γ(0,T0,µ;Ea),

‖(−A)−θBB(·, ω, φ)‖γ(0,T0,µ;H,E) ≤ CγB(1 + ‖φ‖L2

γ(0,T0,µ;Ea)). (9.5.2)

Moreover, for all x ∈ Ea, (t, ω) 7→ (−A)−θBB(t, ω, x) is H-strongly measurableand adapted.

Notice that for type 2 spaces one can check (A3) via Lemma 9.4.1.We prove an existence and uniqueness result for (SE) using fixed point methods.

First we introduce the fixed point space and the corresponding operator and we giveconditions under which the operator is a strict contraction.

Fix T ∈ (0, T0]. For p ∈ [1,∞], r ∈ [1,∞), α ∈ [0, 12) and a ∈ [0, 1] define the

space V p,rα,a (0, T ;E) as the set of all measurable (a.s. pathwise continuous if p = ∞)

and adapted processes φ : [0, T ]× Ω → Ea such that ‖φ‖V p,rα,a (0,T ;E) <∞, where

‖φ‖V p,rα,a (0,T ;E) :=

( ∫ T

0

(E‖s 7→ (t− s)−αφ(s)‖rγ(0,t;Ea)

) prdt

) 1p

+ ‖φ‖Lr(Ω;Lp(0,T ;Ea)).

if p <∞ and

‖φ‖V p,rα,a (0,T ;E) := sup

t∈[0,T ]

(E‖s 7→ (t− s)−αφ(s)‖rγ(0,t;Ea)

) 1r

+ ‖φ‖Lr(Ω;C([0,T ];Ea)).


if p = ∞.Identifying processes which are equal almost everywhere on [0, T ]× Ω this defines

a norm on V p,rα,a (0, T ;E) for p < ∞ which turns V p,r

α,a (0, T ;E) into a Banach space.Identifying indistinguishable processes this defines a norm on V p,r

α,a (0, T ;E) for p = ∞and again V p,r

α,a (0, T ;E) is a Banach space. Our main interest concerns the cases p = ∞and p = r <∞.

Later on we need the following result on weighted γ-spaces. Let µ be the Lebesguemeasure on [0, T ]. For t ∈ [0, T ] and α ∈ [0, 1

2) let µt,α be the measure on ([0, T ],B[0,T ])

defined by

µt,α(B) =

∫ t

0

(t− s)−2α1B(s) dµ(s).

Notice that for a function φ : [0, T ] → E we have φ ∈ γ(0, T, µt,α;E) if and only ifs 7→ (t− s)−αφ(s)1[0,t](s) ∈ γ(0, T ;E). We proceed with an easy lemma.

Lemma 9.5.1. Let E1 and E2 be Banach spaces and let p ∈ [1,∞). Let f : [0, T ] ×E1 → γ(H,E2) be a L2

γ-Lipschitz function. Let t ∈ [0, T ] and α ∈ [0, 12). Then for all

φ1, φ2 ∈ γ(0, T, µt,α;E1) ∩ L∞(0, T ;E1), we have f(·, φ1), f(·, φ2) ∈ γ(0, T, µt,α;H,E2)and ( ∫ T

0

‖f(·, φ1)− f(·, φ2)‖pγ(0,T,µt,α;H,E2) dt) 1

p

≤ Lγf

( ∫ T

0

‖φ1 − φ2‖pγ(0,T,µt,α;E1) dt) 1

p+ LγfCαT

−α+ 12‖φ1 − φ2‖Lp(0,T ;E1)

with the usual modification if p = ∞.

Proof. By Young’s inequality( ∫ T

0

‖φ‖pL2(0,T,µt,α,E1) dt) 1

p ≤ CαT−α+ 1

2‖φ‖Lp(0,T ;E1). (9.5.3)

It follows that φ1 and φ2 are in L2(0, T, µt,α, E1) for a.a. t ∈ [0, T ] and since f is L2γ-

Lipschitz we obtain that f(·, φ1) and f(·, φ2) are in γ(0, T, µt,α;H,E2) for a.a. t ∈ [0, T ]and

‖f(·, φ1)− f(·, φ2)‖γ(0,T,µt,α;H,E2) ≤ Lγf‖φ1 − φ2‖γ(0,T,µt,α;E1) + Lγf‖φ1 − φ2‖L2(0,T,µt,α;E1).

If we take Lp-moments and use (9.5.3), the result follows.

Define the fixed point operator LT : V p,rα,a (0, T ;E) → V p,r

α,a (0, T ;E) as

LT (φ) = t 7→ S(t)u0 + S ∗ F (φ)(t) + S B(φ)(t).

In the next lemma we show that LT is well-defined and that it is a strict contractionon V p,r

α,a (0, T ;E) for T small enough.


Lemma 9.5.2. Let E be a UMD− space with property (α−) and type q ∈ [1, 2] andlet H be a separable Hilbert space. Assume (A1)-(A3). Let r ∈ (2,∞), p ∈ [r,∞]and u0 ∈ Lr(Ω,F0;Ea). Then for every α ∈ (maxa + θB,

1r, 1

2), the operator LT

is well-defined and there exist a constant CT with limT↓0CT = 0 such that for allφ1, φ2 ∈ V p,r

α,a (0, T ;E),

‖LT (φ1)− LT (φ2)‖V p,rα,a (0,T ;E) ≤ CT‖φ1 − φ2‖V p,r

α,a (0,T ;E). (9.5.4)

Moreover, there is a constant C independent of u0 such that for all φ ∈ V p,rα,a (0, T ;E),

‖LT (φ)‖V p,rα,a (0,T ;E) ≤ C(1 + (E‖u0‖rEa

)1r . (9.5.5)

Proof. Below we estimate the different parts in separate paragraphs. The estimatesare combined at the end of the proof.

Initial value part – Let ε ∈ (0, 12). From Lemma 9.3.1 it follows that, the operator

family sεS(s) ∈ B(Ea) : s ∈ [0, T ] is γ-bounded by C1. An application of Proposition3.2.4 shows that

‖s 7→ (t− s)−αS(s)u0‖γ(0,t;Ea) ≤ C1‖s 7→ (t− s)−αs−εu0‖γ(0,t;Ea)

≤ C1‖s 7→ (t− s)−αs−ε‖L2(0,t)‖u0‖Ea

= C2‖u0‖Ea .

For the other part of the norm we note that

‖Su0‖C([0,T ];Ea) ≤ C3‖u0‖Ea .

It follows that‖Su0‖V p,r

α,a (0,T ;E) ≤ C4E‖u0‖Lr(Ω;Ea).

Deterministic convolution –(a): Let (−A)−θFφ ∈ Lp(0, T ;E). We estimate the V p,r

α,a (0, T ;E)-norm of S ∗ φ.Since E has type q it follows from Proposition 9.2.4 and 0 ≤ a+ θF <

32− 1

qthat

‖s 7→(t− s)−αS ∗ φ‖γ(0,t;Ea) ≤ C5T12−α− 1

p‖(−A)−θFφ‖Lp(0,T ;E). (9.5.6)

By Proposition 7.3.2, S ∗φ is continuous in Ea and by Young’s inequality one mayestimate

‖S ∗ φ‖Lp(0,T ;Ea) ≤ T 1−(a+θF )C6‖(−A)−θFφ‖Lp(0,T ;E). (9.5.7)

Now let (−A)−θFφ ∈ Lr(Ω;C([0, T ];E)). By applying (9.5.6) and (9.5.7) pointwiseto φ(·, ω) one obtains that

‖S ∗ φ‖V p,rα,a (0,T ;E) ≤ C7T

β1‖(−A)−θFφ‖Lr(Ω;Lp(0,T ;E)), (9.5.8)

where β1 = min12− α− 1

p, 1− (a+ θF ) and C7 depends on E, S, p, q, a, θF , α, T ∨ 1.


(b): Let φ1, φ2 ∈ V p,rα,a (0, T ;E). Since F is of linear growth,

(−A)−θFF (·, φ1), (−A)−θFF (·, φ2) ∈ Lr(Ω;Lp(0, T ;E)).

From (9.5.8) and the fact that F is Lipschitz continuous we deduce that S ∗ (F (·, φ1)and S ∗ (F (·, φ2) are in V p,r

α,a (0, T ;E) and

‖S ∗ (F (·, φ1)− F (·, φ2))‖V p,rα,a (0,T ;E)

≤ C7Tβ1‖(−A)−θF (F (·, φ1)− F (·, φ2))‖Lr(Ω;Lp(0,T ;E))

≤ C7Tβ1LF‖φ1 − φ2‖V p,r

α,a (0,T ;E).

(9.5.9)

Stochastic convolution –(a): Let (−A)−θBΦ : [0, T ]× Ω → B(H,E) be such that( ∫ T

0

(E‖s 7→ (t− s)−α(−A)−θBΦ(s)‖rγ(0,t;H,E)

) prdt

) 1p<∞, (9.5.10)

if p < ∞ and where we take the supremum norm if p = ∞. We estimate theV p,rα,a (0, T ;E) norm of S Φ. Let t ∈ [0, T ], be such that

E‖s 7→ (t− s)−α(−A)−θBΦ(s)‖rγ(0,t;H,E) <∞.

By Lemma 9.3.5, Remark 9.3.6 and the fact that 0 ≤ a+ θB <12

we obtain that(( ∫ T

0

E‖s 7→ (t− s)−αS Φ(s)‖rγ(0,t;Ea)

) prdt

) 1p

≤ C8T12−(a+θB)

( ∫ T

0

(E‖s 7→ (t− s)−α(−A)−θBΦ‖rγ(0,t;H,E)

) prdt

) 1p,

with the obvious modification if p = ∞.For the other part of the norm (here we use p ≥ r) notice that from Theorem 9.3.2

an Remark 9.3.3, it follows that there is an ε > 0 such that(E‖S Φ‖rLp(0,T ;Ea)

) 1r

≤ T1p

(E‖S Φ‖rC([0,T ];Ea)

) 1r

≤ T1p+εC9

( ∫ T

0


) prdt

) 1p.

We conclude that

‖S Φ‖V p,rα,a (0,T ;E)

≤ T β2C10

( ∫ T

0


) prdt

) 1p,

(9.5.11)


where β2 = min12− (a+ θB), 1

p+ ε

(b): Let φ1, φ2 ∈ V p,rα,a (0, T ;E). Since B is L2

γ-Lipschitz and of linear growth,B(·, φ1) and B(·, φ2) satisfy (9.5.10). It follows from (9.5.11) and Lemma 9.5.1 that

‖S (B(·, φ1)−B(·, φ2))‖V p,rα,a (0,T ;E)

≤ T β2C10

( ∫ T

0

(E‖s 7→ (t− s)−α(−A)−θB(B(s, φ1(s))−B(s, φ2(s)))‖rγ(0,t;H,E)

) prdt

) 1p

= T β2C10

( ∫ T

0

(E‖(−A)−θB(B(·, φ1)−B(·, φ2))‖rγ(0,t,µt,α;H,E)

) prdt

) 1p

≤ T β2C11LγB‖φ1 − φ2‖V p,r

α,a (0,T ;E).

Conclusions – It follows from the above considerations that LT is well-defined onV p,rα,a (0, T ;E) and there exist constants C and β > 0 depending on E, S, q, p, r, α, a, θF ,θB, F, B and T ∨ 1 such that for all φ1, φ2 ∈ V p,r

α,a (0, T ;E) we have

‖LT (φ1)− LT (φ2)‖V p,rα,a (0,T ;E) ≤ CT β‖φ1 − φ2‖V p,r

α,a (0,T ;E). (9.5.12)

The estimate (9.5.5) follows from (9.5.12) and

‖LT (0)‖V p,rα,a (0,T ;E) ≤ C(1 + (E‖u0‖rEa

)1r .

We can now state and proof the first existence and uniqueness result for (SE).

Theorem 9.5.3. Let E be a UMD− space with property (α−) and type q ∈ [1, 2]and let H be a separable Hilbert space. Assume (A1)-(A3). Let r ∈ (2,∞) andu0 ∈ Lr(Ω,F0;Ea).

Then for every α ∈ (maxa + θB,1r, 1

2) there exists a unique mild solution U ∈

V ∞,rα,a (0, T0;E) of (SE). Moreover, there exists a constant C depending on E, q, S,T0, λ, δ, a, θF , θB, r, F and B such that

‖U‖V∞,rα,a (0,T0;E) ≤ C(1 + (E‖u0‖rEa

)1r ). (9.5.13)

Remark 9.5.4. If E does not have property (α−), then under the condition thata + θB < 1

q− 1

2a version of the above result still holds. This can be proved using

Lemma 9.3.7 instead of Lemma 9.3.5. The same holds for other results below.

Proof of Theorem 9.5.3. It follows from Lemma 9.5.2 can find T ∈ (0, T0] such thatCT <

12. It follows from (9.5.4) that LT has a unique fixed point U ∈ V ∞,r

α,a (0, T ;E).This gives an almost surely pathwise continuous and adapted process U : [0, T ]×Ω →Ea such that almost surely for all t ∈ [0, T ], we have

U(t) = S(t)u0 + S ∗ F (·, U)(t) + S B(·, U)(t). (9.5.14)


Construction for arbitrary finite T – By a standard induction argument one mayconstruct a mild solution on each of the intervals [T, 2T ], . . . , [(n − 1)T, nT ], [nT, T0]for an appropriate integer n. The induced U solution on [0, T0] is the mild solution of(SE). Moreover, from (9.5.5) we deduce that

‖U‖V p,rα,a (0,T0;E) = ‖LT0(U)‖V∞,r

α,a (0,T0;E) ≤ C(1 + ‖u0‖Lr(Ω;Ea))

and (9.5.13) follows.Uniqueness: For small T ∈ (0, T0] this follows from the uniqueness of the fixed

point. For larger T the result follows from an induction argument as in Step 2.

In the next two corollaries we deduce regularity properties of the solution U fromTheorem 9.5.3. The regularity results are formulated for U − Su0, but if u0 is regu-lar enough, information about the regularity of U can be deduced. More regularityproperties of the paths of the solution will be obtained in Section 9.6.

Corollary 9.5.5. In the situation of Theorem 9.5.3 for every α ∈ (a + θB,12) and

every r ∈ (2,∞) such that α − (a+ θB)− 1r> 0, there exists a constant C depending

on E, q, S, T0, λ, δ, α, a, θF , θB, r, F and B such that(E‖U − Su0‖rCλ([0,T0];Ea+δ)

) 1r ≤ C(1 + (E‖u0‖rEa

)1r , (9.5.15)

for all λ > 0, δ ≥ 0 such that λ+ δ < minα− (a+ θB)− 1r, 1− (a+ θF ).

Proof. Let U ∈ V ∞,rα,a (0, T0;E) be the mild solution from Theorem 9.5.3. It follows

from Proposition 7.3.2 that we may take a version of S ∗ F (·, U) with

E‖S ∗ F (·, U)‖rCλ([0,T0];Ea+δ) ≤ C1E‖(−A)−θFF (·, U)‖rL∞(0,T0;E).

Similarly, by Theorem 9.3.2 we may take a version of S ∗B(·, U) with

E‖S B(·, U)‖rCλ([0,T0];Ea+δ) ≤ C2 supt∈[0,T0]

E‖s 7→ (t− s)−α(−A)−θBB(·, U(s))‖rγ(0,t;H,E).

Define U : [0, T0]× Ω → Ea as

U(t) = S(t)u0 + S ∗ F (·, U)(t) + S B(·, U)(t),

where we take the versions of the convolutions as above. By uniqueness we have almostsurely U ≡ U . Since F and B are of linear growth we may deduce that

E‖U − Su0‖rCλ([0,T0];Ea+δ) ≤ C3(1 + ‖U‖rV∞,rα,a (0,T0;E)).

Now (9.5.15) follows from (9.5.13).


9.6 Lipschitz coefficients and general initial values

Fix T ∈ (0, T0]. For p ∈ [1,∞), α ∈ [0, 12) and a ∈ R define the space V p,loc

α,a (0, T ;E)as all φ : [0, T ]×Ω → Ea such that φ is strongly measurable and adapted and almostsurely ∫ T

0

‖s 7→ (t− s)−αφ(s)‖pγ(0,t;Ea) dt+ ‖φ‖pLp(0,T ;Ea) <∞.

Notice that V p,rα,a (0, T ;E) ⊂ V p∧r,loc

α,a (0, T ;E).

Theorem 9.6.1. Let E be a UMD− space with property (α−) and type q ∈ [1, 2] and letH be a separable Hilbert space. Assume (A1)-(A3). Let u0 : Ω → Ea be F0-measurable.

Then for every pair α ∈ (0, 12) and p ∈ (2,∞) such that α ∈ (a + θB,

12− 1

p) and

a+ θF <32− 1

q− 1

p, there exists a unique mild solution U ∈ V p,loc

α,a (0, T0;E) of (SE).

For the proof we need the following uniqueness result.

Lemma 9.6.2. Under the conditions of Theorem 9.5.3 let U and V in V ∞,rα,a (0, T0;E)

be the mild solutions of (SE) with initial values u0 and v0 in Lr(Ω,F0;Ea). Thenalmost surely on the set u0 = v0 we have U ≡ V .

Proof. Let Γ = u0 = v0. First consider small T ∈ (0, T0] as in Step 1 in the proof ofTheorem 9.5.3. Since Γ is F0-measurable we have

‖U1Γ − V 1Γ‖V∞,rα,a (0,T ;E) = ‖LT (U)1Γ − LT (V )1Γ‖V∞,r

α,a (0,T ;E)

= ‖(LT (U1Γ)− LT (V 1Γ))1Γ‖V∞,rα,a (0,T ;E)

≤ 1

2‖U1Γ − V 1Γ‖V∞,r

α,a (0,T ;E),

hence almost surely U |[0,T ]×Γ ≡ V |[0,T ]×Γ.For [0, T0] one may proceed as in the proof of Theorem 9.5.3.

Proof of Theorem 9.6.1.Existence: Let r ∈ [p,∞) be such that α > 1

r. Define (un)n≥1 in Lr(Ω,F0;Ea)

as un = 1‖u0‖≤nu0. By Theorem 9.5.3, for each n ≥ 1, there is a solution Un ∈V ∞,rα,a (0, T0;E) of (SE) with initial value un. Lemma 9.6.2 implies that for 1 ≤ m ≤ n

almost surely on the set ‖u0‖ ≤ m, for all t ∈ [0, T0], Un(t) = Um(t). It follows thatalmost surely, for all t ∈ [0, T0], limn→∞ Un(t) exists in Ea. Define U : [0, T0]×Ω → Eaas U(t) = limn→∞ Un(t) if this limit exists and 0 otherwise. Clearly, U is stronglymeasurable and adapted. Moreover, almost surely on ‖u0‖ ≤ n, for all t ∈ [0, T0],U(t) = Un(t) and hence U ∈ V p,loc

α,a (0, T0;E). Furthermore, for all t ∈ [0, T0], s 7→S(t − s)F (·, U(s)) is in L1(0, t;Ea) almost surely, and s 7→ S(t − s)B(·, U(s)) is inγ(0, t;H,Ea) almost surely.

We show that U is a solution of (SE). Since the deterministic convolution isdefined pathwise, it follows that almost surely on ‖u0‖ ≤ n, for all t ∈ [0, T0],


S ∗ F (·, U)(t) = S ∗ F (·, Un)(t). In particular, almost surely for all t ∈ [0, T0], S ∗F (·, U)(t) = limn→∞ S ∗ F (·, Un)(t).

From Proposition 4.2.3, we deduce that for all m ≤ n, for all t ∈ [0, T0], almostsurely on ‖u0‖ ≤ n, S B(·, Um)(t) = S B(·, Un)(t) and by path continuity weobtain that almost surely on ‖u0‖ ≤ n for all m ≤ n, S B(·, Um) = S B(·, Un). Itfollows that we may define ζ : [0, T0]×Ω → Ea pathwise as limn→∞ S B(·, Un) if thislimit exists and 0 otherwise. Then ζ has almost all paths in C([0, T0];Ea), and againby Proposition 4.2.3 for all t ∈ [0, T0], almost surely in ‖u0‖ ≤ n,

S B(·, U)(t) = S B(·, Un)(t) = ζ(t).

Taking ζ as a version of S B(·, U), we obtain that almost surely, S B(·, U) =limn→∞ S B(·, Un) in C([0, T0];Ea).

We may conclude that almost surely for all t ∈ [0, T0],

U(t) = limn→∞

Un(t) = limn→∞

(S(t)un + S ∗ F (·, Un)(t) + S B(·, Un)(t)

)= S(t)u0 + S ∗ F (·, U)(t) + S B(·, U)(t).

This proves that U is a mild solution of (SE).Uniqueness: Let U1, U2 ∈ V p,loc

α,a (0, T0;E) be mild solutions of (SE). For each n ≥ 1let the stopping times µ1

n and ν1n for U1 be defined as

µ1n = inf

r ∈ [0, T0] :

∫ T0

0

‖s 7→ (t− s)−αU1(s)1[0,r](s)‖pγ(0,t;Ea) dt ≥ n,

ν1n = inf

r ∈ [0, T0] :

∫ T0

0

‖U1(s)1[0,r](s)‖pEads ≥ n

.

Similarly, we define the stopping times µ2n and ν2

n for U2. For each n ≥ 1 let

τn = µ1n ∧ ν1

n ∧ µ2n ∧ ν2

n,

and let U1n = U11[0,τn] and U2

n = U21[0,τn]. Then for all n ≥ 1, U1n and U2

n are inV p,pα,a (0, T ;E). One easily checks that

U1n = 1[0,τn](LT (U1

n))τn and U2

n = 1[0,τn](LT (U2n))

τn ,

where LT is the mapping introduced before Lemma 9.5.2 and

(LT (U in))

τn(t) := (LT (U in))(t ∧ τn).

By Lemma 9.5.2 we can find T ∈ (0, T0] such that CT ≤ 12

and therefore

‖U1n − U2

n‖V p,pα,a (0,T ;E) = ‖1[0,τn](LT (U1

n)− LT (U2n))

τn‖V p,pα,a (0,T ;E)

≤ ‖LT (U1n)− LT (U2

n)‖V p,pα,a (0,T ;E)

≤ 1

2‖U1

n − U2n‖V p,p

α,a (0,T ;E).


We obtain that U1n = U2

n in V p,pα,a (0, T ;E), hence U1

n(t, ω) = U2n(t, ω) for almost all

(t, ω) ∈ [0, T ]× Ω. Choosing n ≥ 1 arbitrary large, we may conclude that U1(t, ω) =U2(t, ω) for almost all (t, ω) ∈ [0, T ] × Ω. This proves the result. By an inductionargument as before one can show that the same result holds on [0, T0].

Corollary 9.6.3. In the situation of Theorem 9.6.1 the mild solution U of (SE) has aversion such that almost all paths satisfy U − Su0 ∈ Cλ([0, T0];Ea+δ), where one maytake λ > 0 and δ ≥ 0 according to λ+ δ < min1

2− (a+ θB), 1− (a+ θF ).

Proof. Let λ, δ ≥ 0, λ+ δ < min12− (a+ θB), 1− (a+ θF ) be arbitrary and choose

λ + δ < α < min12− (a + θB), 1 − (a + θF ) arbitrary. Let r be so large that

λ+ δ < α− (a+ θB)− 1r. Let U be the solution constructed in Theorem 9.6.1. Define

(un)n≥1 in Lr(Ω,F0;Ea) as un = 1‖u0‖≤nu0. By Corollary 9.5.5, for each n ≥ 1, wecan find a solution Un of (SE) with initial value un such that Un − Sun has paths inCλ([0, T0];Ea+δ). Since by the proof of Theorem 9.6.1, almost surely on u0 = un wehave U ≡ Un, the result follows.

9.7 Locally Lipschitz coefficients


(A2)′ Assume that E has type q ∈ [1, 2]. Let a, θF ∈ [0, 1] be such that 0 ≤ a + θF <32− 1

q. The function (−A)−θFF : [0, T0] × Ω × Ea → E is a locally Lipschitz

function uniformly in [0, T0]×Ω, i.e. for every R > 0 there is a constant LRF suchthat for all t ∈ [0, T0], ω ∈ Ω and ‖x‖Ea , ‖y‖Ea ≤ R,

‖(−A)−θF (F (t, ω, x)− F (t, ω, y))‖E ≤ LRF‖x− y‖Ea .

Moreover, for all x ∈ Ea, (t, ω) 7→ (−A)−θFF (t, ω, x) is strongly measurable andadapted and there is a constant CF,0 such that for all t ∈ [0, T0] and ω ∈ Ω

‖(−A)−θFF (t, ω, 0)‖E ≤ CF,0.

(A3)′ Let 0 ≤ a + θB < 12. The function (−A)−θBB : [0, T0] × Ω × Ea → γ(H,E) is a

locally L2γ-Lipschitz function uniformly in Ω, i.e. there exists a sequence of L2

γ-Lipschitz functions (Bn)n≥1 such that (−A)−θBBn : [0, T0]× Ω× Ea → γ(H,E)and (−A)−θBB(·, x) = (−A)−θBBn(·, x) for all ‖x‖Ea < n. There is a constantCB,0 such that for all measures µ on ([0, T0],B[0,T0]),

‖t 7→ (−A)−θBB(t, ω, 0)‖γ(0,T0,µ;H,E) ≤ CB,0, ω ∈ Ω.

Moreover, for all x ∈ Ea, (t, ω) 7→ (−A)−θBB(t, ω, x) is H-strongly measurableand adapted.


One may check that the locally Lipschitz version of Lemma 9.4.1 holds as well. Thisgives an easy way to check (A3)′.

Fix a stopping time τ with values in [0, T0]. For p ∈ [1,∞), α ∈ [0, 12) and a ∈ R

define the space V p,admα,a (0, τ ;E) as all Ea-valued admissible processes (φ(t))t∈[0,τ) such

that there exists an increasing sequence of stopping times (τn)n≥1 with τ = limn→∞ τnand almost surely∫ T

0

‖s 7→ (t− s)−αφ(s)1[0,τn](s)‖pγ(0,t;Ea) dt+ ‖φ1[0,τn]‖pLp(0,T ;Ea) <∞.

Notice that in the case that for almost all ω, τn(ω) = T for n large enough,

L0(Ω;C([0, T ];Ea)) ∩ V p,locα,a (0, T ;E) = V p,adm

α,a (0, T ;E).

Theorem 9.7.1. Let E be a UMD− space with property (α−) and type q ∈ [1, 2] andlet H be a separable Hilbert space. Assume (A1), (A2)′ and (A3)′. Then for everyα ∈ (0, 1

2) and p ∈ (2,∞) such that α ∈ (a + θB,

12− 1

p) and a + θF < 3

2− 1

q− 1

p

there exists a unique maximal local mild solution (U(t))[0,τ) in V p,admα,a (0, τ ;E) of (SE).

Moreover, U has a version such that for almost all ω ∈ Ω,

t 7→ U(t, ω)− S(t)u0(ω) ∈ Cλloc([0, τ(ω));Ea+δ),

where one may take λ > 0 and δ ≥ 0 according to λ+δ < min12−(a+θB), 1−(a+θF ).

If, additionally F and B are of linear growth, i.e. (9.5.1) and (9.5.2) hold, then theabove function U is the unique global mild solution of (SE) in V p,adm

α,a (0, T0;E) and thefollowing assertions hold:

(1) The solution U satisfies the statements of Theorem 9.6.1 and Corollaries 9.6.3.

(2) If r ∈ (2,∞) and u0 ∈ Lr(Ω,F0;Ea), then for every α > 0 such that α ∈(maxa + θB,

1r, 1

2) the solution U is in V ∞,r

α,a (0, T0;E) and (9.5.13) and thestatements of Corollary 9.5.5 holds.

Before we proceed, we prove the following local uniqueness result.

Lemma 9.7.2. Assume that the conditions of Theorem 9.7.1 are satisfied. Assumethat (U1(t))t∈[0,τ1) in V p,adm

α,a (0, τ1;E) and (U2(t))t∈[0,τ2) in V p,admα,a (0, τ2;E) are local mild

solution of (SE) with initial values u10 and u2

0. Let Γ = u10 = u2

0. Then almost surelyon Γ, U1|[0,τ1∧τ2) ≡ U2|[0,τ1∧τ2). Moreover, if τ1 is an explosion time for U1, then almostsurely on Γ, τ1 ≥ τ2. If τ1 and τ2 are explosion times for U1 and U2, then almost surelyon Γ, τ1 = τ2 and U1 ≡ U2.

Proof. Let τ = τ1 ∧ τ2. Let (µn)n≥1 be an increasing sequences of bounded stop-ping times such that limn→∞ µn = τ and for all n ≥ 1, U11[0,µn] and U21[0,µn] are inV p,locα,a (0, T0;E). Let

ν1n = inft ∈ [0, T0] : ‖U1(t)‖Ea ≥ n and ν2

n = inft ∈ [0, T0] : ‖U2(t)‖Ea ≥ n


and let σin = µn ∧ νin and let σn = σ1n ∧ σ2

n. On x ∈ Ea : ‖x‖Ea ≤ n we may replaceF and B by Fn (for a possible definition of Fn, see the proof of Theorem 9.7.1) andBn which satisfy (A2) and (A3). As in the proof of Theorem 9.6.1 it follows that

‖Uσn1 1[0,σn]×Γ − Uσn

2 1[0,σn]×Γ‖V p,pα,a (0,T ;E)

= ‖(LT (Uσn1 1[0,σn]×Γ)− LT (Uσn

2 1[0,σn]×Γ))1[0,σn]×Γ‖V p,pα,a (0,T ;E)

≤ ‖LT (Uσn1 1[0,σn]×Γ)− LT (Uσn

2 1[0,σn]×Γ)‖V p,pα,a (0,T ;E)

≤ CT‖Uσn1 1[0,σn]×Γ − Uσn

2 1[0,σn]×Γ‖V p,pα,a (0,T ;E),

where CT satisfies limT↓0CT = 0. For T small enough it follows that Uσn1 1[0,σn]×Γ =

Uσn2 1[0,σn]×Γ in V p,p

α,a (0, T ;E). By an induction argument this holds on [0, T0] as well.By path continuity it follows that almost surely, U1 ≡ U2 on [0, σn] × Γ. Since τ =limn→∞ σn we may conclude that almost surely, U1 ≡ U2 on [0, τ)× Γ.

If τ1 is an explosion time, then as in [122, Lemma 5.3] this yields τ1 ≥ τ2 on Γalmost surely. Indeed, if for some ω ∈ Γ, τ1(ω) < τ2(ω), then we can find an n suchthat τ1(ω) < ν2

n(ω). We have U1(t, ω) = U2(t, ω) for all 0 ≤ t ≤ ν1n+1(ω) < τ1(ω). If

we combine both assertions we obtain that

n+ 1 = ‖U1(ν1n+1(ω), ω)‖Ea = ‖U2(ν

1n+1(ω), ω)‖Ea ≤ n.

This is a contradiction. The final assertion is now obvious.

Proof of Theorem 9.7.1. For n ≥ 1 let Γn = ‖u0‖ ≤ n2 and un = u01Γn . Let (Bn)n≥1

be the sequence of L2γ-Lipschitz functions from (A3)′. Let r = p. Fix an integer n ≥ 1.

Let (−A)−θFFn : [0, T0]× Ω× Ea → E be defined by

(−A)−θFFn(·, x) = (−A)−θFF (·, x) for ‖x‖Ea ≤ n,

and Fn(·, x) = F(·, nx‖x‖Ea

). Clearly, Fn and Bn satisfy (A2) and (A3). It follows from

Theorem 9.5.3 that there exists a mild solution Un ∈ V ∞,rα,a (0, T0;E) of (SE) with u0,

F and B replaced by un, Fn and Bn. In particular, Un has a version with continuouspaths. Let τn be the stopping time defined by

τn(ω) = inft ∈ [0, T0] : ‖Un(t, ω)‖Ea ≥ n.

It follows from Lemma 9.7.2 that for all 1 ≤ m ≤ n, almost surely, Um ≡ Un on[0, τm ∧ τn] × Γm. By path continuity this implies τm ≤ τn. Therefore, we can defineτ = limn→∞ τn and on Γn, U(t) = Un(t) for t ≤ τn. By approximation and Proposition4.2.3 it is clear that U ∈ V p,adm

α,a (0, τ ;E) is a local mild solution of (SE). Moreover, τis an explosion time. Indeed, if ω ∈ Ω is such that τ(ω) < T0, then

lim supt↑τ(ω)

‖U(t, ω)‖Ea ≥ lim supn→∞

‖U(τn(ω), ω)‖Ea = lim supn→∞

n = ∞.

Maximality: If (U(t))t∈[0,τ) in V p,admα,a (0, τ ;E) is a local mild solution of (SE), then

it follows from Lemma 9.7.2 that τ ≤ τ and U |[0,τ ] ≡ U almost surely. Therefore,(U(t))t∈[0,τ) is a maximal local mild solution.


Holder regularity: By Corollary 9.5.5, each Un satisfies the regularity statement.Therefore, the construction yields the required regularity properties of the paths of U .

Global mild solution in the case that F and B are of linear growth. First we show(2). Let (Un)n≥1 be as before. As in the proof of Lemma 9.5.2 one can check that bythe linear growth assumption,

‖Un‖V∞,rα,a (0,T ;E) = ‖LT (Un)‖V∞,r

α,a (0,T ;E) ≤ CT‖Un‖V∞,rα,a (0,T ;E) + C + C‖un‖Lr(Ω;Ea),

where the constants do not depend on n and u0 and we have limT↓0CT = 0. Since‖un‖Lr(Ω;Ea) ≤ ‖u0‖Lr(Ω;Ea), it follows that for T small we have

‖Un‖V∞,rα,a (0,T ;E) ≤ C(1 + ‖u0‖Lr(Ω;Ea)),

where C is a constant independent of n and u0. Repeating this inductively, we obtain aconstant CT0 independent of n and u0 such that ‖Un‖V∞,r

α,a (0,T0;E) ≤ CT0(1+‖u0‖Lr(Ω;Ea)).In particular,

E sups∈[0,T0]

‖Un(s)‖rEa≤ Cr

T0(1 + ‖u0‖Lr(Ω;Ea))

r.

It follows thatP( sup

s∈[0,T0]

‖Un(s)‖Ea ≥ n) ≤ CrT0n−r.

Since∑

n≥1 n−r <∞, the Borel-Cantelli Lemma implies that

P( ⋂k≥1

⋃n≥k

sup

s∈[0,T0]

‖Un(s)‖Ea ≥ n)

= 0.

This gives that almost surely, for all n large enough τn = T0, where τn is as before. Inparticular, τ = T0 and by Fatou’s lemma

‖U‖V∞,rα,a (0,T0;E) ≤ lim inf

n→∞‖Un‖V∞,r

α,a (0,T0;E) ≤ CT0(1 + ‖u0‖Lr(Ω;Ea)).

Via an approximation argument one can check that U is a global mild solution. Thefinal statement in (2) can be obtained as in Corollary 9.5.5.

For the proof of (1) one may repeat the construction from Theorem 9.6.1, usingLemma 9.7.2 instead of Lemma 9.6.2.

9.8 Examples with bounded generator

We start with the case of a bounded operator A (equivalently A = 0).

Example 9.8.1. Let E be a UMD− space with property (α). Consider the equation

dU(t) = F (t, U(t)) dt+B(U(t)) dWE(t), t ∈ [0, T ],

U(0) = u0.(9.8.1)

9.9. Laplacian in Lp 193

Here we assume that F : [0, T ] × Ω × E → E satisfies (A2) with a = θF = 0 andB ∈ B(E,B(E)). Furthermore WE is an E-valued Brownian motion and u0 : Ω → Eis F0-measurable.

Then for every p, α > 0 such that α < 12− 1

pthere exists a unique strong and mild

solution U : [0, T ] × Ω → E of (9.8.1) in V p,locα,a (0, T ;E). Moreover, for every λ < 1

2,

U has a version with paths in Cλ([0, T ];E).

Proof. It is well-known that we can find a separable Hilbert spaceH be and an operatori ∈ γ(H,E) such that 〈WE, x

∗〉 = WHi∗x∗ (cf. [100, Example 3.2]). Clearly, A = 0 on

E satisfies (A1). Let B ∈ B(E, γ(H,E)) be given by B(x)h = B(x)ih.Via property (α) one may check that B satisfies (A3) with a = θB = 0. Therefore,

the result follows from Theorem 9.6.1 and Corollary 9.5.5.

If B is an L2γ-Lipschitz function then one does not require property (α), then the

result is true under different assumptions on the space E as well. Namely the followingholds:

Remark 9.8.2. Assume E is a UMD− space with non-trivial type and that B is anL2γ-Lipschitz function. Then the result of Example 9.8.1 holds.

Proof. This may be proved in the same way as before, but now using Lemma 9.3.7instead of Lemma 9.3.5 in the proof of Lemma 9.5.2 (also see Remark 9.5.4).

9.9 Laplacian in Lp

Let S be an open subset (not necessarily bounded) of Rd. Consider the followingperturbed heat equation with Dirichlet boundary values:

∂u

∂t(t, s) = ∆u(t, s) + f(t, s, u(t, s))

+∑n≥1

bn(t, s, u(t, s))∂Wn(t)

∂t, s ∈ S, t ∈ (0, T ],

u(t, s) = 0, s ∈ ∂S, t ∈ (0, T ], (9.9.1)

u(0, s) = u0(s), s ∈ S.

The functions f, bn : [0, T ]×Ω× S ×R → R for n ≥ 1 are jointly measurable, andadapted in the sense that for each t ∈ [0, T ], f(t, ·), g(t, ·) is Ft⊗BS ⊗BR-measurable.We assume that (Wn)n≥1 is a sequence of independent standard Brownian motionsand u0 : S × Ω → R is a BS ⊗ F0-measurable initial value condition. We say thatu : [0, T ] × Ω × S → R is a mild solution of (9.9.1) if the corresponding functionalanalytic model (SE) has a mild solution.

Let p ∈ [1,∞) be fixed and let E = Lp(S). It is well-known that the DirichletLaplacian ∆p generates an exponentially stable and analytic C0-semigroup (Sp(t))t≥0


on Lp(S) and under regularity assumption on S one can identify D(∆p) as W 2,p(S) ∩W 1,p

0 (S). Consider the following p-dependent condition:

(C) There exist constants Lf , Lbn and Cbn such that

|f(t, ω, s, x)− f(t, ω, s, y)| ≤ Lf |x− y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and x, y ∈ R,

|bn(t, ω, s, x)−bn(t, ω, s, y)| ≤ Lbn|x−y|, t ∈ [0, T ], ω ∈ Ω, s ∈ S and x, y ∈ R.

The function f satisfies the following boundedness condition:

ess. sup ‖s 7→ f(t, ω, s, 0)‖Lp(S) <∞,

where the essential supremum is taken over all t ∈ [0, T ] and ω ∈ Ω.

The functions (bn)n≥1 satisfy the following boundedness condition: for all finitemeasures µ on [0, T ],

ess. sup∥∥∥s 7→ ( ∫ T

0

∑n≥1

|bn(t, ω, s, 0)|2 dµ(t)) 1

2∥∥∥Lp(S)

<∞,

where the essential supremum is taken over all ω ∈ Ω.

Of course it suffices to consider the measures µt,α from Lemma 9.5.1 in this condi-tion.

Theorem 9.9.1. Let S be an open subset of Rd and let p ∈ [1,∞). Assume thatcondition (C) holds and

∑n≥1 L

2bn< ∞ and

∑n≥1C

2bn< ∞. If a.s. ‖u0‖Lp(S) < ∞,

then for all α ∈ (0, 12) and r ∈ (2,∞) such that α < 1

2− 1

r, (9.9.1) has a unique mild

solution u ∈ V r,locα,0 (0, T ;Lp(S)). Moreover, for all λ + δ < 1

2, there is a version of U

such that for almost all ω ∈ Ω,

t 7→ u(t, ω)− Sp(t)u0(ω) ∈ Cλ([0, T ], (Lp(S), D(∆p))δ,1).

Remark 9.9.2. Under regularity conditions on S and for p ∈ (1,∞) one has

Lp(S), D(∆p))δ,1 = x ∈ B2δp,1(S) : x = 0 on ∂S if 2δ − 1

p> 0.

for all δ ∈ (0, 1) such that 2δ − 1p6= 0.

Proof. We check the conditions of Theorem 9.6.1. It is already noted that (A1) isfulfilled. Let F : E → E be defined as F (t, x)(s) = f(t, s, x(s)). Then one easily checksthat F satisfies (A2) with θF = a = 0. Let H = l2 and let B : [0, T ]×Ω×E → B(H,E)

9.10. Second order equation with colored noise 195

be defined as (B(t, ω, x)en)(s) = bn(t, ω, s, x(s)). Then for all Borel measures µ on[0, T ], φ1, φ2 ∈ γ(0, T, µ;H,E),

‖B(·, φ1)−B(·, φ2)‖γ(0,T,µ;H,E)

hp

∥∥∥s 7→ ( ∫ T

0

∑n≥1

|bn(t, s, φ1(t)(s))− bn(t, s, φ2(t)(s))|2 dµ(t)) 1

2∥∥∥E

≤∥∥∥s 7→ ( ∫ T

0

∑n≥1

L2bn|φ1(t)(s)− φ2(t)(s)|2 dµ(t)

) 12∥∥∥E

hp L‖φ1 − φ2‖γ(0,T,µ;E),

where L =( ∑

n≥1 L2bn

) 12. Moreover,

‖B(·, 0)‖γ(0,T,µ;H,E)) hp

∥∥∥s 7→ ( ∫ T

0

∑n≥1

|bn(t, s, 0)|2 dµ(t)) 1

2∥∥∥E<∞.

From these two estimates one can obtain (A3).

9.10 Second order equation with colored noise

In this Section we return to the problem (8.6.1) from Section 8.6. Only this time weassume that A doe not depend on time.

Theorem 9.10.1. Let p ∈ (1,∞) be such that n < p and let a ∈ [ n2p, 1

2). Under the

above conditions for each u0 which is in (Lp(S), D(Ap))a,1 a.s. and α ∈ (0, 12) and r ∈

(2,∞) such that α < 12− 1

r, there exists a unique mild solution u ∈ V r,loc

α,a (0, T ;Lp(S))of (8.6.1) and it has paths in C([0, T ]; (Lp(S), D(Ap))a,1).

If u0 ∈ (Lp(S), D(Ap)) 12,1 a.s. then for all λ, δ ≥ 0 such that δ + a + λ < 1

2, u has

a version with paths in Cλ([0, T ]; (Lp(S), D(Ap))a+δ,1). Moreover, for all r ∈ (2,∞)such that δ + a+ λ < 1

2− 1

rthere exists a constant C not depending on u0, such that

E‖u‖rCλ([0,T ];(Lp(S),D(Ap))a+δ,1) ≤ C(1 + E‖u0‖r(Lp(S),D(Ap)) 12 ,1

).

Proof. One can proceed as in Theorem 8.6.1, but this time we use Theorem 9.7.1. Bythe choice of a we still have (Lp(S), Dp)a,1 → C(S).

We only have to check that B is locally L2γ-Lipschitz and of linear growth. First

we check B is locally L2γ-Lipschitz. Let C be the norm of the Sobolev embedding

(Lp(S), Dp)a,1 → C(S). For n ≥ 1, let bn : R → R be defined as bn(ξ)) = b(ξ) if|ξ| ≤ Cn and bn(ξ) = b(ξ/|ξ|) otherwise. For n ≥ 1, let Bn : Ea → γ(H,E) bedefined as (Bn(x)h)(s) = (ih)(s)bn(x(s)). Then Bn(x) = B(x) for all ‖x‖Ea ≤ n.


Moreover, each Bn is L2γ-Lipschitz. For for all finite measures µ on ([0, T ],B[0,T ]), and

φ1, φ2 ∈ L2γ(0, T, µ;Ea), we have

‖(Bn(·, ω, φ1)−Bn(·, ω, φ2)‖γ(0,T,µ;H,E)

(i)hp

∥∥∥( ∑m≥1

∫ T

0

|(bn(t, ω, ·, φ1)− bn(t, ω, ·, φ2))ihm|2) 1

2dµ(t)

∥∥∥E

≤∥∥∥( ∑

m≥1

|ihm|2) 1

2∥∥∥E

∥∥∥( ∫ T

0

|(bn(t, ω, ·, φ1)− bn(t, ω, ·, φ2))|2) 1

2dµ(t)

∥∥∥L∞(S)

(ii)

.p Lbn‖i‖γ(H,E)

∥∥∥( ∫ T

0

|φ1 − φ2|2) 1

2dµ(t)

∥∥∥L∞(S)

(iii)

. Lbn‖i‖γ(H,E)‖φ1 − φ2‖γ(0,T,µ;L∞(S)) .p Lbn‖i‖γ(H,E)‖φ1 − φ2‖γ(0,T,µ;Ea).

For (i) one can use the Kahane-Khinchine inequalities as in Corollary 3.4.5. (ii) alsofollows from the Kahane-Khinchine inequalities. The inequality in (iii) is similar to aone-sided inequality used in Corollary 3.4.5 which is valid for arbitrary Banach functionspaces.

In the same way one can show that B is of linear growth.

9.11 Elliptic equations with space-time white noise

We consider the equation from Section 8.7 again. However, we assume that A doesnot dependent on time. Below we show that the equations can be solved in Lp forp ∈ (1, 2) as well.

Theorem 9.11.1. Assume dm< 2. Assume (C1) from Section 8.7 and let p ∈ (1,∞)

be such that d2mp

< 12− d

4m. Assume that A does not depend on time.

(1) If a ∈ [ d2mp

, 12− d

4m) is such that (8.7.2) holds for (a, p), and for almost all ω ∈ Ω,

u0(·, ω) ∈ B2map,1,Bj(S), then for all r ∈ (2,∞) and α ∈ (a+ d

4m, 1

2− 1

r) there exists

a unique maximal mild solution (u(t))t∈[0,τ) of (8.7.1) in V r,admα,a (0, τ ;Lp(S)) such

that almost surely t 7→ u(t, ·) is continuous as an B2map,1,Bj(S)-valued process.


2

p,1,Bj(S), then almost surely

u has paths in Cλloc([0, τ);B

2mδp,1,Bj(S)) for all δ > d

2mpand λ > 0 that satisfy

δ + λ < 12− d

4mand (8.7.2) for (δ, p).

Furthermore, if condition (C2) holds as well, then

(3) If a ∈ [ d2mp

, 12− d

4m) is such that (8.7.2) holds (a, p) and for almost all ω ∈ Ω,

u0(·, ω) ∈ B2map,1,Bj(S), then for all r ∈ (2,∞) and α ∈ (a+ d

4m, 1

2− 1

r) there exists

a unique global mild solution process (u(t))t∈[0,T ] of (8.7.1) in V r,locα,a ((0, T ;Lp(S)))


such that almost surely t 7→ u(t, ·) is continuous as an B2map,1,Bj(S))-valued

process.


2

p,1,Bj(S), then almost surely

u has paths in Cλ([0, T ];B2mδp,1Bj(S)) for all δ > d

2mpand λ > 0 that satisfy

δ + λ < 12− d

4mand (8.7.2) for (δ, p).

Proof of Theorem 9.11.1. We may argue in the same way as in Theorem 8.7.1, butnow using Theorem 9.7.1.

We only have to show that B is L2γ-Lipschitz and of linear growth in the case that

p ∈ (1, 2). This may be deduced from the case that p = 2. Indeed, for each n define(−Ap)−θBBn : [0, T ] × Ω × Ea → γ(H,E) as (−Ap)−θBBn(x) = (−Ap)−θBB(x) for all‖x‖Ea ≤ n and (−Ap)−θBBn(x) = (−Ap)−θBBn

(nx‖x‖

)otherwise. Define (−A2)

−θBB∞n :

[0, T ] × Ω × L∞(S) → γ(H,H) as (−A2)−θBB∞

n (x) = (−A2)−θBBn(x). Replacing E

with L2(S) in the above calculation it follows that B∞n is a Lipschitz function uniformly

on [0, T ] × Ω. Since H has type 2, (−A2)−θBB∞

n is L2γ-Lipschitz. Fix a finite Borel

measure µ on (0, T ) and fix φ1, φ2 ∈ L2γ(0, T, µ, Ea). Since H → E, it follows that

‖(−Ap)−θB(Bn(t, ω, φ1)−Bn(t, ω, φ2))‖γ(0,T,µ;H,E)

.S,µ,p,θB‖(−A2)

−θB(B∞n (t, ω, φ1)−B∞

n (t, ω, φ2))‖γ(0,T,µ;H,H)

.θBCn,B‖φ1 − φ2‖L2(0,T,µ,L∞(S))

.p,S,a Cn,B‖φ1 − φ2‖L2(0,T,µ,Ea).

Similarly, one can show that B has linear growth in the sense (9.5.2) if g has lineargrowth.


The definition of Λsp,q(I;E) is taken from [69, Section 3.b] by Konig. There he defines

Λsp,q(S;E) for general domains S ⊂ Rd and s > 0 and he calls this the vector-valued

Besov space. One may also define the vector-valued Besov space Bsp,q(S;E) by tak-

ing restrictions in the definition in Section 2.7. In many situations Λsp,q(S;E) and

Bsp,q(S;E) coincide. For S = Rd this has been proved in [109, Proposition 3.1] by

Pelczynski and Wojciechowski and in [120, Theorem 4.3.3] by Schmeisser in aslightly different setting. In the case that E = R the definitions coincide if S is smoothenough (cf. [126]). We do not know what happens in general.

In the case that δ = 1 and θ = 0, Lemma 9.2.3 is a maximal regularity result forBesov spaces which might be well-known to experts. Together with the embeddingresults from Section 3.3 it allows us to estimate a weighted γ-norm of convolutions interms of a Bochner norm.

Theorem 9.3.2 is a generalization of Theorem 8.2.1 to UMD− Banach spaces in theautonomous case. In Corollary 9.3.4 we specialized the result in the case that E has


type q ∈ [1, 2). There are versions of the results for evolution families. To obtain suchresults one needs R-bounded properties of evolution operators.

Property (α) has been introduced by Pisier in [113]. He shows that every Banachfunction space with finite cotype satisfies property (α). It is independent of the UMDproperty, because the Schatten classes Cp for p ∈ (1,∞) are UMD spaces withoutproperty (α) (if p 6= 2) and L1(0, 1) has property (α) and is not a UMD space. It isalso independent of the UMD− property, because Garling was able to construct aBanach function space which is not UMD−, but has finite cotype (see [51]). The one-sided properties (α−) and (α+) were introduced in [99] by van Neerven and Weis.They show that these properties are related to the boundedness of the embeddings

γ(L2(0, T ;H), E) → γ(L2(0, T ), γ(H,E)))

andγ(L2(0, T ), γ(H,E))) → γ(L2(0, T ;H), E).

It is also noted in [99] that Cp satisfies (α−) for p ∈ [2,∞) and (α+) for p ∈ [1, 2].It can be difficult to check whether a function is an L2

γ-Lipschitz function. However,in some case it is possible to calculate the γ-norm (see Corollary 3.4.5). In type 2 spacesa function f : E1 → γ(H,E2) is Lipschitz if and only if it is L2

γ-Lipschitz. This is nolonger true if f depends on S as well, where (S,Σ, µ) is some finite measure space. In[93] a slightly stronger Lipschitz condition has been considered. Let us call a functionf : E1 → γ(H,E2) γ-Lipschitz if it extends to a Lipschitz function from γ(S;E1) toγ(S;H,E2) for all finite measure spaces (S,Σ, µ). The difference with L2

γ-Lipschitz isthat there we considered γ(S;E1) ∩ L2(S;E1). In [93, Theorem 1] the following hasbeen proved.

Theorem 9.12.1. Assume that E1 is infinite-dimensional and dim(H) ≥ 1. EveryLipschitz function f : E1 → γ(H,E2) is γ-Lipschitz if and only if E1 has cotype 2 andE2 has type 2.

The proof of this result is based on the Dvoretzky theorem which says that l2 isfinitely representable in every infinite-dimensional Banach space. One can still applythe proof of the result to obtain the following result for L2

γ-Lipschitz functions.

Theorem 9.12.2. Assume that E1 is infinite-dimensional and dim(H) ≥ 1. EveryLipschitz function f : E1 → γ(H,E2) is L2

γ-Lipschitz if and only if E2 has type 2.

Comparing the methods in Section 9.5 with those in Section 8.3, the main difficultythat arises is that the fixed point space is more complicated. The reason for this is thatthe γ-norm is needed for the stochastic convolution. This was not needed in Section8.3, because in type 2 the embedding of Proposition 3.3.1 holds. However, there aresituations where even in type 2 spaces one needs γ-spaces. Such a situation can occurfor instance in the example in Section 9.9. With the methods of Chapter 8 problemscould occur if bn(t, ω, s, 0) is not regular enough.

Chapter 10

Strong solutions

10.1 Introduction

Let (E, ‖ · ‖) be a UMD Banach space. Let (Ω,A,P) is a complete probability spaceand (Wn)

Nn=1 be independent standard Brownian motions with respect to a complete

filtration F = (Ft)t≥0 on (Ω,A,P).In this Chapter we study the following linear equation:


BnU(t) dWn(t), t ∈ [0, T ],

U(0, x) = u0.

(10.1.1)

Here the linear operators A(t) are closed and densely defined on a Banach space E, theoperators (Bn)

Nn=1 are generators of strongly continuous groups on E, and (Wn)

Nn=1 are

real-valued independent F -adapted Brownian motions. The main problem here is that(Bn)

Nn=1 are unbounded with D(A(t)) ⊂

⋂Nn=1D(B2

n) and that we want the solution tobe a strong solution in the sense of Definition 7.5.1. Typically (A(t))t∈[0,T ] is a familyof second order differential operators and the (Bn)n≥1 are first order operators. Theproblem is related to maximal regularity questions for stochastic equations. However,we will follow the method of Da Prato, Iannelli and Tubaro from [34] and DaPrato and Zabczyk from [37, Section 6.5]

We will be working under the Acquistapace-Terreni conditions and under the Tan-abe conditions. The former will be explained briefly in Section 10.2. The latter isexplained in Section 7.2. The main difference beween the two theories is that in theAcquistapace-Terreni theory the domains are allowed to be time-dependent.

In Section 10.3 we will explain the hypothesis we will be working under. Also weextend the definition of strong solutions to half open intervals (0, T ]. In Section 10.4we will use Ito’s formula to reduce (10.1.1) to a deterministic problem. For instancewe apply the formula to the case where f : E∗×E → R is the duality between E andE∗ (see (4.8.4)).

In Section 10.5 we study the deterministic problem under the Acquistapace-Terreniconditions. An example will be given in Section 10.6. In Sections 10.7 and 10.8 we

199

200 Chapter 10. Strong solutions

will consider the deterministic problem under the Tanabe conditions and we apply thecorresponding result to an instance of Zakai’s equation.


10.2 Acquistapace-Terreni conditions

Let (A(t), D(A(t)))t∈[0,T ] be a family of closed and densely defined linear operators ona Banach space E. The Acquistapace-Terreni conditions are:

(AT1) There are constants w ∈ R, K ≥ 0, and ϕ ∈ (π2, π) such that for all t ∈ [0, T ],

Σ(ϕ,w) ⊂ %(A(t)) and for all λ ∈ Σ(ϕ,w) and t ∈ [0, T ],

‖R(λ,A(t))‖ ≤ K

1 + |λ− w|.

(AT2) There exists a k ≥ 1 and constants L ≥ 0, α1, . . . , αk, and β1, . . . , βk ∈ R with0 ≤ βi < αi ≤ 2 such that for all λ ∈ Σ(ϕ,w) and s, t ∈ [0, T ] we have

‖Aw(t)R(λ,A(t))[A−1w (t)− A−1

w (s)]‖ ≤ Lk∑i=1

|t− s|αi(1 + |λ− w|)βi−1.

Below it will be convenient to denote

κ = κ(α1, . . . , αk, β1, . . . , βk) = minαi − βi, i = 1, . . . , k ∈ (0, 1]. (10.2.1)

Clearly, (AT1) is the same as (T1) from Section 7.2. In [4, Section 7] it has beenshown that if the domains are constant, then (T1) and (T2) imply (AT2) with k = 1,α1 = µ and β1 = 0.

Under the assumptions (AT1) and (AT2) the following result holds (see [4, Theorem6.1-6.4] and [1, Theorems 2.3 and 5.2])

Theorem 10.2.1. If conditions (AT1) and (AT2) hold, then there exists a uniquestrongly continuous evolution family (P (t, s))0≤s≤t≤T that solves (7.2.1) on D(A(s))and for all x ∈ E, P (t, s)x is the unique classical solution of (7.2.1). Moreover,(P (t, s))0≤s≤t≤T is continuous on 0 ≤ s < t ≤ T and strongly continuous on 0 ≤ s ≤t ≤ T .

10.3 The abstract problem

On E we consider


BnU(t)dWn(t), t ∈ [0, T ],

U(0) = u0.

(10.3.1)

10.3. The abstract problem 201

The processes Wn = (Wn(t))t∈[0,T ] are independent F -adapted standard Brownianmotions. The initial value u0 : Ω → E is assumed to be strongly F0-measurable.Concerning the operators A(t) : D(A(t)) ⊂ E → E and Bn : D(Bn) ⊂ E → E weassume that the following hypotheses hold.

(H1) The operators A(t) are closed and densely defined.

(H2) The operators Bn generate commuting strongly continuous groups Gn on E.

(H3) For all t ∈ [0, T ] we have D(A(t)) ⊂⋂Nn=1D(B2

n).

Defining D(C(t)) := D(A(t)) and C(t) := A(t)− 12

∑Nn=1B

2n, we further assume that

(H4) There exists a λ ∈ R with λ ∈ %(A(t)) ∩ %(C(t)) for all t ∈ [0, T ], such that thefunctions t 7→ B2

nR(λ,A(t)) and t 7→ B2nR(λ,C(t)) are strongly continuous on

[0, T ].

Hypothesis (H4) is automatically fulfilled in the case that A(t) is independent of t.Below we show that it is fulfilled in several time-dependent situation as well.

Next, we extend the definition of a strong solution from Section 7.5. Let E be aUMD− space. An E-valued process U = U(t)t∈[0,T ] is called an strong solution of(10.3.1) on the interval (0, T ] if U ∈ C([0, T ];E) a.s., U(0) = u0 a.s., and for all ε > 0the following conditions are satisfied:

(1) t 7→ A(t)U(t) is in L0(Ω;L1(ε, t;E))

(2) For n = 1, . . . , N the process BnU is in L0F(Ω; γ(ε, T ;E),

(3) almost surely, we have

U(t) = U(ε) +

∫ t

ε

A(s)U(s) ds+N∑n=1

∫ t

ε

BnU(s) dWn(s) for all t ∈ [ε, T ].

Note that by path continuity, the exceptional sets may be chosen independently ofε ∈ (0, T ]. The case that ε = 0 in (1), (2) and (3) is contained in Definition 7.5.1. Inthis case we will say that U is a strong solution on [0, T ].

Assuming Hypotheses (H1)–(H4), in the Hilbert space setting the existence ofstrong solutions has been established in [34] (see also [37, Section 6.5]) by reducingthe stochastic problem to a deterministic one and then solving the latter by parabolicmethods. Here we shall extend this method to the setting of UMD spaces using thebilinear Ito formula of Section 4.9. However, the formula in Section 4.8 turns out tobe sufficient.


10.4 Reduction to a deterministic problem

Define G : RN → B(E) as

G(a) :=N∏n=1

Gn(an).

Note that each G(a) is invertible with inverse G−1(a) := (G(a))−1 = G(−a). Fort ∈ [0, T ] and ω ∈ Ω we define the operators CW (t, ω) : D(CW (t, ω)) ⊂ E → E by

D(CW (t, ω)) := x ∈ E : G(W (t, ω))x ∈ D(C(t)),CW (t, ω) := G−1(W (t, ω))C(t)G(W (t, ω)),

where W = (W1, . . . ,WN).In terms of the random operators CW (t) we introduce the following pathwise prob-

lem:V ′(t) = CW (t)V (t), t ∈ [0, T ],

V (0) = u0.(10.4.1)

Notice that (10.4.1) is a special case of (10.3.1) with A(t) replaced by CW (t) and withBn = 0. In particular, the notions of strong solutions apply pathwise.

Note that if V is a strong solution of (10.4.1) on (0, T ], then almost surely we haveG(W (t))V (t) ∈ D(C(t)) = D(A(t)) ⊂

⋂Nn=1D(B2

n) for almost all t ∈ [0, T ].

Before the theorem is stated we prove a small lemma which will be needed. Itshows that in Theorem 4.5.9 the condition of being scalarly in L0(Ω;L2(R+;H)) canbe weakened.

Lemma 10.4.1. Let E be a UMD Banach space, let H be a separable Hilbert spaceand let F be a dense subspace of E∗. Let Φ : R+ × Ω → B(H,E) be an H-stronglymeasurable and adapted process such that for all x∗ ∈ F , Φ∗x∗ ∈ L2(R+;H) almostsurely. If there exists process ζ ∈ L0(Ω;Cb(R+;E)) such that for all x∗ ∈ F we have

〈ζ, x∗〉 =

∫ ·

0

Φ∗(s)x∗ dWH(s) in L0(Ω;Cb(R+)), (10.4.2)

then Φ ∈ L0F(Ω; γ(R+;H,E)), so that Φ is stochastically integrable with respect to WH

and

ζ =

∫ ·

0

Φ(s) dWH(s) in L0(Ω;Cb(R+;E)).

Proof. By Theorem 4.5.9, it suffices to show that Φ∗x∗ ∈ L2(R+;H) almost surely andthat (10.4.2) holds for all x∗ ∈ E∗. To do so, fix x∗ ∈ E∗ arbitrary and choose elementsx∗n ∈ F such that x∗ = limn→∞ x∗n in E∗. Clearly we have 〈ζ, x∗〉 = limn→∞〈ζ, x∗n〉 inL0(Ω;Cb(R+)). An application of [63, Proposition 17.6] shows that the processes Φ∗x∗ndefine a Cauchy sequence in L0(Ω;L2(R+;H)). By a standard argument we obtain

10.4. Reduction to a deterministic problem 203

that Φ∗x∗ ∈ L0(Ω;L2(R+;H)) and limn→∞ Φ∗x∗n = Φ∗x∗ in L0(Ω;L2(R+;H)). Byanother application of [63, Proposition 17.6] we conclude that∫ ·

0

Φ∗(s)x∗ dWH(s) = limn→∞

∫ ·

0

Φ∗(s)x∗n dWH(s) = limn→∞

〈ζ, x∗n〉 = 〈ζ, x∗〉

in L0(Ω;Cb(R+)).

The next theorem relates the problems (10.3.1) and (10.4.1).

Theorem 10.4.2. Let E be a UMD Banach space and assume (H1)–(H4). For astrongly measurable and adapted process V : [0, T ] × Ω → E the following assertionsare equivalent:

(1) G(W )V is a strong solution of (10.3.1) on (0, T ] (resp. on [0,T]);

(2) V is a strong solution of (10.4.1) on (0, T ] (resp. on [0,T]).

Proof. First we claim that⋂Nn=1D(B∗2

n ) is norm-dense in E∗. Since E is reflexive it is

sufficient to prove that⋂Nn=1D(B∗2

n ) is weak∗-dense in E∗. Fix an x ∈ E, x 6= 0, and

some λ ∈⋂Nn=1 %(Bn), and put y :=

∏Nn=1R(λ,Bn)

2x. Since by (H2) the resolvents

R(λ,Bn) commute we have y ∈⋂Nn=1D(B2

n). Since y 6= 0 we can find y∗ ∈ E∗ such

that 〈y, y∗〉 6= 0. Then x∗ :=∏N

n=1R(λ,B∗n)

2y∗ ∈⋂Nn=1D(B∗2

n ) and it is obvious that〈x, x∗〉 6= 0. This proves the claim.

We will denote

GW (t, ω) := G(W (t, ω)).

Clearly GW and G−1W are pathwise strongly continuous and there exists a nonnegative

random variable M : Ω → R such that

supt∈[0,T ]

‖GW (t)‖ ≤M, supt∈[0,T ]

‖G−1W (t)‖ ≤M. (10.4.3)

We will now turn to the proof of the equivalence of strong solution on (0, T ]. Theequivalence of strong solution on [0, T ] follows by taking ε = 0 in the proofs below.

(1) ⇒ (2): Let ε > 0 be arbitrary. Since U := GWV is a strong solution of(10.3.1) on (0, T ], almost surely we have GW (t)V (t) ∈ D(C(t)) for almost all t ∈ [ε, T ].Moreover for n = 1, . . . , N ,

B2nU(t) = B2

nR(λ,A(t))(λ− A(t))U = B2nR(λ,A(t))λU(t) +B2

nR(λ,A(t))A(t)U(t).

Therefore, (H4) implies that B2nGWV = B2

nU is in L1(ε, T ;E) almost surely. Wemay conclude that t 7→ C(t)GW (t)V (t) belongs to L1(ε, T ;E) almost surely. Hencet 7→ CW (t)V belongs to L1(ε, T ;E) almost surely.

Let x∗ ∈⋂Nn=1D(B∗2

n ) be fixed. By Theorem 4.8.2 (applied to the Banach spaceE∗ and the Hilbert space H = RN and f : RN → E∗ defined by f(a) = G∗(−a)x∗) it


follows that for all n = 1, . . . , N the processes G−1∗W B∗

nx∗ are stochastically integrable

with respect to Wn on [ε, T ] and that almost surely, for all t ∈ [ε, T ],

G−1∗W (t)x∗−G−1∗

W (ε)x∗

= −N∑n=1

∫ t

ε

G−1∗W (s)B∗

nx∗ dWn(s) +

1

2

N∑n=1

∫ t

ε

G−1∗W (s)B2∗

n x∗ ds.

(10.4.4)

By (4.8.4) applied to U and G−1∗W x∗ we obtain that almost surely, for all t ∈ [ε, T ],

〈V (t), x∗〉 − 〈V (ε), x∗〉= 〈U(t), G−1∗

W (t)x∗〉 − 〈U(ε), G−1∗W (ε)x∗〉

=

∫ t

ε

1

2

N∑n=1

〈U(s), G−1∗W (s)B∗2

n x∗〉+ 〈A(s)U(s), G−1∗

W (s)x∗〉 ds

+N∑n=1

∫ t

ε

−〈U(s), G−1∗W (s)B∗

nx∗〉+ 〈BnU,G

−1∗W (s)x∗〉 dWn(s)

−N∑n=1

∫ t

ε

〈BnU(s), G−1∗W (s)B∗

nx∗〉 ds

=

∫ t

ε

〈G−1W (s)C(s)U(s), x∗〉 ds

=

∫ t

ε

〈CW (s)V (s), x∗〉 ds.

Since CWV has paths in L1(ε, T ;E) almost surely, it follows that, almost surely, forall t ∈ [ε, T ],

〈V (t), x∗〉 − 〈V (ε), x∗〉 =⟨ ∫ t

ε

CW (s)V (s) ds, x∗⟩. (10.4.5)

By approximation we may extend (10.4.5) to all x∗ ∈ E∗. By strong measurabilityand the Hahn-Banach theorem, this shows that almost surely, for all t ∈ [ε, T ],

V (t)− V (ε) =

∫ t

ε

CW (s)V (s) ds.

(2) ⇒ (1): Put U := GWV . Let ε > 0 be arbitrary. Since V is a strong solution of(10.4.1) on (0, T ], as before (H4) implies that almost surely we have U(t) ∈ D(A(t))for all t ∈ [0, T ] and t 7→ A(t)U(t) belongs to L1(ε, T ;E).

Let x∗ ∈⋂Nn=1D(B∗2

n ) be fixed. In the same way as before one obtains that for alln = 1, . . . , N the processes G∗

WB∗nx

∗ are stochastically integrable with respect to Wn

on [ε, T ] and almost surely, for all t ∈ [ε, T ], we have

G∗W (t)x∗ −G∗

W (ε)x∗

=N∑n=1

∫ t

ε

G∗W (s)B∗

nx∗ dWn(s) +

1

2

N∑n=1

∫ t

ε

G∗W (s)B∗2

n x∗ ds.

(10.4.6)

10.5. The deterministic problem for time-varying domains 205

By assumption we have CWV ∈ L1(ε, T ;E) almost surely. Hence we may apply (4.8.4)with V and G∗

Wx∗. It follows that almost surely, for all t ∈ [ε, T ], we have

〈U(t), x∗〉 − 〈U(ε), x∗〉= 〈V (t), G∗

W (t)x∗〉 − 〈V (ε), G∗W (t)x∗〉

=

∫ t

ε

1

2

N∑n=1

〈V (s), G∗W (s)B∗2

n x∗〉+ 〈G−1

W (s)C(s)GW (s)V (s), G∗W (s)x∗〉 ds

+N∑n=1

∫ t

ε

〈V (s), G∗W (s)B∗

nx∗〉 dWn(s)

=

∫ t

ε

〈A(s)GW (s)V (s), x∗〉 ds+N∑n=1

∫ t

ε

〈BnGW (s)V (s), x∗〉 dWn(s)

=

∫ t

ε

〈A(s)U(s), x∗〉 ds+N∑n=1

∫ t

ε

〈BnU(s), x∗〉 dWn(s).

Since G−1W CU = CWV ∈ L1(ε, T ;E) almost surely, we have CU ∈ L1(ε, T ;E) almost

surely, and therefore by (H4) we also have AU ∈ L1(ε, T ;E) almost surely. Also, Vhas continuous paths almost surely, and therefore the same is true for U = GWV .Thanks to the claim we are now in a position to apply Lemma 10.4.1 on the interval[ε, T ] (for the Hilbert space H = RN and the process ζ = U −U(ε)−

∫ ·εA(s)U(s) ds).

We obtain that the processes BnU are stochastically integrable with respect to Wn on[ε, T ] and that almost surely, for all t ∈ [ε, T ], we have

U(t)− U(ε) =

∫ t

ε

A(s)U(s) ds+N∑n=1

∫ t

ε

BnU(s) dWn(s). (10.4.7)

10.5 The deterministic problem for time-varying

domains

Recall the non-autonomous Cauchy problem from section 7.2:

du

dt(t) = C(t)u(t) t ∈ [0, T ],

u(0) = x,(10.5.1)

where C(t) : D(C(t)) ⊂ E → E are closed operators. We study this equation assumingthe Acquistapace-Terreni conditions (see Section 7.2).

Assuming Hypothesis (H2), we study the problem

du

dt(t) = Ch(t)u(t) t ∈ [0, T ],

u(0) = x.(10.5.2)


Here Ch(t) = G−1(h(t))C(t)G(h(t)), with D(Ch(t)) = x ∈ E : G(h(t))x ∈ D(C(t)),G is as in Section 10.3, and h : [0, T ] → RN is a measurable function. Notice thatpathwise, (10.4.1) may be seen as the special case of (10.5.2), where C = C and h = W .

The following condition has been introduced in [37, Theorem 6.30] (see also [34,Proposition 1]) in the time independent case. Let (C(t))t∈[0,T ] be densely defined andsuch that 0 ∈ ρ(C(t)) for all t ∈ [0, T ]. Assuming Hypothesis (H2) we consider thefollowing Hypothesis (K) (which may be weakened somewhat (cf. [3, Remark 1.2])).

(K) We have 0 ∈ %(C(t)) for all t ∈ [0, T ] and there exist uniformly bounded functionsKn : [0, T ] → B(E) such that for all t ∈ [0, T ], all n = 1, . . . , N , and allx ∈ D(Bn) we have BnC−1(t)x ∈ D(C(t)) and

C(t)BnC−1(t)x = Bnx+Kn(t)x.

The latter may be rewritten as the commutator condition:

[C(t), Bn]C−1(t)x = Kn(t)x.

In many cases it is enough to consider only x ∈ D(C(t)) instead of x ∈ D(Bn) (cf. [3,Proposition A.1]).

Assume that (AT1) and (AT2) hold for the operators C(t). If (K) holds for theoperators C(t), then the uniform boundedness of t 7→ R(λ, C(t)) can be used to checkthat for all λ > 0, (K) holds for the operators C(t)− λ for all λ > 0.

The following lemma lists some consequences of Hypothesis (K).

Lemma 10.5.1. Let (C(t))t∈[0,T ] be closed densely defined operators such that 0 ∈ρ(C(t)) for all t ∈ [0, T ]. Assume Hypotheses (H2) and (K).

(1) For all n = 1, . . . , N , s ∈ R and t ∈ [0, T ], Gn(s) leaves D(C(t)) invariant and

C(t)Gn(s)C−1(t) = es(Bn+Kn(t)).

(2) For all R ≥ 0 there is a constant MR ≥ 0 such that for all n = 1, . . . , N , |s| ≤ Rand t ∈ [0, T ] we have

‖C(t)Gn(s)C−1(t)−Gn(s)‖ ≤MR|s|.

Proof. The first assertion follows from the proof of [34, Proposition 1] and the secondfrom a standard perturbation result (cf. [47, Corollary III.1.11]).

The following perturbation type result for operators satisfying (AT1) and (AT2)gives a way to relate the problems (10.5.1) and (10.5.2).


Proposition 10.5.2. Let (C(t))t∈[0,T ] be closed densely defined operators such that0 ∈ ρ(C(t)) for all t ∈ [0, T ]. Assume Hypotheses (H2) and (K). Let h : [0, T ] →RN be Holder continuous with parameter α ∈ (0, 1] and define the similar operators(Ch(t))t∈[0,T ] as

Ch(t) = G−1(h(t))C(t)G(h(t)) with D(Ch(t)) = x ∈ E : G(h(t))x ∈ D(C(t)).

If the operators C(t) satisfy (AT1) and (AT2) on Σ(ϕ, 0) with (αi, βi)ki=1, then the

operators Ch(t) satisfy (AT1) and (AT2) on Σ(ϕ, 0) with (αi, βi)k+1i=1 , where αk+1 = α

and βk+1 = 0.

Proof. We denote Gh(t) = G(h(t)). For all t ∈ [0, T ] and λ ∈ %(C(t)) we clearly haveλ ∈ %(Ch(t)) and R(λ, Ch(t)) = G−1

h (t)R(λ, C(t))Gh(t). By the strong continuity of Git follows that for all t ∈ [0, T ] and λ ∈ Σ(ϕ, 0),

‖R(λ, Ch(t))‖ ≤M2‖R(λ, C(t))‖,

where M = supG(±h(t)) : t ∈ [0, T ]. Hence (Ch(t))t∈[0,T ] satisfies (AT1).Next we check (AT2). Fix t, s ∈ [0, T ] and λ ∈ Σ(ϕ, 0). We have

‖Ch(t)R(λ, Ch(t))[C−1h (t)− C−1

h (s)]‖= ‖G−1

h (t)C(t)R(λ, C(t))[C−1(t)Gh(t)−Gh(t)G−1h (s)C−1(s)Gh(s)]‖

≤M‖C(t)R(λ, C(t))[C−1(t)Gh(t)− C−1(s)Gh(t)]‖+M‖C(t)R(λ, C(t))[C−1(s)Gh(t)−Gh(t)G

−1h (s)C−1(s)Gh(s)]‖.

We estimate the two terms on the right-hand side separately. Since (C(t))t∈[0,T ] satisfies(AT2), it follows for the first term that

‖C(t)R(λ, C(t))[C−1(t)Gh(t)− C−1(s)Gh(t)]‖≤M‖C(t)R(λ, C(t))[C−1(t)− C−1(s)]‖

≤ML

k∑i=1

|t− s|αi|λ|βi−1.

(10.5.3)

For the second term we have

‖C(t)R(λ, C(t))[C−1(s)Gh(t)−Gh(t)G−1h (s)C−1(s)Gh(s)]‖

≤M‖C(t)R(λ, C(t))C−1(s)[Gh(t)G−1h (s)− C(s)Gh(t)G

−1h (s)C−1(s)]‖

= M‖C(t)R(λ, C(t))C−1(s)[G(h(t)− h(s))− C(s)Gh(t)G−1h (s)C−1(s)]‖.

(10.5.4)

By an induction argument and Lemma 10.5.1 as in the proof of [37, Theorem 6.30],the Holder continuity of h implies that,

‖G(h(t)− h(s))− C(s)Gh(t)G−1h (s)C−1(s)‖ ≤MαN |t− s|α, (10.5.5)


where

Mα =N∑n=1

supt,s∈[0,T ]

|hn(t)− hn(s)||t− s|α

.

On the other hand it follows from (AT1) and (AT2) that

‖C(t)R(λ, C(t))C−1(s)‖≤ ‖C(t)R(λ, C(t))[C−1(s)− C−1(t)]‖+ ‖C(t)R(λ, C(t))C−1(t)‖

≤ L

k∑i=1

|t− s|αi|λ|βi−1 +K|λ|−1.

(10.5.6)

Combining (10.5.4), (10.5.5) and (10.5.6) gives

‖C(t)R(λ, C(t))[C−1(s)Gh(t)−Gh(t)G−1h (s)C−1(s)Gh(s)‖

≤MLMαNk∑i=1

|t− s|αi+α|λ|βi−1 +MKMαN |t− s|α|λ|−1.(10.5.7)

We conclude from (10.5.3), (10.5.7), and the trivial estimate |t− s|αi+α ≤ CT |t− s|αi

that ∥∥Ch(t)R(λ, Ch(t))[C−1h (s)− C−1

h (t)]∥∥ ≤ L

k+1∑i=1

|t− s|αi|λ|βi−1

for a certain constant L and αk+1 = α, βk+1 = 0.

We can now formulate and prove the main result of this section.

Theorem 10.5.3. Let E be a UMD Banach space and assume that Hypotheses (H1),(H2), (H3) and (H4) are fulfilled and that (AT1), (AT2) and (K) are satisfied forC(t)− µ for all µ ∈ R large enough.

(1) The problem (10.3.1) admits a unique strong solution U on (0, T ] for whichAU ∈ C((0, T ];E) almost surely.

(2) If u0 ∈ (E,D(A(0)))1−σ,∞ almost surely, then the problem (10.3.1) admits aunique strong solution U on [0, T ] for which AU ∈ C((0, T ];E) almost surely.Moreover AU ∈ Lp(0, T ;E) for all 1 ≤ p < σ−1.

(3) If u0 ∈ D(A(0)) almost surely, the problem (10.3.1) admits a unique strongsolution U on [0, T ] for which AU ∈ C([0, T ];E) almost surely.

Proof. If Uλ is a solution of (10.3.1) with A(t) replaced by A(t)− λ, then it is easy tosee that t 7→ eλtUλ(t) is a solution of (10.3.1). It follows from this that without lossof generality we may assume that λ = 0 in the assumptions above.

(1): By the standing assumption made in Section 10.3, the initial value u0 is anF0-measurable random variable. By Proposition 10.5.2 and the Holder continuity of


Brownian motion, the operators CW (t) satisfy (AT1) and (AT2). Hence by Theorem10.2.1, almost surely the problem (10.4.1) admits a unique classical solution V . It iseasy to see that V is adapted. Since V has continuous paths almost surely, it followsthat V is strongly measurable. Since continuous functions are integrable, the solutionV is a strong solution on (0, T ]. Hence by Theorem 10.4.2, U = GWV is a strongsolution of (10.3.1) on (0, T ]. The pathwise regularity properties of V carry over toU , thanks to (H4). The pathwise uniqueness of V implies the uniqueness of U againvia Theorem 10.4.2 and (H4).

(2): If u0 ∈ (E,D(A(0)))1−σ,∞ almost surely, then by [4, Theorem 6.3] we haveAV ∈ Lp(0, T ;E) and hence V is a strong solution of (10.4.1) on [0, T ]. Therefore,Theorem 10.4.2 implies that U is a strong solution of (10.3.1) on [0, T ]. The pathwiseregularity properties of V carry over to U as before.

(3): If u0 ∈ D(A(0)) almost surely, then it follows from Theorem 10.2.1 that Vis a strong solution of (10.4.1) on [0, T ], and from Theorem 10.4.2 we see that U isa strong solution of (10.3.1) on [0, T ]. The pathwise regularity properties of V carryover to U as before.

Remark 10.5.4. If (A(t))t∈[0,T ] satisfies (AT1) and (AT2), then under certain condi-tions the perturbation result in [44, Lemma 4.1] may be used to obtain that (C(t))t∈[0,T ]

satisfies (AT1) and (AT2) as well. In particular, this is the case if the (Bn)Nn=1 are

assumed to be bounded.

Remark 10.5.5. Assume that E is reflexive (e.g. E is a UMD space). If the Bn arebounded and commuting and the closed operators A(t) and C(t) satisfy (AT1), (AT2),then (H1) - (H4) are fulfilled.

Proof. It is trivial that (H2) and (H3) are satisfied. For (H1) one may use Kato’sresult (cf. [138, Section VIII.4]) to check the denseness of the domains. For (H4)notice that for λ ∈ R large enough (AT1) and (AT2) imply that t 7→ R(λ,A(t)) andt 7→ R(λ,C(t)) are continuous (cf. [125, Lemma 6.7]). Since Bn are assumed to bebounded this clearly implies (H4).

Remark 10.5.6. Assume that the operators B1, B2, . . . , BN are bounded and commut-ing. Then each etBn is continuously differentiable, so G(W ) is Holder continuous withexponent µ ∈ (0, 1

2). As a consequence, time regularity of the solution V of (10.4.1)

translates in time regularity of the solution U = G(W )V of (10.3.1). We will illustratethis in two ways below.

As in [121, p. 5] it can be seen that if almost surely u0 ∈ D((w−A(0))α for someα ∈ (0, 1], then almost surely V is Holder continuous with parameter α. We concludethat under the condition that almost surely, u0 ∈ D((w−A(0))α) for some α ∈ (0, 1

2),

U is Holder continuous with parameter α.Assume that u0 ∈ D(A(0)) and A(0)u0 ∈ (E,D(A))α,∞ almost surely for some

α ∈ (0, κ] ∩ (0, 1), where κ is as in (10.2.1). Then we deduce from [4, Theorem 6.2]that almost surely, CWV has paths in Cα([0, T ];E). If α < 1

2, then we readily obtain,

almost surely, AU has paths in Cα([0, T ];E).


10.6 Second order equation with time-dependent

domains

We consider the problem

Dtu(t, x) = A(t, x,D)u(t, x) +N∑n=1

Bn(x)u(t, x)DtWn(t), t ∈ [0, T ], x ∈ S

V (t, x,D)u = 0, t ∈ [0, T ], x ∈ ∂S,u(0, x) = u0(x), x ∈ S.

(10.6.1)Here

A(t, x,D) =d∑

i,j=1


qi(t, x)Di + r(t, x), Bn(x) = bn(x),

and

V (t, x) =d∑i=1

vi(t, x,D)Di + v0(t, x).

The set S ⊂ Rd is a bounded domain with boundary of class C2 being locally on oneside of S and outer unit normal vector n(x). We assume that ∂S consists of two closed(possibly empty) disjoint subsets Γ0 and Γ1. Moreover the coefficients are real andaij, qi, r ∈ Cα([0, T ], C(S)), where α ∈ (1

2, 1) if Γ1 6= ∅ and α ∈ (0, 1) if Γ1 = ∅ and

the matrix (aij(·, x))i,j is symmetric and strictly positive definite uniformly in time,i.e. there exists an ν > 0 such that for all t ∈ [0, T ] we have

d∑i,j=1

aij(t, x)λiλj ≥ νd∑i=1

λ2i , x ∈ S, λ ∈ Rd.

The boundary coefficients are assumed to be real and vi, v0 ∈ Cα([0, T ], C1(∂S)),v0 = 1 and vi = 0 on Γ0 and there is a constant δ > 0 such that for all x ∈ Γ1 andt ∈ [0, T ] we have

∑di=1 vi(t, x)ni(x) ≥ δ. Finally we assume that bn ∈ C2(S) and

d∑i=1

vi(t, x)Dibn(x) = 0, t ∈ [0, T ], x ∈ ∂S. (10.6.2)

We say that u : [0, T ]×Ω×S → R is a strong solution of (10.6.1) if the correspondingfunctional analytic model (10.3.1) has a strong solution.

Theorem 10.6.1. Let p ∈ (1,∞) and u0 ∈ L0(Ω;F0;Lp(S)). Under the assump-

tions, there exists a unique strong solution u of (10.6.1) on (0, T ] for which Au ∈C((0, T ];Lp(S)) almost surely.

10.7. The deterministic problem for constant domains 211

Furthermore, if almost surely we have u0 ∈ W 2,p(S) and

V (0, x)u0 = 0 x ∈ ∂S,

then there exists a unique strong solution u of (10.6.1) on [0, T ] for which Au ∈C([0, T ];Lp(S)) almost surely.

Notice that Remark 10.5.6 can be used to obtain time regularity of u and Au undersuitable conditions on u0.

Proof. We check the conditions in Theorem 10.5.3. In [121] it has been shown that(AT1) and (AT2) hold for (A(t))t∈[0,T ] and (C(t))t∈[0,T ], with k = 1 and coefficientsα1 = α and β1 = 1

2in case Γ1 6= ∅ and β1 = 0 in case Γ1 = ∅. Since the operators Bn

are bounded, Remark 10.5.5 applies and we conclude that (H1)–(H4) hold.Let λ ∈ R large enough be fixed. The only thing that is left to be checked is

condition (K) for the operators C(t) − λ. It follows from (10.6.2) that for all x ∈ E,BnR(λ,C(t))x ∈ D(C(t)). For n = 1, 2, . . . , N and t ∈ [0, T ] define

Kn(t) = (C(t)− λ)Bn(C(t)− λ)−1 −Bn.

One can check that Kn(t) = [C(t), Bn]R(λ,C(t)), where [C(t), Bn] is the commutatorof C(t) and Bn. Since [C(t), Bn] is a first order operator, each Kn(t) is a boundedoperator. To prove their uniform boundedness in t, we note that from the assumptionson the coefficients it follows that there are constants C1, C2 > 0 such that for allt ∈ [0, T ] and j = 1, . . . , d,

‖R(λ,C(t))‖ ≤ C1 and ‖DjR(λ,C(t))‖ ≤ C2.

Indeed, the first estimate is obviously true, and the second one follows from the Agmon-Douglis-Nirenberg estimates (see [6]).

10.7 The deterministic problem for constant do-

mains

In this section we will consider the Tanabe conditions (T1) and (T2) (see Section 7.2).It is clear that under conditions (H1) and (H3), the operators A(t) satisfy (T2) if

and only if the operators C(t) satisfy (T2).

Lemma 10.7.1. Assume (H1), (H3) and that (A(t))t∈[0,T ] and that (C(t))t∈[0,T ] satisfythe Tanabe conditions, then (H4) holds.

Proof. SinceD(A(0)) ⊂ D(B2n), there is a constant Cn such that ‖B2

nx‖ ≤ Cn‖Aw(0)x‖for all x ∈ D(A(0)). It follows from the uniform boundedness of Aw(0)A−1

w (t) : t ∈[0, T ] and (7.2.2) that for all t, s ∈ [0, T ] we have

‖B2nA

−1w (t)−B2

nA−1w (s)‖ ≤ Cn‖Aw(0)A−1

w (t)− Aw(0)A−1w (s)‖

≤ CnC‖Aw(t)A−1w (t)− Aw(t)A−1

w (s)‖≤ CnC‖Aw(s)A−1

w (s)− Aw(t)A−1w (s)‖ ≤ CnCL|t− s|µ.


This shows that t 7→ B2nA

−1w (t) is Holder continuous. In the same way one can show

that t 7→ B2nC

−1w (t) is Holder continuous. We conclude that (H4) holds.

In the same way as in Proposition 10.5.2, one can show that if C satisfies (T1)and (T2) with parameter µ ∈ (0, 1], then Ch satisfies (T1) and (T2) with parameterµ ∧ α. Thus in the case where the domains D(A(t)) are constant, the more difficultAcquistapace-Terreni theory is not needed.

If the operators B1, . . . , BN are bounded we have the following consequence ofTheorem 10.5.3. Note that the assumptions are made on the operators A(t) ratherthan on C(t).

Proposition 10.7.2. Let E be a UMD space and D(A(t) = D(A(0)) for all t ∈ [0, T ].Assume that (A(t))t∈[0,T ] satisfies (T1) and (T2) and let B1, . . . , BN ∈ B(E) be boundedcommuting operators which leave D(A(0)) invariant. Consider the problem

dU(t) = A(t)U(t) dt+N∑n=1

BnU(t)dWn(t), t ∈ [0, T ],

U(0) = u0.

(10.7.1)

(1) If u0 ∈ E almost surely, the problem (10.7.1) admits a unique strong solutionU ∈ C([0, T ];E) on (0, T ] for which AU ∈ C((0, T ];E).

(2) If u0 ∈ (E,D(A(0)))1−σ,∞ almost surely, then the problem (10.3.1) admits aunique strong solution U ∈ C([0, T ];E) on [0, T ] with AU ∈ C((0, T ];E). More-over AU ∈ Lp(0, T ;E) for all 1 ≤ p < σ−1.

(3) If u0 ∈ D(A) almost surely, the problem (10.3.1) admits a unique strong solutionU ∈ C([0, T ];E) on [0, T ] for which AU ∈ C([0, T ];E).

Proof. We check the conditions of Theorem 10.5.3. It follows from Remark 10.5.5 that(H1), (H2) and (H3) are satisfied. Lemma 10.7.1 implies that (H4) is satisfied.

By the bounded perturbation theorem (cf. [47]), the operators C(t) = A(t) −12

∑Nn=1B

2n satisfy (T1). Hence condition (T2) for (C(t))t∈[0,T ] follows from (T2) for

the operators (A(t))t∈[0,T ].Finally to check (K), by the assumption on the operators Bn we have D(A(0)) =

D(C(0)), and by the closed graph theorem we have ‖Bnx‖D(C(0)) ≤ cn‖x‖D(C(0)) forsome constant cn. This implies that ‖C(0)Bnx‖ ≤ cn‖Cw(0)x‖. We check that theoperators Kn(t) = Cw(t)BnC

−1w (t)−Bn are uniformly bounded. By the remark follow-

ing (7.2.2), the family Cw(0)C−1w (t) : t ∈ [0, T ] is uniformly bounded, say by some

constant k, and therefore

‖Cw(t)BnC−1w (t)‖ ≤ ‖Cw(t)C−1

w (0)Cw(0)BnC−1w (0)Cw(0)C−1

w (t)‖≤ k2‖Cw(0)BnC

−1w (0)‖ ≤ cn.

10.8. Zakai’s equation 213

10.8 Zakai’s equation

We consider the problem

Dtu(t, x) = A(t, x,D)u(t, x) +B(x,D)u(t, x)DtW (t), t ∈ [0, T ], x ∈ Rd

u(0, x) = u0(x), x ∈ Rd.(10.8.1)

Here

A(t, x,D) =d∑

i,j=1


qi(t, x)Di + r(t, x),

B(x,D) =d∑i=1

bi(x)Di + c(x).

All coefficients are real-valued and we take aij, qi, r ∈ B([0, T ];C1b (R

d)), where Bstands for bounded. The coefficients aij, qi and r are µ-Holder continuous in time forsome µ ∈ (0, 1], uniformly in Rd. Furthermore we assume that the matrices (aij(t, x))i,jare symmetric, and there exists a constant ν > 0 such that for all t ∈ [0, T ]

d∑i,j=1

(aij(t, x)−

1

2bi(x)bj(x)

)λiλj ≥ ν

d∑i=1

λ2i , x ∈ Rd, λ ∈ Rd. (10.8.2)

Finally, we assume that bi, c ∈ C2b (Rd).

Theorem 10.8.1. Let p ∈ (1,∞) and u0 ∈ L0(Ω,F0;Lp(Rd)) and assume the above

conditions. Then there exists a unique strong solution u of (10.8.1) on (0, T ] withalmost all paths in C([0, T ];E) ∩ C((0, T ];W 2,p(Rd)).

If moreover u0 ∈ B2(1−σ)p,∞ (Rd) almost surely, then there exists a unique strong so-

lution u of (10.8.1) on [0, T ] for which u ∈ C((0, T ];W 2,p(Rd)) almost surely andAu ∈ Lq(0, T ;Lp(S)) for all 1 ≤ q < σ−1. If u0 ∈ W 2,p(Rd) almost surely, then thereexists a unique strong solution u of (10.8.1) on [0, T ] with paths in Cα([0, T ];E) ∩C([0, T ];W 2,p(Rd)) for all α ∈ (0, 1

2).

Proof. Let E = Lp(Rd), where p ∈ (1,∞). Let D(A(t)) = W 2,p(Rd) and A(t)f) =A(t, ·, D)f for all t ∈ [0, T ]. Let D(B0) = W 1,p(Rd) and B0f = B(·, D)f , and let(B,D(B)) be the closure of (B0, D(B0)).

We check the conditions of Theorem 10.5.3. We begin with the Hypotheses (H1) -(H3). That (H1) holds is clear, and (H2) follows as in [12, Example C.III.4.12]. Finally(H3) follows from D(A(t)) ⊂ D(B2).

As is well-known, the operators (A(t))t∈[0,T ] and (C(t))t∈[0,T ] satisfy condition (T1).Furthermore it can be checked that they satisfy (T2). Now Condition (H4) follows from

(10.7.1). Notice that (E,D(A))1−σ,∞ = B2(1−σ)p,∞ (Rd) (cf. [126, Section 2.4.2, Remark

4]).To check (K) for the operators C(t) − λ, put K(t) = [C(t), B]R(λ,C(t)). Since

the third order derivatives in the commutator [C(t), B] cancel and aij(t), qi(t), r(t) ∈


C1b (R

d) and bi, c ∈ C2b (Rd), the operators K(t) are bounded for each t ∈ [0, T ]. More-

over,

K(t) = [C(t), B]R(λ,C(t)) = [C(t)− λ,B]R(λ,C(t)) = (C(t)− λ)B(C(t)− λ)−1 +B

on W 1,p(Rd), and this identity extends to D(B) (see [3, Proposition A.1]). To checkthat K is uniformly bounded, note that by the uniform boundedness of the family(λ− C(0))R(λ,C(t)) : t ∈ [0, T ] it suffices to check that there is a constant C suchthat for all t ∈ [0, T ] and f ∈ W 2,p(Rd),

‖[C(t), B]f‖ ≤ C‖f‖W 2,p(Rd).

But this follows from the assumptions aij, qi, r ∈ L∞([0, T ];C1b (R

d)) and bi, c ∈ C2b (Rd).

Finally, we show that if u0 ∈ W 2,p(Rd) a.s., then U has paths in Cα([0, T ];E) a.s.for all α ∈ (0, 1

2). One can check that for all x ∈ D(A(0)), G(t)x is continuously

differentiable and there are constants C1, C2 such that for all x ∈ D(A(0)) and s, t ∈[0, T ],

‖G(t)x−G(s)x‖ ≤ C1|t− s|‖x‖D(A(0)) ≤ C2|t− s|‖x‖D(CW (0)).

On the other hand it follows from Theorem 10.2.1 that (10.4.1) has a unique strictsolution V . It follows that there exist mapsM,Mα : Ω → R such that all for s, t ∈ [0, T ]

‖U(t)− U(s)‖ ≤ ‖GW (t)V (t)−GW (s)V (s)‖≤ ‖GW (t)V (t)−GW (t)V (s)‖+ ‖GW (t)V (s)−GW (s)V (s)‖≤M‖V (t)− V (s)‖+Mα|t− s|α‖V (s)‖D(CW (0)).

The first term can be estimated because V is continuously differentiable. We alreadyobserved that (CW (s))s∈[0,T ] satisfies (T2). In particular (CW (0)−w)(CW (s)−w)−1 :s ∈ [0, T ] is uniformly bounded. Since s 7→ CW (s)V (s) and V are uniformly bounded,we may conclude that ‖V (s)‖D(CW (0)) is uniformly bounded. The result follows fromthis.


The Acquistapace-Terreni theory is developed in [4] by Acquistapace and Terreni.They also consider domains which are not necessarily dense. We did not need such agenerality here.

The methods we have used to solve (10.3.1) have been inspired by the results of DaPrato, Iannelli and Tubaro from [34] (also see [36, Section 6.5] by Da Prato andZabczyk). The authors have studied the autonomous case of the equation in a Hilbertspace setting. Versions of the hypotheses we have stated in Section 10.3 can be foundin these papers. Theorem 10.4.2 is the generalization of [34, Theorem 1] (also see [36,Theorem 1]) and [37, Proposition 6.29] to the non-autonomous setting in UMD spaces.


In [3] Acquistapace and Terreni have studied the non-autonomous case as well, butonly in Hilbert spaces. They use the theory of Kato and Tanabe [124, Section 5.3] foroperators A(t) with time-dependent domains. In this approach, a technical difficultyarises due to the fact that in the associated deterministic problem, certain operator-valued functions are only Holder continuous, whereas the Kato-Tanabe theory requirestheir differentiability. This difficulty is overcome by approximation arguments. Theauthors have also noted that for the case where the domains D(A(t)) do not dependon time, the methods from [34] can be extended using the Tanabe theory [124, Section5.2]. We explained this in Section 10.8.

Rather than using the Kato-Tanabe theory for operators A(t) with time-depen-dent domains, in Section 10.5 we have used the more recent Acquistapace-Terrenitheory. The basis for this is the perturbation result in Proposition 10.5.2 which is ageneralization of [34, Proposition 1] and [37, Theorem 6.30]. The non-stochastic caseof the example in Section 10.6 has been studied in [1, 121, 136].

The Zakai equation from Section 10.8 has been studied by many authors and hasapplications in filter theory (cf. [37, 139] and the references therein). The result inthis section is a generalization of [37, Example 6.31].

Using a different approach based on fixed point arguments, a general class of prob-lems including (10.1.1) has been studied by Brzezniak [19] in the setting of martin-gale type 2 spaces E. For E = Lp(Rd) with 2 ≤ p <∞, the existence of solutions forthe Zakai equation with paths in L2(0, T ;W 2,p(Rd)) was obtained. We do not knowwhether the techniques of [34] can be extended to the setting of martingale type 2spaces E. It would require an extension of the Ito formula for the duality mapping.Here the problem arises that E and E∗ have martingale type 2 if and only if E isisomorphic to a Hilbert space.

Another approach was taken by Krylov [70], who developed an Lp-theory for ageneral class of time-dependent parabolic stochastic partial differential equations onRd by analytic methods. The equations he has considered are more general than theZakai equation we have studied. The equations are non-linear and many coefficientsare only assumed to be measurable. In this generality he has been able to obtain asolution with paths in Lp(0, T ;W 2,p(Rd)) with p ∈ [2,∞).

List of symbols

Spaces

Bsp,q(S), Besov space.

Bsp,q(Rd;E), see p. 27.

Bsp,q,0(S), see p. 157.

Bsp,1,Bj(S), see p. 160

B(E,F ), bounded linear operators fromE into F .C, complex numbers.C(S;E), continuous functions.C(R+;E), see p. 55.CL([0, T ];E), see p. 21.CsBj(S), see p. 161.

Cp(H1, H2), Schatten class operators.C1(H1, H2), trace class operators.C2(H1, H2), Hilbert-Schmidt operators.(E, ‖ · ‖), (real) Banach space.E∗, dual space of E.(E,F )θ,p, real interpolation space.[E,F ]θ, complex interpolation space.Ea, space of type Ja between E andD(A).Γt, see p. 134.Υt, see p. 133.γ(H,E), see p. 34.γ(S;H,E), see p. 37.(H, [·, ·]), Hilbert space.Hs,p(S), Bessel potential space.Hs,p

0 (S), see p. 157.Hs,pBj(S), see p. 160.

L0(S;E), see p. 20.L2γ, see p. 178.

Lp(S), Lebesgue space.LpF(Ω; γ(L2(R+;H), E)), see p. 58.L0F(Ω; γ(L2(R+;H), E)), see p. 66.

L(WH ;E), see p. 72.Λsp,q(S;E), see p. 166.

Mc,loc0 (Ω;E), see p. 55.

N = 0, 1, 2, 3, . . ..Q, rational numbers.R, real numbers.R+ = [0,∞).Σ(ϕ,w), see p. 125.V r,pa (0, T ;E), see p. 146.V p,loca (0, T ;E), see p. 149.V adma (0, τ ;E), see p. 153.V p,rα,a (0, T ;E), see p. 181.V p,locα,a (0, T ;E), see p. 187.V p,admα,a (0, τ ;E), see p. 190.W s,p(S), Sobolev space.Z = . . . ,−2,−1, 0, 1, 2, . . . , ,

Probabilistic notations

a.a., almost all.a.e., almost everywhere.a.s., almost surely.E, expectation with respect to P.E, expectation with respect to P.F = (Ft)t≥0, filtration on (Ω,A,P).F = (Ft)t≥0, filtration on (Ω, A, P).FWH , see p. 78.(γn)n≥1, see p. 21.i.i.d., independent and identicallydistributed.M , continuous local martingale.[M ], quadratic variation.M τ = (Mτ∧t)t∈R+ .µt,α, see p. 182.(Ω,A,P), probability space.

217

218 List of symbols

(Ω, A, P), copy of (Ω,A,P).(rn)n≥1, see p. 22.[0, τ)× Ω, see p. 152.(S,Σ, µ), measure space.τ , stopping time.WH , see p. 43.WH , see p. 57.

Operations

(A,D(A)), closed operator.A∗, adjoint operator.A, see p. 135.Aw = A− w.βp(E), see p. 23.β−p (E), see p. 96.β+p (E), see p. 96.CW , see p. 202.Cq(E), see p. 25.Cγq (E), see p. 26.

Di, i-th partial derivative.Fub, see p. 51.γ(T ), see p. 27.G(a), see p. 202.IΦ, see p. 37.IWH , see p. 59.LT , see p. 146 and p. 182.MCq(E), see p. 26.MTp(E), see p. 26.(P (t, s))0≤s≤t≤T , see p. 124.P Φ, see p. 134.P ∗ φ, see p. 134.φ, see p. 27.Rα, see p. 126.R(T ), see p. 27.S Φ, see p. 134.S ∗ φ, see p. 134.TrR,S, see p. 81.Tr(Φ1,Φ2), see p. 87.Tp(E), see p. 25.T γp (E), see p. 26.

Miscellaneous

a .Q b, a ≤ CQb.&Q, CQa ≥ b.hQ, cQb ≤ a ≤ CQb.〈·, ·〉, Banach space duality.[·, ·], inner product.δmn = 1 if m = n and 0 otherwise.a ∨ b = maxa, b.a ∧ b = mina, b.→, continuous embedding.

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Index

E-strongly measurable, 20H-cylindrical Brownian motion, 43H-strongly measurable, 20, 49H∞-calculus, 164Lp-martingale, 75

local, 75Lp-space, 23R-boundedness, 3, 27, 29, 172

semi, 107γ-Fubini isomorphism, 50γ-Lipschitz function, 198γ-boundedness, 27γ-norm, 4, 33, 37γ-radonifying operator, 3, 34γ-space, 4, 33, 37L2γ-Lipschitz functions, 178, 181

γ-boundedness, 27

Acquistapace-Terreni conditions, 200adapted

scalarly, 57admissible, 152almost summing operator, 4, 51

Besov space, 5, 28, 38biconcave, 3blow up, 152BMO-norm, 106Borel measurable, 19bounded H∞-calculus, 144, 164bounded imaginary powers, 144Brownian motion, 22

H-cylindrical, 43Brownian representation theorem, 6Burkholder-Davis-Gundy inequalities, 6,

54, 61, 70, 110, 112

Cameron-Martin space, 23, 43class C1,2, 82classical solution, 124complex interpolation, 125cotype, 25

non-trivial, 25covariance operator, 21

decoupled stochastic integral, 57decoupling inequality, 3, 24, 28, 57, 98

elementary, 44, 54, 57, 110equation

colored noise, 156, 195perturbed heat, 193space-time white noise, 14, 159, 196Zakai, 16, 213

evolution family, 124explosion time, 152

factorization method, 10, 126, 129, 142,173

finitely representable, 104, 106, 198Fourier transform, 27fractional domain space, 164Fubini isomorphism, 50function

γ-Lipschitz, 198L2γ-Lipschitz, 178, 181

biconcave, 3class C1,2, 82elementary, 44Lipschitz, 146locally L2

γ-Lipschitz, 189locally Lipschitz, 153

Gaussian covariance operator, 21

229

230 INDEX

Gaussian process, 22Gaussian random variable, 21Gaussian sequence, 3, 21generalized square-function, 4, 33global mild solution, 152

Hardy space, 23Hilbert transform, 3, 28Hilbert-Schmidt operator, 34homeomorphism

Ito, 68homogenous Wiener process, 164

ideal property, 34independent copy, 24inequality

Burkholder-Davis-Gundy, 6, 54, 61,70, 110, 112

decoupling, 3, 24, 28, 57, 98Kahane-Khinchine, 22vector-valued Stein, 76, 93, 106

isomorphismγ-Fubini, 50Ito, 59

Ito formula, 8, 81, 86, 99, 119Ito homeomorphism, 68Ito isometry, 1, 2, 4, 43, 45Ito isomorphism, 59Ito map, 98

Kahane-Khinchine inequalities, 22

LERMT, 7, 106Lipschitz functions, 146Littlewood-Paley decomposition, 27, 28local Lp-martingale, 75local mild solution, 152locally L2

γ-Lipschitz functions, 189locally Lipschitz functions, 153locally stochastically integrable, 72

martingalePaley-Walsh, 7, 23

martingale cotype, 26martingale difference sequence, 23

Paley-Walsh, 23tangent, 3, 28

martingale representation theorem, 6martingale solution, 10martingale type, 26maximal Lp-regularity, 3, 29maximal local mild solution, 152maximal regularity, 144mean, 21measurable, 19

E-strongly, 20H-strongly, 20, 49Borel, 19strongly, 19weakly, 20

Mikhlin multiplier theorem, 3mild solution, 15, 134, 157, 159, 193multiplier, 3, 37, 74

non-trivial cotype, 25non-trivial type, 25

operatorR-bounded, 3, 27, 29, 172γ-bounded, 27γ-radonifying, 3, 34almost summing, 4, 51bounded H∞-calculus, 144, 164bounded imaginary powers, 144covariance, 21Hilbert transform, 3, 28Hilbert-Schmidt, 34Pettis integral, 37radonifying, 3, 34semi-R-bounded, 107trace class, 81, 87

Paley-Walsh martingale, 7, 23Pettis integral, 37predictable sequence, 24process

H-cylindrical Brownian motion, 43admissible, 152Brownian motion, 22

INDEX 231

elementary, 54, 110Gaussian, 22quadratic variation, 54, 65

progressive, 59strongly, 57

property (α), (α+), (α−), 175, 198

quadratic variation, 54quadratic variation process, 65, 109

R-boundedness, 3, 27, 29, 172semi, 107

Rademacher sequence, 22radonifying operator , 3, 34random variable, 21randomized norm, 4, 33randomized UMD, 7, 95real interpolation, 125representation, 36, 37, 49, 72, 111reproducing kernel Hilbert space, 23, 43rough path analysis, 11

scalarly, 4, 5, 36, 110scalarly adapted, 57Schatten class, 23semi-R-boundedness, 107separable measure space, 36sequence

Gaussian, 3, 21predictable, 24Rademacher, 22tangent, 3, 29

series expansion, 72solution

classical, 124martingale, 10maximal, 152mild, 11, 15, 134, 157, 159, 193

global, 152local, 152

strict, 124strong, 10, 11, 16, 133, 201, 210variational, 11, 133weak, 10, 11, 138

spaceγ-radonifying operators, 3, 34Besov, 28, 38Cameron-Martin, 23, 43complex interpolation, 125cotype, 25fractional domain, 164Hardy space, 23Lebesgue space, 23LERMT, 7, 106martingale cotype, 26martingale type, 26of type Ja, 166property (α), (α+), (α−), 175, 198randomized UMD, 95real interpolation, 125reproducing kernel, 23, 43Schatten class, 23trace class operators, 81, 87type, 25UERMT, 7, 106UMD, 3, 23UMD+, 7, 95, 106UMD−, 7, 95, 106uniformly convex, 27uniformly smooth, 27

square-function, 4, 33, 35, 47stochastic integral

characterization, 60, 69, 111decoupled, 57decoupling inequality, 57, 98function, 44local, 72process, 5, 55, 110series expansion, 72

stopping time change, 114strict solution, 124strong solution, 10, 16, 133, 201, 210strongly measurable, 19strongly progressive, 57super-reflexive, 23, 96

Tanabe conditions, 125

232 INDEX

tangent sequences, 3, 28, 29tempered distributions, 28theorem

Brownian representation, 78Dambis, Dubins and Schwartz, 114Doob regularization, 20, 28Dvoretzky, 198Fernique, 21Hoffmann-Jørgensen and Kwapien,

65Kazamaki, 114Kwapien, 26martingale representation, 78Maurey-Pisier, 52, 104Pettis measurability, 19

time change, 114trace, 8, 81, 87trace class operators, 81, 87type, 25

non-trivial, 25

UERMT, 7, 106UMD, 3, 23

randomized, 7, 95UMD+, 7, 106UMD−, 7, 106uniformly convex, 27uniformly smooth, 27

variational solution, 133vector-valued Stein inequality, 76, 93,

106

weak solution, 10, 138weakly measurable, 20

Zakai equation, 16, 213

Summary

In this thesis we study stochastic integration in Banach spaces. Our motivation for thiscomes from the theory of stochastic partial differential equations. These equations canoften be modelled as stochastic differential equations in an infinite-dimensional space.To study such equations one needs a good stochastic integration theory for processeswith values in an infinite-dimensional space. In Part I of this thesis, consisting ofChapters 3-6, we study the integration theory and in Part II in Chapters 7-10 weapply the theory to equations.

In Chapter 3 we study a randomized norm together with the corresponding spacesγ(0, T ;H,E) and γ(0, T ;E), which play a crucial role in this thesis. Here E is aBanach space and H is a separable Hilbert space. We give several embedding resultsfor γ-spaces and the corresponding relation with type and cotype of the Banach spaceE.

In Chapter 3, as a first step in the development of the integration theory we considerstochastic integrals of the form

∫ T

0Φ(t) dWH(t), where the integrand Φ : [0, T ] →

B(H,E) is an operator-valued function and WH is a cylindrical Brownian motionwhich is defined on a probability space (Ω,A,P). The integrable functions can becharacterized as the elements of γ(0, T ;H,E). In Chapter 4 we extend the integrationtheory to adapted operator-valued processes Φ : [0, T ] × Ω → B(H,E). This is doneusing decoupling inequalities for random sums in UMD Banach spaces E. We giveseveral characterizations for integrability and we prove the following two-sided estimatefor the stochastic integral for all p ∈ (1,∞):

cp,EE‖Φ‖pγ(0,T ;H,E) ≤ E supt∈[0,T ]

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p ≤ Cp,EE‖Φ‖pγ(0,T ;H,E).

Here cp,E and Cp,E do not depend on the process Φ. Some sufficient conditions forstochastic integrability are given. Furthermore, there is an extensive discussion on theIto formula for the stochastic integral.

In Chapter 5 we consider randomized UMD conditions. We show that under weakerassumptions on the space E one can get a stochastic integration theory with one-sidedestimates in terms of γ-norms. This can be applied, for example, in the case ofL1-spaces which are not UMD spaces. It is shown that the integration theory is insome sense restricted to randomized UMD spaces by proving that if for a space Ethe decoupling inequalities for the stochastic integral (or estimates for the stochasticintegral in terms of γ-spaces) hold, then randomized UMD properties hold for the

233

234 Summary

space E. In particular this shows that two-sided estimates hold if and only if thespace E has the UMD property.

In the last Chapter 6 of Part I, we consider stochastic integration of progressivelymeasurable E-valued processes φ : [0, T ] × Ω → E with respect to a continuouslocal martingale M : [0, T ]× Ω → R. With the results of Chapter 4 and time changetechniques, we show that also in this situation, integrability can again be characterizedin terms of γ-spaces and we prove two-sided estimates.

In Chapter 7 we discuss some basic concepts needed in the following chapters. Wegive results for non-autonomous Cauchy problems under the Tanabe conditions. Forinstance, convolutions with evolution systems and regularity results are considered.Finally, we introduce several solution concepts for stochastic differential equations inBanach spaces. The equations we study are of the form

(SDE)


U(0) = u0.

The solution concepts we discuss are: strong, variational, mild and weak solutions. Incertain situations we can prove that some solution concepts coincide.

In Chapter 8 we study existence and uniqueness results for the parabolic case of(SDE). We assume that (A(t))t∈[0,T ] satisfies the Tanabe conditions. In this chapter werestrict ourselves to the class of Banach spaces with type 2 and UMD. With a factor-ization method we obtain regularity results for stochastic convolutions with evolutionfamilies. By fixed point methods we find existence, uniqueness and regularity resultsfor (SDE), under the assumption that F and B satisfy certain local Lipschitz andlinear growth conditions. The theory is applied to a second order partial differentialequation on a bounded domain S ⊂ Rd, perturbed by colored noise. This equation ismodelled in Lp(S) with p ∈ [2,∞). A second application is a general elliptic partialdifferential equation on a bounded domain S ⊂ Rd, perturbed by space-time whitenoise. We show that whenever the order of the elliptic operator is higher than thedimension d, there exists a unique regular solution.

In Chapter 9 we extend some of the results of Chapter 8 to randomized UMDBanach spaces E. To avoid technical problems, we only consider the case where Adoes not depend on time and generates an analytic semigroup. We prove resultsfor deterministic and stochastic convolutions with analytic semigroups. To do fixedpoint arguments, we define a technical fixed point space using the γ-norm. Underrandomized Lipschitz conditions we find existence and uniqueness of the solution inthis fixed point space. The theory is applied to several partial differential equations,including the equations of Chapter 8. The equations are modelled in Lp-spaces withp ∈ [1,∞).

Finally in Chapter 10, we consider a special linear case of (SDE) in a UMD space.The Ito formula of Chapter 4 allows us to reduce the stochastic equation to a de-terministic problem. Under the Acquistapace-Terreni conditions for non-autonomousequations, we solve the deterministic equation. This gives existence and uniqueness

Summary 235

of strong solutions and several regularity properties. The theory is applied to a sec-ond order stochastic partial differential equation on a bounded domain S ⊂ Rd withtime-dependent boundary conditions. In the last application we consider the Zakaiequation, which arises in filtering theory. The stochastic partial differential equationsare modelled in Lp-spaces with p ∈ (1,∞).

236 Summary

Samenvatting

In dit proefschrift bestuderen we stochastische integratie in Banachruimten. Onzemotivatie hiervoor komt uit de theorie van stochastische partiele differentiaalvergelij-kingen. Deze vergelijkingen kunnen vaak gemodelleerd worden als stochastische dif-ferentiaalvergelijkingen in een oneindigdimensionale ruimte. Om zulke vergelijkingente bestuderen is een goede stochastische integratietheorie vereist voor processen metwaarden in een oneindigdimensionale ruimte. In Deel I van dit proefschrift bestaatuit de Hoofdstukken 3-6 waarin we de integratietheorie bestuderen. In Deel II, dat deHoofdstukken 7-10 beslaat, passen we de theorie toe op vergelijkingen.

In Hoofdstuk 3 bestuderen we een gerandomiseerde norm en de bijbehorende ruim-ten γ(0, T,H,E) en γ(0, T ;E). Deze spelen een cruciale rol in dit proefschrift. Hierbijis E een Banachruimte en H een separabele Hilbertruimte. We geven verschillendeinbedding resultaten voor γ-ruimten en de bijbehorende relatie met type en cotypevan de Banachruimte E.

Als eerste stap voor de integratietheorie beschouwen we in Hoofdstuk 3 de integraal∫ T

0Φ(t) dWH(t). Hierbij is de integrand Φ : [0, T ] → B(H,E) een operatorwaardige

functie en WH is een cilindrische Brownse beweging WH die gedefinieerd is op een kans-ruimte (Ω,A,P). De integreerbare functions laten zich karakteriseren als de elemen-ten van de ruimte γ(0, T ;H,E). De integratietheorie wordt in Hoofdstuk 4 uitgebreidnaar aangepaste operatorwaardige processen Φ : [0, T ] × Ω → B(H,E). Dit wordtgedaan met behulp van ontkoppelingsongelijkheden voor random sommen in UMDBanachruimten E. We geven verschillende karakteriseringen voor integreerbaarheiden laten zien dat de volgende tweezijdige afschatting voor de integraal geldt voor allep ∈ (1,∞):

cp,EE‖Φ‖pγ(0,T ;H,E) ≤ E supt∈[0,T ]

∥∥∥∫ t

0

Φ(s) dWH(s)∥∥∥p ≤ Cp,EE‖Φ‖pγ(0,T ;H,E).

Hierbij hangen cp,E en Cp,E niet af van het proces Φ. Er worden meerdere conditiesgegeven voor stochastische integreerbaarheid. Verder is er een uitgebreide discussieover de Ito formule voor de stochastische integraal.

In Hoofdstuk 5 beschouwen we gerandomiseerde UMD condities. We laten zienhoe een stochastische integratietheorie onder zwakkere aannamen op de ruimte E kanworden verkregen met eenzijdige afschattingen in termen van γ-normen. Dit is bij-voorbeeld van toepassing op de L1-ruimten, die geen UMD ruimten zijn. We tonenaan dat de integratietheorie beperkt is tot gerandomiseerde UMD ruimten. Dit doen

237

238 Samenvatting

we door te laten zien dat als er voor een ruimte E ontkoppelingsongelijkheden voorstochastische integralen (of afschattingen voor stochastische integralen in termen vanγ-normen) gelden, de gerandomiseerde UMD eigenschappen voor de ruimte E gelden.In het bijzonder volgt hieruit dat er tweezijdige afschattingen gelden dan en slechtsdan als de ruimte E de eigenschap UMD heeft.

In het laatste Hoofdstuk 6 van Deel 1 beschouwen we stochastische integratie vanprogressief meetbare E-waardige processen φ : [0, T ] × Ω → E tegen een continuelokale martingaal M : [0, T ] × Ω → R. Met behulp de resultaten van Hoofdstuk4 en tijdstransformaties laten we zien dat de integreerbaarheid ook in deze situatiegekarakteriseerd kan worden in termen van γ-ruimten en we bewijzen dat ook hiertweezijdige afschattingen gelden.

In Hoofdstuk 7 bespreken we enige basisconcepten die we nodig hebben in ver-volghoofdstukken. We geven resultaten voor niet-autonome Cauchy problemen dievoldoen aan de Tanabe condities. We beschouwen hiervoor convoluties met evolutie-systemen en regulariteitsresultaten hiervoor. Als laatste introduceren we verschillendeoplossingsbegrippen voor stochastische differentiaalvergelijkingen in Banachruimten.De vergelijkingen die we beschouwen zijn van de vorm

(SDV)


U(0) = u0.

De oplossingingsbegrippen die we bestuderen zijn: sterke, variationele, milde en zwak-ke oplossingen. In bepaalde situaties kunnen we bewijzen dat sommige oplossingsbe-grippen overeenstemmen.

In Hoofdstuk 8 bewijzen we existentie en eenduidigheidsresultaten voor het para-bolische geval van (SDV). We nemen aan dat (A(t))t∈[0,T ] voldoet aan de Tanabe con-dities. In dit hoofdstuk beperken we de klasse van Banachruimten tot de ruimten mettype 2 en UMD. Met behulp van een factorisatie methode bewijzen we regulariteits-eigenschappen voor stochastische convoluties met evolutiesystemen. Via fixpuntme-thoden vinden we existentie-, eenduidigheids- en regulariteitsresultaten voor (SDV),onder de aanname dat F en B voldoen aan lokale Lipschitz- en lineaire groeicondities.De theorie wordt toegepast op een tweede orde partiele differentiaalvergelijking opeen begrensd gebied S ⊂ Rd, verstoord door gekleurde ruis. Deze vergelijking wordtgemodelleerd in Lp(S) met p ∈ [2,∞). Een tweede toepassing is een algemene ellip-tische partiele differentiaalvergelijking op een begrensd gebied S ⊂ Rd, verstoord doortijd-ruimte witte ruis. We tonen aan dat zodra de orde van de elliptische operatorgroter is dan de dimensie d, er een unieke gladde oplossing bestaat.

In Hoofdstuk 9 breiden we sommige resultaten van Hoofdstuk 8 uit naar geran-domiseerde UMD Banachruimten E. Om technische problemen te voorkomen beschou-wen we (SDV) alleen in het geval waar A niet van de tijd afhangt en een analytischehalfgroep genereert. Om te beginnen bewijzen we regulariteitsresultaten voor de-terministische en stochastische convolutie met analytische halfgroepen. We definiereneen technische fixpuntruimte met behulp van de γ-norm en onder een gerandomiseerde

Samenvatting 239

Lipschitzconditie vinden we existentie en eenduidigheid van de oplossing in deze fix-puntruimte. De theorie wordt toegepast op verschillende partiele differentiaalvergelij-kingen, waaronder de vergelijkingen uit Hoofdstuk 8. De vergelijkingen worden gemo-delleerd in Lp-ruimten met p ∈ [1,∞).

In het laatste Hoofdstuk 10 beschouwen we een speciaal lineair geval van (SDV) ineen UMD ruimte. Via de Ito formule uit Hoofdstuk 4 reduceren we de stochastischedifferentiaalvergelijking tot een deterministisch probleem. Onder aanname van deAcquistapace-Terreni condities voor niet-autonome vergelijkingen lossen we het deter-ministische probleem op. Dit geeft existentie en eenduidigheid van sterke oplossingenen verschillende regulariteitseigenschappen. We passen de theorie toe op een tweedeorde stochastische partiele differentiaalvergelijking op een begrensd gebied S met tijds-afhankelijke randvoorwaarden. Als laatste geven we een toepassing op de zogehetenZakai vergelijking uit de filtertheorie. De stochastische partiele differentiaalvergelij-kingen worden gemodelleerd in Lp-ruimten met p ∈ (1,∞).

240 Samenvatting

Curriculum Vitae

Mark Christiaan Veraar was born in Delft on April 22th, 1980. In 1998 he completedhis high school education at College het Loo in Voorburg. In that same year he startedhis studies of Applied Mathematics at the Delft University of Technology. Under thesupervision of Prof. dr. J.M.A.M. van Neerven, he wrote his Master’s thesis entitled“Stochastic Integration in Banach Spaces”. On June 23th, 2003, Mark obtained hisM.Sc. degree “cum laude” for his research in the area of Probability Theory andFunctional Analysis. As of July 1st, 2003, he continued his research as a Ph.D. studentat the Delft University of Technology. Part of this research was carried out during hisstays at the Universitat Karlsruhe and the University of South-Carolina.

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