a class of infinite dimensional stochastic processes with...

Linköping Studies in Science and Technology. Dissertations.No. 1699

A class of infinite dimensionalstochastic processes with

unbounded diffusion and itsassociated Dirichlet forms

John Karlsson

Department of MathematicsLinköping University, SE–581 83 Linköping, Sweden

Linköping 2015

Linköping Studies in Science and Technology. Dissertations.No. 1699

A class of infinite dimensional stochastic processes with unbounded diffusionand its associated Dirichlet forms

John Karlsson

[email protected]

Mathematical StatisticsDepartment of Mathematics

Linköping UniversitySE–581 83 Linköping

Sweden

ISBN 978-91-7685-966-7ISSN 0345-7524

Copyright c© 2015 John Karlsson, unless otherwise noted

Printed by LiU-Tryck, Linköping, Sweden 2015

Abstract

This thesis consists of two papers which focuses on a particular diffusion type Dirichletform

E(F,G) =

∫〈ADF,DG〉H dν,

whereA =∑∞i=1 λi〈Si, ·〉HSi. Here Si, i ∈ N, is the basis in the Cameron-Martin space,

H, consisting of the Schauder functions, and ν denotes the Wiener measure.In Paper I, we let λi, i ∈ N, vary over the space of wiener trajectories in a way that

the diffusion operator A is almost everywhere an unbounded operator on the Cameron–Martin space. In addition we put a weight function ϕ on the Wiener measure ν and showthat under these changes of the reference measure, the Malliavin derivative and divergenceare closable operators with certain closable inverses. It is then shown that under certainconditions on λi, i ∈ N , and these changes of reference measure, the Dirichlet form isquasi-regular. This is done first in the classical Wiener space and then the results aretransferred to the Wiener space over a Riemannian manifold.

Paper II focuses on the case when λi, i ∈ N, is a sequence of non-decreasing realnumbers. The processX associated to (E, D(E)) is then an infinite dimensional Ornstein-Uhlenbeck process. In this case we show that the distributions of a sequence of certain fi-nite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of theinfinite dimensional Ornstein-Uhlenbeck process. We also investigate the quadratic vari-ation for this process, both in the classical sense and in the recent framework of stochasticcalculus via regularization. Since the process is Banach space valued, the tensor quadraticvariation is an appropriate tool to establish the Itô formula for the infinite dimensionalOrnstein-Uhlenbeck process X . Sufficient conditions are presented for the scalar as wellas the tensor quadratic variation to exist.

v

Populärvetenskaplig sammanfattning

En stokastisk process är en matematisk representation av hur ett slumpmässigt systemutvecklas under tid. Exempelvis är värdet på en aktie en endimensionell process och po-sitionen på en partikel som rör sig slumpmässigt i rummet är en tredimensionell process.Det är svårare att föreställa sig och analysera processer som tar värden i oändligdimen-sionella rum men det finns olika sätt att behandla problemet matematiskt. Ett sätt är attstudera så kallade Dirichletformer. En Dirichletform är ett matematiskt objekt inom om-rådet potentialteori. Genom att använda sig av en sådan framställning får man tillgång tillde verktyg som finns inom potentialteorins ämnesområde vilket kan göra det matematiskaarbetet enklare.

Det visar sig att endast vissa Dirichletformer har en motsvarande stokastisk process.I det första pappret i den här avhandlingen behandlas en viss typ av Dirichletformer därden så kallade diffusionen ökar för varje dimension. Man kan säga att diffusionen ärhastigheten på slumprörelsen. Vi visar hur snabbt diffusionen får öka för att slutprocessenska vara väldefinierad. Pappret behandlar även fallet då processen lever i ett krökt rum,på en mångfald, som exempel kan man tänka sig ytan av en ballong istället för ytan på ettpapper.

I det andra pappret ligger fokus på den associerade oändligdimensionella processensom vi kallar X . Här visas bland annat att man under vissa omständigheter kan approx-imera X med ändligdimensionella processer. Vi beräknar också processens kvadratiskavariation som är ett mått på hur mycket processens värde fluktuerar under tid. Genomatt visa att en speciell typ av kvadratisk variation existerar kan vi även presentera en Itôformel, ett hjälpsamt verktyg för att bedriva analys på processen.

vii

Acknowledgments

I would like to express my thanks to Linköping University and the department of math-ematics for the opportunity I have had to work there. A special thanks goes to my mainsupervisor Jörg-Uwe Löbus and my co-supervisor Torkel Erhardsson for all the help andsupport I have received during this PhD project. Thanks also to the administrative per-sonnel and co-workers at MAI that I have been fortunate to share this department with.

Thanks goes to my fellow PhD students that have been a source of inspiration andgood times. A special mention goes to my friend Marcus Kardell that I have now knownand had countless discussions with for almost five years.

I would also like to thank my family, other friends, and anyone else I might haveforgot to mention, these pages would be full if I were to write down the names of everyonedeserving a mention.

Finally, once again I would like to thank my main supervisor Jörg-Uwe Löbus for allthe effort and time he put into this work, without him this could never have been done.

ix

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPopulärvetenskaplig sammanfattning . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Introduction1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1 Motivation for the papers . . . . . . . . . . . . . . . . . . . . . . 31.2 Paper I specifics . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Paper II specifics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Malliavin calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 General framework . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Hermite polynomials . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Wiener Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Multiple stochastic integrals . . . . . . . . . . . . . . . . . . . . 72.5 Derivative operator . . . . . . . . . . . . . . . . . . . . . . . . . 82.6 Divergence operator . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Stochastic integration via regularization . . . . . . . . . . . . . . . . . . 12

4.1 One dimensional case . . . . . . . . . . . . . . . . . . . . . . . 124.2 Infinite dimensional case . . . . . . . . . . . . . . . . . . . . . . 13

5 The geometric Cameron–Martin formula . . . . . . . . . . . . . . . . . . 155.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Riemannian connection . . . . . . . . . . . . . . . . . . . . . . . 185.3 The orthonormal bundle . . . . . . . . . . . . . . . . . . . . . . 205.4 Brownian motion on a smooth Riemannian manifold . . . . . . . 215.5 Analysis on the path space . . . . . . . . . . . . . . . . . . . . . 235.6 Directional derivative, gradient, divergence and integration by parts 275.7 Tensors and Ricci curvature . . . . . . . . . . . . . . . . . . . . 29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Paper I1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Closability of derivative, divergence, and their inverses . . . . . . . . . . 424 Closability of the bilinear form . . . . . . . . . . . . . . . . . . . . . . . 50

xi

xii CONTENTS

5 Quasi-regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556 Transferring the results to a geometric setting . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Paper II1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751.2 Some basic definitions . . . . . . . . . . . . . . . . . . . . . . . 77

2 Finite dimensional approximation of the infinite dimensional process . . . 783 Scalar quadratic variation . . . . . . . . . . . . . . . . . . . . . . . . . . 854 Tensor quadratic variation . . . . . . . . . . . . . . . . . . . . . . . . . 935 Itô’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99A Appendix: Some lemmas in extreme value theory . . . . . . . . . . . . . 107References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Introduction

1 Outline 3

1 Outline

This thesis consists of two main parts. The first part is an introductory part containingkey concepts of the theory used in the second part. The second part contains two papers,A class of infinite dimensional stochastic processes with unbounded diffusion and Infinitedimensional Ornstein-Uhlenbeck processes with unbounded diffusion – Approximation,quadratic variation, and Itô formula, written at Linköping university during the years2012-2015. Both papers are joint work with Jörg-Uwe Löbus. What follows below is ashort summary of the different topics in part one.

Malliavin calculus

The purpose of this section is to provide a brief introduction to some notions in Malliavincalculus. Concepts such as the Malliavin derivative are essential for the definition of thebilinear Dirichlet form which is the main object of this study. We also present the Wienerchaos decomposition as we make use of it in Paper I.

Dirichlet forms

This section is devoted to the notion of Dirichlet forms. The section contains the ba-sic definitions needed for understanding the theorem of Ma/Röckner which provides theconnection between Dirichlet forms and stochastic processes.

Stochastic integration via regularization

Here we present some main ideas of the recent theory of stochastic integration via reg-ularization. These concepts are used in Paper II in order to obtain a corresponding Itôformula to the stochastic process being studied.

Differential geometry

This is a presentation of basic notions in differential geometry. These concepts are thenused to define Brownian motion on a Riemannian manifold. This theory is used in PaperI in the final section which transfers results of the flat case to a geometric setting.

1.1 Motivation for the papers

The theory of Dirichlet forms is a subject showing that certain bilinear forms can serveas a connection between analysis and probability. This connection to probability wasestablished by the fundamental work of Silverstein and Fukushima during the 1970s (see[Meyer, 2009]). In particular, Fukushima showed that if a Dirichlet form on a locallycompact state space is regular, it is possible to construct an associated Markov processwith right continuous sample paths. By the 1980s, the demand for tools to study Markovprocesses on infinite dimensional (not locally compact) spaces led to various extensionsof Fukushima’s result. In 1992, the general question was settled by the characterizationof Ma and Röckner to the extent that a Dirichlet form on a separable metric space is

4 Introduction

associated with a Markov process with decent sample paths if and only if the form isquasi-regular. This result provides a method to construct Markov processes on separablemetric spaces.

The papers in the thesis are concerned with Dirichlet forms of type

E(F,G) =

∫〈DF,ADG〉H dν. (1)

Here the diffusion operator is given by

A =∞∑i=1

λi 〈Si, ·〉H Si, (2)

Si, i ∈ N are the Schauder functions, H is the Cameron–Martin space C0([0, 1];Rd) andν is the Wiener measure. In the paper [Driver and Röckner, 1992] the classical quasi-regular Dirichlet form, i.e. λi = 1, i ∈ N, is studied and and associated with a diffusionprocess on a compact Riemannian manifold. Similar results but in the case of unboundeddiffusion where studied in [Löbus, 2004].

1.2 Paper I specifics

In the first paper we investigate the case where the Wiener measure ν is replaced with theweighted Wiener measure ϕν. Here the weight functions ϕ is of the form

ϕ(γ) = exp

1∫

0

〈b(γs), dγs〉Rd −1

2

1∫0

|b(γs)|2 ds

.

This particular choice of weight functions is motivated by the geometric setting in the pa-pers [Wang and Wu, 2008, Wang and Wu, 2009] by F.-Y. Wang and B. Wu. These weightfunctions are considered both in a geometric framework and in the flat case. We study theform (1) on the set of smooth cylindrical functions of type

F,G ∈ Y = F (γ) = f(γ(s1), . . . , γ(sk)) : sj is a dyadic point,

and

F,G ∈ Z = F (γ) = f(γ(s1), . . . , γ(sk)) : sj ∈ [0, 1].

Above, γ is a Wiener trajectory. In the paper we formulate necessary and sufficient condi-tions on λ1(γ), λ2(γ), . . . that guarantee closability of (E, Y ) and (E, Z) on L2(ϕν). Thepaper then shows locality, Dirichlet property, and quasi-regularity of the closure (E, Z)on L2(ϕν). The final parts of the paper are devoted to transferring the results into a geo-metric setting. In this case we consider Wiener trajectories on a stochastically completeRiemannian manifold, a generalization of the results of [Driver and Röckner, 1992] and[Löbus, 2004] where compact manifolds are studied.

2 Malliavin calculus 5

1.3 Paper II specifics

The second paper focuses on the process associated with the Dirichlet form (1). In Pa-per II the diffusion coefficients λi of the diffusion operator (2) are considered to be realnumbers. We show that the associated stochastic process has the representation

Xt :=∞∑i=1

Gi(λit) · Si , t ≥ 0,

where, for t ≥ 0, the right-hand side converges in C0([0, 1];Rd). Here Gi, i ∈ N, areindependent one-dimensional Ornstein-Uhlenbeck processes, i.e. dGi(t) = −Gi(t)dt +√

2 dWi(t), t ≥ 0, for a sequence of independent one-dimensional Wiener processesWi, i ∈ N. We show that the distributions of a sequence of certain finite dimensionalOrnstein-Uhlenbeck processes converge weakly to the distribution of the infinite dimen-sional Ornstein-Uhlenbeck process, X , associated with the Dirichlet form (E, D(E)). Wealso investigate the quadratic variation for this process, both in the classical sense and inthe recent framework of stochastic calculus via regularization. Since the process is Ba-nach space valued, the tensor quadratic variation is an appropriate tool to establish the Itôformula for the processX . Sufficient conditions are presented for the scalar as well as thetensor quadratic variation to exist.

2 Malliavin calculus

The French mathematician Paul Malliavin developed a calculus that is extending the cal-culus of variations, from functions to stochastic processes. Among other results it makesit possible to calculate the derivative of random variables. The material in this section isbased on David Nualart’s monographs [Nualart, 2009] and [Nualart, 2006] and can alsobe found in [Karlsson, 2013].

2.1 General framework

We let H be a real separable Hilbert space with inner product 〈·, ·〉H and correspondingnorm ‖ · ‖H. Unless otherwise stated, in this presentation we may consider the Hilbertspace H to be L2([0, 1]) with the Borel σ-algebra B and Lebesgue measure µ.

Definition 2.1. A stochastic process W = W (h), h ∈ H defined in a complete proba-bility space (Ω,F,P) is a Gaussian process on H, if W is a centered Gaussian family ofrandom variables and E[W (g)W (h)] = 〈g, h〉H for all g, h ∈ H.

This can be illustrated by a simple example.

Example .1Let H = L2(R+) and Wt := W (1[0, t]), t ≥ 0. This is standard Brownian motion since

E[WsWt] = E[W (1[0,s])W (1[0,t])] = 〈1[0,s],1[0,t]〉L2 = min(s, t),

i.e. the covariance function for Brownian motion.

6 Introduction

2.2 Hermite polynomials

The Hermite polynomials will play an important role in the upcoming Wiener chaos de-composition of a Gaussian random variable as it turns out they have certain orthogonalityproperties. Here we define the Hermite polynomials Hn as the coefficients of the Taylorexpansion of F (x, t) = exptx− t2/2 in powers of t. In other words

F (x, t) := etx−t2

2 =∞∑n=0

Hn(x) · tn. (3)

We have the relations

H0(x) = 1, H1(x) = x, H2(x) =1

2(x2 − 1), Hn(x) =

(−1)n

n!ex2

2dn

dxn

(e−

x2

2

)and also

d

dxHn(x) = Hn−1(x), (n+ 1)Hn+1(x) = xHn(x)−Hn−1(x)

as well as

Hn(0) =

0 if n odd,(−1)k2kk!

if n even.

2.3 Wiener Chaos

Assume that H (we may think of it asL2([0, 1])) is infinite dimensional and let e1, e2, . . . be an ON-basis in H. We let Λ denote the set of all finite multi-indices a = (a1, a2, . . . ),ai ∈ N for i = 1, 2, . . . , i.e. only finitely many ai:s are non-zero. For any a ∈ Λ wedefine

a! :=∞∏i=1

ai!, |a| :=∞∑i=1

ai, Φa :=√a!∞∏i=1

Hai(W (ei)).

Taking the Hermite polynomials of Gaussian variables we obtain an orthogonality relationin the following way. For a two-dimensional Gaussian vector (X,Y ) s.t. E[X] = E[Y ] =0 and V [X] = V [Y ] = 1, we have that for all m,n ∈ N

E[Hm(X)Hn(Y )] =

0 if m 6= n,1n!E[XY ]n if m = n.

Let Hn denote the closed subspace of L2(Ω,F,P) spanned by Φa : a ∈ Λ, |a| = n.It can be shown that L2(Ω,F,P) =

⊕∞n=0 Hn. We call Hn the nth Wiener Chaos.

Furthermore we let Jn denote the projection operator onto the nth Wiener Chaos. Itfollows for F ∈ L2(Ω,F,P) that

F =∞∑n=0

JnF.

Remark 2.2. The nth Wiener chaos contains polynomials of Gaussian variables of degreen.

While we defined nth Wiener chaos as an expression of ON-basis (ei)∞i=1, it turns out

that the nth Wiener chaos is independent of this choice of ON-basis.


2.4 Multiple stochastic integrals

We will see that there is another representation for the nth Wiener chaos in the formof iterated Itô integrals. We begin with introducing a type of elementary functions firstdescribed by Norbert Wiener.

Definition 2.3. We define En as the set of all elementary functions of form

f(t1, . . . , tn) =k∑

i1,...,in=1

ai1...in1Ai1×···×Ain (t1, . . . , tn), (4)

where ai1...in ∈ R and ai1...in = 0 if two indices coincide and A1, . . . , Ak are pairwisedisjoint subsets of [0, 1] with finite Lebesgue measure.

We see that this is the usual simple functions in [0, 1]n with the condition of being0 on any diagonal described by two equal coordinates. It can be shown that this set hasmeasure 0 and thus the set of elementary functions En is dense in L2([0, 1]n). We cannow define a stochastic integral for these elementary functions. For a function of the form(4) we define

In(f(t1, . . . , tn)) =k∑

i1,...,in=1

ai1,...,inW (Ai1) · . . . ·W (Ain).

Remark 2.4. In(f) can be expressed as an iterated Itô integral

In(f) = n!

1∫tn=0

tn∫tn−1=0

. . .

t2∫t1=0

f(t1, . . . , tn) dWt1 . . . dWtn .

The following proposition shows some of the properties of this stochastic integral.

Proposition 2.5. In has the following properties:(i) In(f) = In(f), where f denotes the symmetrization of f , i.e.

f(t1, . . . , tn) =1

n!

∑Π

f(tΠ(1), . . . , tΠ(n))

where Π runs over all permutations of 1, . . . , n.(ii)

E[Im(f), In(g)] =

0 if m 6= n

n!〈f , g〉L2([0,1]n) if m = n.

It follows that we get an isometry between L2s([0, 1]n) := f : f ∈ L2([0, 1]n) and

Hn, i.e. √n!‖f‖L2([0,1]n) = ‖I(f)‖L2(Ω,F,P).

Since En is dense in L2([0, 1]n) we can extend In to a linear continuous operator fromL2([0, 1]n) to L2(Ω,F,P).

8 Introduction

Remark 2.6. The image of L2s([0, 1]n) under In is the nth Wiener chaos Hn since In(f)

is a polynomial of degree n in W (A1), . . . ,W (An) when f has the form (4), and Propo-sition 2.5(ii) shows that stochastic integrals of different order are orthogonal. The claimthen follows from induction.

Since we can express a random variable in its chaos expansion we note that for F ∈L2(Ω,F,P), there exists fn ∈ L2

s([0, 1]n), n ∈ N such that

F = E[F ] +

∞∑n=1

In(fn).

2.5 Derivative operator

To define the calculus on the Wiener space a natural way of proceeding is to first look ata simple class of functions. Let Z denote the set of smooth cylindrical functions

Z = F : F = f(W (h1), . . . ,W (hn)), h1, . . . , hn ∈ H, f ∈ C∞p

where C∞p denotes smooth functions with polynomial growth.

Definition 2.7. The gradient of a smooth cylindrical function is defined by:

DF :=n∑n=1

∂f

∂xi(W (h1), . . . ,W (hn)) · hi.

Remark 2.8. We have that DW (h) = h.

Clearly we have the product rule D(FG) = DF · G + F · DG and we also getan integration by parts formula as will be shown shortly. We also define the directionalderivative in the following manner.

Definition 2.9. The directional derivative Dh is defined by

DhF := 〈DF, h〉 h ∈ H.

Proposition 2.10. Let F ∈ D1,2 and assume that F has representation

F = E[F ] +∞∑n=1

In(fn)

where fn ∈ L2s([0, 1]n). Then

DtF =∞∑n=1

nIn−1(fn(·, t)), t ∈ [0, 1].

Proof: First let F = In(fn) where

fn(t1, . . . , tn) =k∑

i1,...,in

ai1...in1Ai1×···×Ain (t1, . . . , tn).


Then

F = In(fn) =k∑

i1,...,in=1

ai1...inW (Ai1) · . . . ·W (Ain)

and F ∈ Z. We get

DtF =n∑j=1

k∑i1,...,in=1

ai1...inW (Ai1) · . . . · 1Aij (t) · . . . ·W (Ain) = nIn−1(f(·, t))

by symmetry. The claim follows from the closedness of D1,2.

2.6 Divergence operator

We introduce the divergence operator δ. It turns out that δ is the adjoint operator to D.Let Dom δ be the set of all u ∈ L2(Ω,F,P;H) such that there exists c(u) > 0 with

|E[〈DF, u〉H]| ≤ c‖F‖L2 (5)

for all F ∈ D1,2. Then E[〈DF, u〉H] is a bounded linear functional from D1,2 to R. Nowsince D1,2 is dense in L2(Ω,F,P) Riesz representation theorem states that there exists aunique representing element δ(u), bounded in L2(Ω,F,P). That is

E[〈DF, u〉H] = E[Fδ(u)], ∀F ∈ D1,2. (6)

Just like for the derivative there is a connection between the Wiener chaos expansionand the divergence operator. Recall that for u ∈ L2([0, 1] × Ω), u has a Wiener chaosexpansion of form

u(t) =

∞∑n=0

In(fn(·, t)),

where fn ∈ L2([0, 1]n) and fn is symmetric in the n first variables.

Proposition 2.11. We have u ∈ Dom δ if and only if the series

∞∑n=0

In+1(fn)

converges in L2(Ω,F,P). In this case we have

δ(u) =∞∑n=0

In+1(fn).

10 Introduction

Proof: Suppose G = In(g) where g ∈ L2s([0, 1]n). Now using Proposition 2.10 we get

E[〈u,DG〉H] = E[〈u, nIn−1(g(·, t))〉H] = E[〈In−1(fn−1(·, t)), nIn−1(g(·, t))〉H]

=

∫[0,1]

E[In−1(fn−1(·, t))nIn−1(g(·, t))] dt

= n(n− 1)!

∫[0,1]

〈fn−1, g〉L2([0,1]n−1) dt

= n!〈fn−1, g〉L2([0,1]n = n!〈fn−1, g〉L2([0,1]n)

= E[In(fn−1)In(g)] = E[In(fn−1)G]

where the third and last lines come from the isometry between L2s([0, 1])n and Hn.

One can see from this that δ is an integral. δ(u) is called the Skorokhod stochasticintegral of the process u and if the process is adapted then the Skorokhod integral willcoincide with the Itô integral, i.e.

δ(u) =

1∫0

ut dWt.

3 Dirichlet forms

The theory of Dirichlet forms was first introduced in the works by Beurling and Deny in1958 and 1959. Dirichlet form theory can be used in the area of Markov processes giving adifferent approach to problems. The usual tools for studying diffusion theory are methodsfrom partial differential equations, whereas Dirichlet forms are connected to potentialtheory and energy methods. This section is based on the book [Ma and Röckner, 1992]which contains a much more through treatment of the subject.

Let H be a Hilbert space with corresponding norm ‖ · ‖H and inner product 〈·, ·〉H.

Definition 3.1. A Dirichlet form is a densely defined positive symmetric bilinear formon L2(E,µ) such that

(i) D(E) is a real Hilbert space with inner product 〈u, v〉E := 〈, 〉2E + E(u, v)2,

(ii) For every u ∈ D(E) it holds u+ ∧ 1 ∈ D(E) and E(u+ ∧ 1, u+ ∧ 1) ≤ E(u, u).

Not all Dirichlet forms have an associated process. What follows are the differentnotions needed for the formulation of Ma–Röckner’s theorem.

Definition 3.2. Let H = L2(E,m). A bilinear form E on E is regular if D(E)∩C0(E)is dense inD(E) w.r.t. E1-norm and dense inC0(E) w.r.t. the uniform norm. HereC0(E)denotes all continuous functions on E with compact support.

Next we recall some basic definitions from the theory of stochastic processes.

3 Dirichlet forms 11

Definition 3.3. We say that a function is cadlag if it is right continuous with left limits,i.e.

(i) f(x−) := limtx

f(t) exists,

(ii) f(x+) := limtx

f(t) exists and is equal to f(x).

The corresponding term for left continuous functions is caglad.

Definition 3.4. Two stochastic processesX and Y with a common index set T are calledversions of one another if

t ∈ T, P (ω : X(t, ω) = Y (t, ω)) = 1.

Such processes are also said to be stochastically equivalent.

If in addition the process Xt is left or right continuous then for a version Yt we haveXt = Yt almost surely.

Definition 3.5. (i) An increasing sequence (Fk)k∈N of closed subsets of E is called anE-nest if ∪k≥1D(E)Fk is dense inD(E) w.r.t. ‖·‖E1

, whereD(E)Fk denotes u ∈ D(E) :u = 0 m-a.e. on E \ Fk.(ii) A subset N ⊂ E is called E-exceptional if N ⊂ ∩k≥0F ck for some E-nest (Fk)k∈N.(iii) We say that a property holds E-quasi-everywhere if it holds everywhere outsidesome E-exceptional set.

We can relate the definition of quasi-continuity to a similar notion on the E-nest.

Definition 3.6. An E-quasi-everywhere defined function f is called E-quasi-continuousif there exists an E-nest (Fk)k∈N such that f is continuous on (Fk)k∈N.

Definition 3.7. A Dirichlet form (E, D(E)) on L2(E,m) is called quasi-regular if:

(i) There exists an E-nest (Fk)k∈N consisting of compact sets,

(ii) There exists an E1/21 -dense subset of D(E) whose elements have

E-quasi-continuous m-versions,(iii) There exist un ∈ D(E), n ∈ N, having E-quasi-continuous m-versions un,

n ∈ N, and an E-exeptional set N ⊂ E such that un : n ∈ N separates thepoints of E \N.

Definition 3.8. A process M with state space E is called µ-tight if there exists an in-creasing sequence (Kn)n∈N of compact sets in E such that

Pµ( limn→∞

inft > 0 : Mt ∈ E \Kn <∞) = 0

where inf ∅ :=∞.

We note that it simply means that for every ε > 0 there exists a compact set K suchthat Pµ(Mt ∈ K) > 1− ε.

12 Introduction

Definition 3.9. A cadlag Markov processM with state spaceE and transition semigroup(pt)t>0 is said to be properly associated with (E, D(E)) and its corresponding semigroup(Tt)t>0 if ptf is an E-quasi continuous µ-version of Ttf for all t > 0 and all boundedf ∈ L2(E;µ).

The following theorem is the main purpose of this section. It provides the connectionbetween Dirichlet forms and process theory.

Theorem 3.10. Let (E, D(E)) be a quasi-regular Dirichlet form on L2(E,µ). Then thereexists a pair (M, M) of µ-tight special standard processes which is properly associatedwith (E, D(E)).

Proof: The proof is omitted but can be found in [Ma and Röckner, 1992].

Remark 3.11. It is known that if E is locally compact, i.e. finite dimensional, thenthere exists an associated process to every regular Dirichlet form on L2(E,µ). See[Fukushima et al., 2011].

4 Stochastic integration via regularization

This section contains some notions in the area of stochastic integration via regulariza-tion. Stochastic integration via regularization is a recently developed theory using tech-niques from integration. The different objects in this theory are presented first for finitedimensional stochastic processes, before considering the corresponding objects in the in-finite dimensional case. This section is based on material in [Russo and Vallois, 2007],[Di Girolami and Russo, 2014] and [Di Girolami et al., 2014].

4.1 One dimensional case

In this section let X(t), t ≥ 0, be a real-valued continuous process and Y (t), t ≥ 0, bea real-valued locally integrable process. Below we introduce the key concept of uniformconvergence in probability, abbreviated ucp.

Definition 4.1. A sequence of real-valued processes (Xδt )t∈[0,T ] is said to converge to

(Xt)t∈[0,T ] in the ucp sense as δ → 0, if for all ε > 0

limδ→0

P

(supt∈[0,T ]

|Xδt −Xt| > ε

)= 0.

We may now introduce the following integral and covariation.

Definition 4.2. Provided that the following limits exist in the ucp sense we define theforward integral

∫Y d−X := lim

δ→0

t∫0

YsXs+δ −Xs

δds

4 Stochastic integration via regularization 13

and the covariation

[X,Y ]t := limδ→0

t∫0

(Xs+δ −Xs)(Ys+δ − Ys)δ

ds.

We also denote [X]t := [X,X]t and call it the quadratic variation of X .

Remark 4.3. In e.g. the case of Y being continuous with bounded variation then theforward integral coincides with the usual Itô integral, see [Russo and Vallois, 2007].

In the one dimensional case we have the following Itô formula from [Russo and Vallois, 2007]Proposition 12.

Proposition 4.4. Suppose that [X]t, t ≥ 0 exists and f ∈ C2(R). Then∫f ′(Xs)d

−Xs

exists and

f(Xt) = f(X0) +

t∫0

f ′(Xs) d−Xs +

1

2

t∫0

f ′′(Xs) d[X]s.

4.2 Infinite dimensional case

We now consider the corresponding notions in the infinite dimensional case. We nowconsider X to be a process taking values in a Banach space B. Ucp convergence isdefined in a similar way as above.

Definition 4.5. A sequence of B valued processes (Xδt )t∈[0,T ] is said to converge to

(Xt)t∈[0,T ] in the ucp sense as δ → 0, if for all ε > 0

limδ→0

P

(supt∈[0,T ]

‖Xδt −Xt‖B > ε

)= 0.

In [Di Girolami and Russo, 2014] Definition 5.1 the authors introduce the followingdefinition.

Definition 4.6. Let X be a B valued process and Y be a B∗ valued process such thatX is continuous and

∫ T0‖Ys‖B∗ ds < ∞ a.s.. Provided that the following limit exist in

probability for every t ∈ [0, T ] we define the forward integral

t∫0

B∗〈Ys, d−Xs〉B := limδ→0

t∫0

B∗

⟨Ys,

Xs+δ −Xs

δ

⟩B

ds

if the process t∫0

B∗〈Ys, d−Xs〉B

t∈[0,T ]

admits a continuous version.

14 Introduction

In e.g. the case of B = L1([0, 1]), the duality pairing B∗〈g∗, f〉B has the integralrepresentation

B∗〈g∗, f〉B =

1∫0

f · g dx

for f ∈ B = L1([0, 1]), g∗ ∈ B∗ and some representing element g ∈ L∞([0, 1]) ∼= B∗.In the case of F ∈ B ⊗πB ∼= L1([0, 1]2;Rd2), G∗ ∈ (B ⊗πB)∗ and some representingelement G ∈ L∞([0, 1]2;Rd2) ∼= (B ⊗πB)∗ the pairing duality becomes

(B ⊗πB)∗ 〈G∗, F 〉(B ⊗πB)∗∗ =

∫F (u, v) qG(u, v) dudv,

where the symbol q to denotes the scalar product in Rd2 .The paper [Di Girolami and Russo, 2014] also defines the following notions of co-

variation.

Definition 4.7. Let X and Y be a B1 respectively B2 valued processes. We define thescalar covariation as the ucp limit

[X,Y ]t := limδ→0

t∫0

‖Xs+δ −Xs‖B1‖Ys+δ − Ys‖B2

δds,

provided it exists. We also denote [X]t := [X,X]t and call it the scalar quadratic variationof X .

Definition 4.8. Let X and Y be B1 respectively B2 valued processes. We define thetensor covariation as the ucp limit

[X,Y ]⊗t := limδ→0

t∫0

(Xs+δ −Xs)⊗ (Ys+δ − Ys)δ

ds,

provided it exists. We also denote [X]⊗t := [X,X]⊗t and call it the tensor quadraticvariation of X .

It is worth mentioning that quadratic variation in the sense above is essentially onlysuitable for semimartingale processes, see [Di Girolami et al., 2014]. However in the sit-uation where a process admits having both scalar and tensor quadratic variation we getthe Itô formula in the form of [Di Girolami and Russo, 2014] Theorem 5.2.

Proposition 4.9. Suppose that X is a B valued continuous process admitting a scalarquadratic variation and a tensor quadratic variation. Furthermore let F be a mappingF : [0, T ] × B → R such that F is one time continuously Fréchet differentiable andtwo times continuously Fréchet differentiable in the second argument. That is, denotingthe Fréchet derivative with respect to the second variable by D, for every t ∈ [0, T ] we

5 The geometric Cameron–Martin formula 15

have DF (t, ·) : B → B∗ and D2F (t, ·) : B → (B ⊗πB)∗ continuously. Under theseconditions for every t ∈ [0, T ] the forward integral

t∫0

B∗〈DF (s,Xs), d−Xs〉B exists

and

F (t,Xt) = F (0, X0) +

t∫0

∂tF (s,Xs) ds+

t∫0

B∗〈DF (s,Xs), d−Xs〉B

+1

2

t∫0

(B ⊗πB)∗

⟨D2F (s,Xs), d[X]s

⟩(B ⊗πB)∗∗

.

Remark 4.10. The above proposition makes use of the existence of scalar quadratic vari-ation and a tensor quadratic variation. The statement then follows as a direct conse-quence of [Di Girolami et al., 2014] Remark 5.7 applied to the mentioned Theorem 5.2of [Di Girolami and Russo, 2014].

5 The geometric Cameron–Martin formula

This chapter serves to introduce the concepts needed for construction of Brownian mo-tion on a manifold M . The main part of section is based on the series of lecture notes[Löbus, 1995]. Much of the material can also be found in [Rogers and Williams, 2000].We first recall some basic definitions to introduce the notation used.

5.1 Manifolds

Definition 5.1. A subset M ⊂ RN is called a d-dimensional C∞-manifold in RN if foreach x ∈ M there is an open (in the relative topology) set G ⊂ M with x ∈ G and thereexist F ∈ C∞(Rd −→ RN−d) such that (if necessary, after a permutation of coordinates)

G = (y, F (y)) : y ∈ A,

for an open set A ⊂ Rd.

Definition 5.2. We say that y(t), t ∈ [0, 1] is a smooth curve in Rd if

y(t) =d∑i=1

yi(t)ei, t ∈ [0, 1],

where ei = (0, . . . , 0, 1i, 0, . . . , 0), yi ∈ C∞([0, 1]), i = 1, . . . , d, such that

d∑i=1

(yi(t))2 6= 0, t ∈ [0, 1].

16 Introduction

With the notation used in the previous definition, (x(t), t ∈ [0, 1]) defined as

x(t) :=(y(t), F

(y(t)

)), t ∈ [0, 1],

is called a smooth curve on M .

We recall the definition of tangents and tangent space.

Definition 5.3. We say that v(t) = y(t) is a tangent to (y(s), s ∈ [0, 1]) (at t). Let Iddenote the d× d identity matrix and J(y) denote the Jacobian matrix(

Jri (y))i,r=1,...,d

=(∂F r∂yi

)i,r=1,...,d

.

We use the notation (Id, J)T for the matrix

1 · · · 0. . .

0 · · · 1∂F 1

∂y1 · · · ∂F 1

∂yd

......

∂Fn

∂y1 · · · ∂Fn

∂yd.

We call

u(t) := x(t) =(

Id, J(y(t)

))Tv(t) (7)

a tangent to (x(s), s ∈ [0, 1]) at t.

Definition 5.4. For a fixed y ∈ Rd and the corresponding point x = (y, F (y)) ∈ M .We define Tx := x+ t : x ∈M, t = (Id, J(y))T v for some v ∈ Rd which we identifywith all the tangents to M at the point x and call it the tangentspace to M at x.

We may induce a metric on the tangent space from the Euclidean metric in RN . Letu1, u2 ∈ Tx where x = (y, F (y)), y ∈ Rd, and v1, v2 ∈ Rd where u1 = (Id, J(y))T v1,u2 = (Id, J(y))T v2. Then

〈u1, u2〉RN = vT1 (Id, J(y))(Id, J(y))T v2. (8)

We introduce the notation

g(y) := (Id, J(y))(Id, J(y))T = Id +J(y)J(y)T (9)

and write (8) as〈u1, u2〉Tx = vT1 g(y)v2 =: g〈v1, v2〉.

Remark 5.5. The length of a smooth curve (x(t), t ∈ [0, 1]) on M is calculated by

1∫0

(y(t)T g(y(t))y(t)

)1/2dt,

where x(t) =(y(t), F

(y(t)

)), t ∈ [0, 1].


We now recall the notion of parallel transport along a curve.

Definition 5.6. For a smooth curve (y(s), s ∈ [0, 1]) in Rd and the corresponding smoothcurve (x(s), s ∈ [0, 1]) on M , i.e. x(s) =

(y(s), F

(y(s)

)), s ∈ [0, 1], let t1, t2 ∈ [0, 1]

and u(t1) ∈ Tx(t1). Assume 0 ≤ t1 ≤ t2 ≤ 1 and let v(t) ∈ Tx(t) such that w(t) ∈ Rd,t ∈ [t1, t2] and

v(t) =(

Id, J(y(t)

))Tw(t), t ∈ [t1, t2].

Let v(t) ∈ RN denote the rate of change of v(t) and let Px(t)(·) denote the orthogonalprojection onto the tangent space Tx(t). We say that v(t) moves parallel in t ∈ [t1, t2],along x(s), if for all t ∈ [t1, t2] we have Px(t)v(t) = 0.

Remark 5.7. In this case we have

Px(t)(·) =(

Id, J(y(t)

))Tg−1

(y(t)

)(Id, J

(y(t)

))(·),

since for all w ∈ Tx(t) and all z ∈ RN we have for w =(

Id, J(y(t)

))Tq⟨

Px(t)z, w⟩RN

=⟨

(Id, J)T g−1(Id, J)z, (Id, J)T q⟩RN

=(g−1(Id, J)z

)T(gq)

=⟨z, (Id, J)T q

⟩RN

= 〈z, w〉RN .

It follows that v(t) moves parallel if

0 = Px(t)v(t) =(

Id, J(y(t)

))Tg−1

(y(t)

)(Id, J

(y(t)

))v(t), t ∈ [t1, t2]. (10)

Definition 5.8. Let v(t), t ∈ [0, 1] be the unique solution to the ODE (10) with boundaryconditions v(t1) = u(t1) ∈ Tx(t1). We say that u(t2) := v(t2) ∈ Tx(t2) is the paralleltransport of the vector u(t1) along (x(s), s ∈ [0, 1]).

For notational clarity the rest of this section uses the Einstein summation convention.

Remark 5.9. Take v(t) and w(t), t ∈ [t1, t2], as above and denote (gij)i,j=1,...,d := g−1.We say that w(t) is the induced parallel transport along (y(s), s ∈ [0, 1]). That is w(t)solves the equation

0 = g−1(y(t)

)(Id, J(y(t)

)[(Id, J(y(t))Tw(t)]

q= w(t) + g−1

(y(t)

)(Id, J(y(t)

)[(Id, J(y(t))]

qw(t)

= w(t) +

(gil∑r

Jrl DjJrk

)(y(t)

)yj(t)wk(t)

i=1,...,d

.

Using the notation

Γ ijk := gil∑r

Jrl DjJrk = gil

∑r

Jrl DjDkFr, i, j, k ∈ 1, . . . , d, (11)

the above equation turns into

0 = w(t) +Γ ijk(y(t)

)yj(t)wk(t)

i=1,...,d

. (12)

18 Introduction

5.2 Riemannian connection

In this section we use a different definition of Γ and instead consider it in the followingway.

Definition 5.10. A function

Γ =(Γ ijk(y)

)i,j,k=1,...,d

y ∈ Rd,

Γ ∈ C∞(Rd × 1, . . . , d3 → R) is called a connection i.e. Γ is a collection of d3

smooth functions on Rd. Let M = (y, F (y)) : y ∈ Rd be a C∞-manifold of RN andg(y) := (Id, J(y))(Id, J(y))T , y ∈ Rd. The pair (Rd, g) is called a Riemannian spaceand the pair (M, g) is called a Riemannian manifold.

Definition 5.11. Let Γ be a connection on (Rd, g) and let (w(s), s ∈ [0, 1]) ∈ C∞([0, 1]→Rd) be a vector field along a smooth curve (y(s), s ∈ [0, 1]). If equation (12) holds i.e.

0 = w(t) +Γ ijk(y(t)

)yj(t)wk(t)

i=1,...,d

t ∈ [t1, t2], (13)

we say that w(t) is parallel transported along (y(s), s ∈ [0, 1]) under Γ , for t ∈ [t1, t2].

Remark 5.12. The objects Γ ijk, i, j, k ∈ 1, . . . , d are called Christoffel symbols.

Definition 5.13. A connection Γ on a Riemannian space (Rd, g) is called Riemannian ifthe following holds.

(i) For all i ∈ 1, . . . , d we have Γ ijk = Γ ikj , j, k ∈ 1, . . . , d.

(ii) For vector fields (w(s), s ∈ [0, 1]), (v(s), s ∈ [0, 1]) parallel transported along a smoothcurve (y(s), s ∈ [0, 1]), it holds that

g〈w(t1), v(t1)〉 = g〈w(t2), v(t2)〉, t1, t2 ∈ [0, 1].

If the parallel transport under Γ preserves inner products in the above sense we saythat Γ is compatible with g.

The following theorem explains why we defined Γ by (11).

Theorem 5.14. There is exactly one Riemannian connection on (Rd, g). This connectionis given by

Γ ijk =1

2gil(Djglk +Dkglj −Dlgjk

), i, j, k ∈ 1, . . . , d.

Proof: Step 1: (Existence) Let M = (y, F (y)) : y ∈ Rd be a d-dimensional C∞-manifold in RN , J the corresponding Jacobian matrix

J =(Jri (y)

)i=1,...,d; r=1,...,n

=

(∂F r

∂yi

)i=1,...,d; r=1,...,n


and g(y) =(

Id, J(y))(

Id, J(y))T

, y ∈ Rd. Set

Γ ijk = gil∑r

Jrl DjDkFr, i, j, k ∈ 1, . . . , d,

i.e. (11). Under this assumption the symmetry condition (i) is satisfied. Let (w(s), s ∈ [0, 1])and (v(s), s ∈ [0, 1]) be vector fields along a smooth curve (y(s), s ∈ [0, 1]) in Rd, whichare parallel transported along this curve under Γ . Furthermore define

q(s) :=(

Id, J(s))Tw(s), p(s) :=

(Id, J(s)

)Tu(s), s ∈ [0, 1].

We haveg〈w(s), v(s)〉 = q(s)T p(s),

and (g〈w(s), v(s)〉

) q= q(s)T p(s) + q(s)p(s)T , s ∈ [0, 1].

Due to (12) and (10), q(s) ∈ Tx(s) as well as p(s) ∈ Tx(s) are parallel transported alongx(s) =

(y(s), F (y(s))

)then per definition q(s) ⊥ Tx(s) and p(s) ⊥ Tx(s). It follows

that (g〈w(s), v(s)〉

) q= 0,

i.e. the isometry condition (ii) is satisfied.Step 2: (Uniqueness) Let Γ be a connection and (y(s), s ∈ [0, 1]), (v(s), s ∈ [0, 1]) and(w(s), s ∈ [0, 1]) be as above. Then from (ii) it holds(

wi(s)gij(s)vj(s)

) q= 0.

Using (12) i.e. w(s) = −Γ ijk(y(s))yj(s)wk(s)i=1,...,d as well as a similar relation forv(s), s ∈ [0, 1] we get

0 = −Γ iklywlgijvj + wiykDkgijvj − wigijΓ iklykvl

=(− Γ iklgij +Dkglj − gliΓ ikj

)wlykvj .

Thus it holdsΓ iklgij + Γ ikjgli = Dkglj .

We swap cyclically k → l→ j → k to obtain

−(Γ iljgik + Γ ilkgji

)= −Dlgjk,

and

Γ ijkgil + Γ ijlgki = Djgkl.

Adding these last three equations together, using the symmetry (i) and the symmetry of(gij) we get

2Γ ijkgil =(Djgkl +Dkglj −Dlgjk

).

The statement follows.

Remark 5.15. The matrix g defined by (9) uniquely defines Γ ijk in Theorem 5.14. Thus ggives a complete description of the geometry of the smooth Riemannian manifold (M, g).

20 Introduction

5.3 The orthonormal bundle

Having defined parallel transport it is natural to define the concept of moving bases.

Definition 5.16. Let(y(s), s ∈ [0, 1]

)be a smooth curve in Rd and (x(s), s ∈ [0, 1]) a

smooth curve on M = (y, F (y)) : y ∈ Rd with x(s) =(y(s), F (y(s))

), s ∈ [0, 1].

Let E = e1, . . . , ed be an orthonormal basis (or short ON-basis) in Tx(0) and let

f1, . . . , fd ∈ Rd where el =(

Id, J(y(0)))Tfl, l = 1, . . . , d. Also let (fl(s), s ∈ [0, 1])

be vector fields along (y(s), s ∈ [0, 1]) satisfying the differential equationfs(s) =

Γ ijk(y(s))yj(s)fkl (s)

i=1,...,d

s ∈ [0, 1],

f(0) = fl,l = 1, . . . , d. (14)

We defineel(s) :=

(Id, J(y(s))

)Tfl(s), s ∈ [0, 1], l = 1, . . . , d.

We have that e1(s), . . . , ed(s) is an orthonormal basis in Tx(s), s ∈ [0, 1] and say thate1(s), . . . , ed(s) is a moving basis along x(s).

These final notions describe how curves behave on a manifold.

Definition 5.17. The family

(x, e1, . . . , ed) : x ∈M, e1, . . . , ed is an ON-basis on Tx,

is called the orthonormal bundle and is denoted O(M).

Definition 5.18. We say that a curve (u(s), s ∈ [0, 1]) in O(M) is horizontal if u(s) =(x(s), E(s), s ∈ [0, 1]), where E(s) is a moving basis along (x(s), s ∈ [0, 1]). The curve(u(s), s ∈ [0, 1]), is called the horizontal lift of (x(s), s ∈ [0, 1]).

Proposition 5.19. Let (w(s), s ∈ [0, 1]) be a smooth curve in Rd. Then there is a hori-zontal curve u(s) on O(M) satisfying

w(s) =(r1(s), . . . , rd(s)

)T,

x(s) = E(s)r(s),

x(s) =(y(s), F (y(s))

),

s ∈ [0, 1]. For given initial condition u0 = (x0, e1, . . . , ed) ∈ O(M) the horizontal curveu(s) =

((x(s), E(s)), s ∈ [0, 1]

), with u(0) = u0 is the unique solution to the system of

differential equationsfl(s) +

Γ ijk(y(s))yj(s)fkl (s)

i=1,...,d

= 0, l = 1, . . . , d,

y(s) = rl(s)fl(s), s ∈ [0, 1].(15)

We call the map w(s)→ u(s) the development map.

Remark 5.20. As a motivation for (15) consider a curve (w(s), s ∈ [0, 1]), in Rd. Nowlet r(s) be the velocity along w(s) i.e. r(s) = w(s). Solving (15) gives y(s), a curvein Rd, and fl(s), l = 1, . . . , d, basis vectors in Rd. These objects provide an equivalentdescription of the curve x(s) on M , with basis vectors el(s), l = 1, . . . , d, on Tx(s).


5.4 Brownian motion on a smooth Riemannian manifold

In order to construct Brownian motion on smooth Riemannian manifolds we first remindof the following relation between the Itô and Stratonovich integral. Below the multiplica-tions are made in the sense of scalar products in Rm.

Definition 5.21. Let X,Y be continuous semimartingales in Rm. The Stratonovich in-tegral

∫Y ∂X is defined by

St :=

t∫0

Ys ∂Xs :=

t∫0

Ys dXs +1

2[Y,X]t,

where t ∈ T (= [0, 1], [0,∞), . . .),∫Y dX is the usual Itô integral and [·, ·]t denotes the

covariation. We may write

∂S = Y ∂X = dS = Y dX +1

2d〈Y,X〉 = Y dX +

1

2dY dX

using differential notation.

Remark 5.22. The Stratonovich calculus is compatible with the classical non-stochasticdifferential calculus. For example for a smooth function, F ∈ C∞(Rm), we have

(i) ∂(F (X)) = F ′(X)∂X,

(ii) ∂(XY ) = Y ∂X +X∂Y.

We consider the system of differential equations (15)y(s) = fm(s)Wm(s),

fl(s) =− Γ ijk(y(s))f jm(s)fkl (s)

i=1,...,d

Wm(s), l = 1, . . . , d, s ∈ [0, 1],

with initial conditions

(y(0), f1(0), . . . , fd(0)) = (y0, f10 , . . . , fd0) (16a)

x0 := (y, F (y0)), el0 :=(

Id, J(y0))Tfl0 , l = 1, . . . , d, (16b)

such that(x0, e10 , . . . , ed0) ∈ O(M). (16c)

For m = 1, . . . , d use the notation

ϕm = ϕm(y(s), f1(s), . . . , fd(s)

):=− Γ ijk(y(s))f jm(s)fkl (s)

i,l=1,...,d

.

Note that ϕm, m = 1, . . . , d, is a d2-dimensional vector whose entries corresponds to thed different d-dimensional parallel transports of the basis vectors fi, i = 1, . . . , d. Let

Φ(y(s), f1(s), . . . , fd(s)

):=

| |f1 · · · fd| |−− −− −−| |ϕ1 · · · ϕd| |

︸︷︷︸

d

22 Introduction

With this notation (15) takes the form(y(s), f1(s), . . . , fd(s)

)T= Φ

(y(s), f1(s), . . . , fd(s)

)W (s), s ∈ [0, 1].

The main idea for constructing Brownian motion on a manifold is to first get a descriptionon how to transfer smooth curves to a manifold via differential calculus. We then wish toreplace these smooth curves with the jagged curves of Brownian motion. This is done byreplacing the usual differentials with Stratonovich ones. Work has to be done to ensurethat this procedure yields a well defined result. We consider the Stratonovich SDE

(∂y, ∂f1, . . . , ∂fd) = Φ(y, f1, . . . , fd)∂γ,

deterministic initial condition (16),(17)

where (γs,Fγs , s ∈ [0, 1]) is a d-dimensional Brownian motion on a probability space

(Ω,F,P) equipped with the filtration (Fs)s∈[0,1]. Let

Ψ :=( i∑q=1

DiΦjq

)i,j=1,...,d(d+1)

.

Since there exist K > 0 such that for all ξ, η ∈ Rd(d+1)

‖Φ(ξ)− Φ(η)‖+ ‖Ψ(ξ)− Ψ(η)‖ ≤ K‖ξ − η‖

and‖Φ(ξ)‖2 + ‖Ψ(ξ)‖2 ≤ K(1 + ‖ξ‖2)

we may formulate the following theorem.

Theorem 5.23. The Itô SDEd(y, f1, . . . , fd) = Φ(y, f1, . . . , fd)dγ + 1

2Ψ(y, f1, . . . , fd)dt,

(16),(18)

has a unique strong solution. This implies that the Stratonovich SDE (17) has a uniquestrong solution namely the solution to the Itô SDE (18).

Definition 5.24. (a) Let(((

y(s), f1(s), . . . , fd(s)), s ∈ [0, 1]

), (Fs)s∈[0,1],P

)be a so-

lution to (17). Furthermore let

γ(s) :=(y(s), F (y(s))

), el(s) :=

(Id, J

(y(s)

))Tfl(s), l = 1, . . . , d, s ∈ [0, 1].

The process (γ, E) =((γ(s), e1(s), . . . , ed(s)

), s ∈ [0, 1], (Fs)s∈[0,1],P

)is called hori-

zontal Brownian motion.(b) We call γ :=

((γ(s), s ∈ [0, 1]), (Fs)s∈[0,1],P

)Brownian motion on M .

(c) Let 0 ≤ s1, s2 ≤ 1 and v = viei(s1) ∈ Tγ(s1). The quantity

T γs1→s2v := vT ei(s2),


is called stochastic parallel transport from Tγ(s1) to Tγ(s2) along γ.(d) Let γ0 ∈ M . The set of curves Pγ0(M) := γ ∈ C([0, 1];M), γ(0) = γ0 is calledthe path space over M .(e) The mapping γ := I(γ)

C([0, 1];Rd) 3 γ(·, ω)→ γ(·, ω) ∈ Pγ(0)(M),P-a.e. ω ∈ Ω

is called the Itô map.(f) Let ν denote the Wiener measure on ϕ ∈ C([0, 1];Rd) : ϕ(0) = 0. The imagemeasure ν := I∗ν under I , is called Wiener measure on Pγ0(M).(g) The mapping γ → (γ, E) is called the stochastic development map and the mappingH : γ → (γ, E) is called stochastic horizontal lift.

The stochastic parallel transport has the following properties.

Proposition 5.25. Let 0 ≤ s1, s2, s3 ≤ 1, v1 = vi1ei(s1) ∈ Tγ(s1), and v2 = vi2ei(s1) ∈Tγ(s1). Then P-a.s.

T γs2→s3Tγs1→s2v1 = T γs1→s3v1, (19)

and〈v1, v2〉Tγ(s1)

= 〈T γs1→s2v1, Tγs1→s2v2〉Tγ(s2)

. (20)

Proof: Step 1: From Definition 5.24, (19) is clear for fixed 0 ≤ s1, s2, s3 ≤ 1, P-a.s. TheP-a.s. continuity of e1, . . . , ed implies P-a.s. continuity in the left hand side of (19) ins1, s2, s3 as well as for the right hand side of (19) for s1, s3. Therefore (19) holds P-a.s.for all 0 ≤ s1, s2, s3 ≤ 1 and v1 ∈ Tγ(s1).Step 2: Since e1, . . . , ed is P-a.s. continuous it follows that vi1ei, v

i2ei is continuous as

well. It is therefore sufficient to prove (20) for fixed 0 ≤ s1, s2 ≤ 1. We note

〈v1, v2〉Tγ(s1)=∑i,j

vi1vj2〈ei(s1), ej(s1)〉Tγ(s1)

=∑i

vi1vi2,

and

〈T γs1→s2v1, Tγs1→s2v2〉Tγ(s2)

= 〈vi1ei(s2), vj2ej(s2)〉Tγ(s2)=

d∑i=1

vi1vi2,

i.e. the left and right hand sides of (20) coincide.

5.5 Analysis on the path space

This section contains a brief overview of the calculus on the path space of Brownianmotion on a Riemannian manifold. Detailed calculations and more results can be foundin [Driver and Röckner, 1992], [Hsu, 1995], [Hsu, 2002]. Let (Ω,F,P) be a probabilityspace and γ(ω) be a d-dimensional Brownian motion on (Ω,F,P) with filtration Fs,s ∈ [0, 1] and corresponding measure ν. In addition let O(d) denote the space of d × dorthogonal matrices.

24 Introduction

Proposition 5.26. Let A =((A(s), s ∈ [0, 1]

), (Fs)s∈[0,1],P

)be an O(d)-valued pro-

cess and α =((α(s), s ∈ [0, 1]

), (Fs)s∈[0,1],P

)be an Rd-valued process such that

E[ ∫|α|2 ds

]<∞, and E

[ ∫‖A‖2 ds

]<∞. Introduce

W (s) =

s∫0

A(r) dγ(r) +

s∫0

α(r) dr, s ∈ [0, 1].

We let ν := W∗ν denote the image measure under the map γ 7→W . We have

dν

dν(γ) = exp

( 1∫0

α(r)A(r) dγ(r)− 1

2

1∫0

|α(r)|2 dr).

Proof: The process γ′(s) =∫ s0A(r) dγ(r), s ∈ [0, 1] is a Brownian motion with respect

to (Fs)s∈[0,1], since γ′ is a continuous local martingale with quadratic variation ·∫0

Adγ,

·∫0

Adγ

s

=

s∫0

ATAd[γ, γ

]r

= Id ·s, s ∈ [0, 1].

That is ν(γ′(ω) : ω ∈ Ω

)= 1 and W = γ′ +

∫αdr. The statement then follows from

the Girsanov formula.

We use the notation

H :=h ∈ C0

([0, 1];Rd

): h absolutely continuous

∫|h′(s)|2 ds <∞

to denote the d-dimensional Cameron–Martin space.

Theorem 5.27. Let h ∈ H. There exist families (A(t, ·))t∈R, (α(t, ·))t∈R of processessuch that the following holds.

(i) The conditions of Proposition 5.26 are satisfied for every t ∈ R.

(ii) It holds that α(·, s) ∈ C1(R;Rd), A(·, s) ∈ C1(R;O(d)), s ∈ [0, 1], P-a.s. as wellas Rd 3 α(·, 0) = 0 and O(d) 3 A(·, 0) = 0.

(iii) Let

W (t, s) =

s∫0

A(t, r) dγ(r) +

s∫0

α(t, r) dr,

and define

ξ(t, s) := I(W (t, s)), u(t, s) := H(ξ(t, s)), t ∈ R, s ∈ [0, 1],


where I andH are the Itô mapping respectively horizontal lift from Definition 5.24.Then ξ has the representation

ξ(t, s) = γ(s) +

t∫0

u(r, s)h(s) dr, (21)

and is a solution to the geometric flow equationddtξ(t, s) = u(t, s)h(s) = H(ξ(t, s))h(s),

ξ(0, s) = γ(s), s ∈ [0, 1], t ∈ R.

is satisfied with the solution

Here γ is a Brownian motion on M .

In (21) u(r, s) is to be understood by the mapping Rd → Tξ(r,s), u(r, s)k := kiei(W (r, s)

)where ei

(W (r, s)

)denotes the walking basis vector ei along W (r, s).

Proof: The proof is omitted but can be found in the papers [Hsu, 1995] and [Hsu, 2002].

We denoteνt := W (t, ·)∗P, t ∈ R,

where ν0 = ν. Furthermore we let γ′(t) =(

(γ′(t, s), s ∈ [0, 1]), (Fs)s∈[0,1],P)

givenby

γ′(t, s) :=

s∫0

A(t, r) dγ(r), t ∈ R.

According to the proof of Proposition 5.26 γ′(t, ·) is a Brownian motion with respect tothe filtration Fs, s ∈ [0, 1]. The map γ(ω)→ γ′(t)(ω) induces an injective map b : Ω →Ω.

The flow equation is used to calculate the directional derivative on the manifold, see(25) below.

Theorem 5.28. (a) (Quasi invariance) The measures νt, t ∈ R are equivalent and wehave for P-a.e. ω ∈ Ω

dνtdν

(γ(ω)) = exp

∫α(t, r)A(t, r) dγ(r)− 1

2

∫|α(t, r)|2 dr

(ω). (22)

(b) Almost surely the Radon–Nikodym derivative dνtdν (γ) belongs to C1(R) and

d

dt

∣∣∣∣t=0

dνtdν

(γ) =

∫d

dt

∣∣∣∣t=0

α(t, r) dγ(r), t ∈ R. (23)

26 Introduction

Proof: (a) Let

νt :=

γ(s)(·) +

s∫0

α(t, r)(b−1·) dr

∗

ν .

The Girsanov formula yields

dνtdν

(γ(ω)) = exp

1∫0

α(t, r)(b−1(ω)) dγ(r)(ω)− 1

2

1∫0

|α(t, r)(b−1(ω))|2 dr.

On the other hand we have νt = b∗νt, t ∈ R. Taking into account that γ and γ′ areBrownian motions it follows ν(dγ(ω)) = ν(dγ′(t)(ω)) and

dνtdν

(γ(ω)

)=νt(dγ(ω)

)ν(dγ(ω)

) =νt(dγ′(t)(ω)

)ν(dγ(ω)

) =ν(dγ′(t)(ω)

)ν(dγ′(t)(ω)

) =dνtdν

(γ(b(ω)

))

= exp

1∫0

α(t, r)(ω) dγ′(r)(ω)− 1

2

∫|α(t, r)(ω)|2 dr

= exp

1∫0

α(t, r)(ω)A(t, r)(ω) dγ(ω)− 1

2

∫|α(t, r)(ω)|2 dr

.

(b) Using the stochastic version of dominated convergence theorem, see [Protter, 2005],and taking under consideration α(0) = 0, A(0) = Id, we obtain (23) by differentiating(22).

Corollary 5.29. (a) (Quasi invariance) The measures νt := I∗νt, t ∈ R are equivalentand we have for P-a.e. ω ∈ Ω

dνtdν

(γ(ω)

)=dνtdν

(γ(ω)

).

(b) Almost surely the Radon–Nikodym derivative dνtdν (γ) ∈ C1(R) and we have

d

dt

∣∣∣∣t=0

dνtdν

(γ) =d

dt

∣∣∣∣t=0

dνtdν

(γ).

Definition 5.30. A vector field on an open set N ⊂ M is a smooth map S : N →TN =

⋃x∈N Tx where S(x) ∈ Tx, x ∈ N . Let N1 be an open subset of M and let

N ⊂ N1. If S is a vector field on N1 then S|N is a vector field on N . In particular ifN is a smooth curve, N := (x(s), s ∈ [0, 1]) on M , we say that S is a vector field along(x(s), s ∈ [0, 1]).

Remark 5.31. Let S be a vector field on N ⊂ M . If x = F (y), y ∈ Rd, and S(x) =(I, J(y))V (y), we can identify S with the first order differential operator vi ∂

∂yi.


5.6 Directional derivative, gradient, divergence and integrationby parts

In this section we introduce the geometrical objects corresponding to the usual differentialnotions. This is done by means of specifying the objects on a class of smooth functions.

Definition 5.32. Let γ0 ∈M . We define

Z :=Φ(γ) = ϕ

(γ(s1), . . . , γ(sk)

), γ ∈ Pγ0(M) :

0 < s1 < . . . < sk = 1, ϕ ∈ C∞b (Mk), k ∈ N,

to be the set of smooth cylindrical functions on M .

For f ∈ C∞b (M) we define the gradient on the tangent space

∇f := (Id, J)T g−1∇f F,

here recall that F denotes the function describing the manifold embedding in RN . Withthis definition and for ψ ∈ C1(R;M) it follows that

d

dt(f ψ(t)) =

d

dt

(f F F−1 ψ(t)

)=⟨∇f F, d

dtF−1ψ(t)

⟩Rd

= g⟨g−1∇f F, d

dtF−1ψ(t)

⟩=⟨∇f(x),

d

dtψ(t)

⟩TxM

.

Definition 5.33. Let Φ ∈ Z where Φ(γ) = ϕ(γ(s1), . . . , γ(sk)). We define

DsΦ(γ) :=k∑i=1

χ[0,si](s)T γsi→s(∇iϕ)(γ(s1), . . . , γ(sk))

, s ∈ [0, 1],

to be the gradient of Φ. Here ∇i denotes the operator ∇ corresponding to the ith variablex(si).

Remark 5.34. We mention that the difference between Ds in Definition 5.33 and Section6 in Paper I, is a parallel transport from Tγ(s) to Tγ(0).

Recall that H denotes the d-dimensional Cameron–Martin space.

Definition 5.35. Let (γ, E) :=((

(γ(s), e1(s), . . . , ed(s)), s ∈ [0, 1]), (Fs)s∈[0,1],P

)be the process constructed from the solution of the system of equations (15) for fixedinitial conditions (γ(0), e1(0), . . . , ed(0)) ∈ O(M). For h ∈ H and Φ ∈ Z we define thedirectional derivative in the direction of h as

DhΦ(γ) :=

1∫0

⟨DsΦ(γ),

d

dshjsej(s)

⟩Tγ(s)

ds.

Note that DhΦ(γ) is independent of (γ(0), e1(0), . . . , ed(0)).

28 Introduction

We obtain

DhΦ(γ) =

1∫0

⟨DsΦ(γ),

d

dshjsej(s)

⟩Tγ(s)

ds

=k∑i=1

1∫0

χ[0,si](s) ·d

dshjs

⟨T γsi→s∇iϕ, ej(s)

⟩Tγ(s)

ds

=

k∑i=1

1∫0

χ[0,si](s) ·d

dshjs

⟨T γsi→0∇iϕ, ej(0)

⟩Tγ0

ds

=k∑i=1

hjsi ·⟨T γsi→0∇iϕ, ej(0)

⟩Tγ0

=k∑i=1

hjsi ·⟨∇iϕ, ej(si)

⟩Tγ(si)

(24)

as another representation of DhΦ(γ). Finally for the solution ξh of (21) we have P-a.s.

d

dt

∣∣∣∣t=0

Φ ξh(t, ·) =k∑i=1

⟨∇ϕ, d

dt

∣∣∣∣t=0

ξh(t, si)

⟩Tγ(si)

=

k∑i=1

⟨∇ϕ, u(0, si)h(si)

⟩Tγ(si)

= DhΦ(γ). (25)

Remark 5.36. The limitlimt→0

1

t

(Φ ξh − Φ(γ)

),

exist in L2 = L2(Pγ0(M), ν). Therefore,

limt→0

1

t

(Φ ξh − Φ(γ)

)= DhΦ(γ) in L2. (26)

Let ν denote the Wiener measure on Pγ0(M), γ0 ∈M . We study γ → ξh(t) trajecto-rywise as a mapping Cγ0([0, 1];M) → Cγ0([0, 1];M). For Φ as before and Ψ ∈ Z suchthat Ψ(γ) = ψ(γ(s1), . . . , γ(sk)), γ ∈ Pγ0(M), ψ ∈ C∞b (M) we have

〈DhΦ, Ψ〉L2 =

⟨d

dt

∣∣∣t=0

Φ ξh, Ψ⟩L2

= limt→0

1

t

[〈Φ ξh(t), Ψ〉L2 − 〈Φ, Ψ〉L2

]= limt→0

1

t

[ ∫Φ ξh(t)(γ)Ψ(γ) ν(dγ)− 〈Φ, Ψ〉L2

]= limt→0

1

t

[ ∫ΦΨ ξh(−t)(γ) ν

(dξh(−t)(γ)

)− 〈Φ, Ψ〉L2

]= limt→0

1

t

[ ∫ΦΨ ξh(−t)(γ)

ν(dξh(−t)(γ)

)ν(dγ)

ν(dγ)− 〈Φ, Ψ〉L2

]. (27)


We use the abbreviations

dξh(−t)∗νdν

=dν−tdν

and zh :=d

dt

∣∣∣∣t=0

dν−tdν

.

From (26) and (27) we obtain the integration by parts formula

⟨DhΦ, Ψ

⟩L2 =

⟨Φ,

d

dt

∣∣∣∣t=0

[Ψξh(−t)(γ)

ν(dξh(−t)(γ)

)ν(dγ)

]⟩L2

=⟨Φ,−DhΨ+zhΨ

⟩L2 .

Let g ∈ L2([0, 1] → Rd) and h :=∫ q0gs ds. We note that h ∈ H. Furthermore let Φ

and Ψ be cylindrical functions. We have

∫ 1∫0

⟨DsΦ, Ψg

jej(s)⟩T·(s)

ds dν =

∫ 1∫0

⟨DsΦ, g

jej(s)⟩T·(s)

ds Ψdν =⟨DhΦ, Ψ

⟩L2

=⟨Φ,−DhΨ + zhΨ

⟩L2 =

⟨Φ,−

k∑i=1

hj(si)⟨∇iψ, ej(si)

⟩T·(si)

+ zhΨ

⟩L2

,

where for the second last equality we have used the above integration by parts formula.

Definition 5.37. We define

δ(Ψgjej) := −k∑i=1

d∑j=1

hj(si)⟨∇iψ, ej(si)

⟩T·(si)

+ zhΨ

and call it the divergence of Ψgjej .

Remark 5.38. The divergence is the adjoint operator to the gradient. This is true also forthe non-geometric Malliavin gradient and divergence. That is one of the most importantrelations in Malliavin calculus.

5.7 Tensors and Ricci curvature

This presentation of the theory is based on the book [Kühnel, 2002].

Definition 5.39. An (r, s)-tensor, A, at a point x ∈M , is a multilinear map

(Tx)∗ × . . .× (Tx)∗︸︷︷︸r

× (Tx)× . . .× (Tx)︸︷︷︸s

→ R.

Here (Tx)∗ denotes the dual space to the tangent space at x.

Let us a consider a vector field S on M . Let x = F (y), y = (y1, . . . , yd) ∈ Rd, andS(x) = (I; J(y))V (y). We can then identify S with the first order differential operator

S(x) =d∑i=1

vi(y)∂

∂yi

∣∣∣∣x

.

30 Introduction

Using the Christoffel symbols of (11) we may introduce the Riemann curvature tensor Rin coordinate form as the object

Rsijk :=∂Γ sik∂xj

−∂Γ sij∂xk

+ Γ rikΓsrj − Γ rijΓ srk. (28)

We define

R

(∂

∂xj,∂

∂xk

)∂

∂xi=

d∑s=1

Rsijk∂

∂xs.

Thus for vector fields, X = (ξ1, . . . , ξd), Y = (η1, . . . , ηd), and Z = (ζ1, . . . , ζd) on M ,we get the curvature tensor

R(X,Y )Z = R

d∑i=1

ξi∂

∂yi,d∑j=1

ηj∂

∂yj

d∑k=1

ζk∂

∂yk

=d∑

i,j,k=1

ξiηjζkR

(∂

∂yi,∂

∂yj

)∂

∂yk

=

d∑i,j,k,s=1

ξiηjζkRskij

∂

∂xs.

Note that the Riemann curvature tensor is a (1, 3)-tensor. Using the Riemann curvaturetensor we now define the Ricci tensor.

Definition 5.40. The Ricci tensor is a (0, 2)-tensor given by the trace of the curvaturetensor. That is for an ON-basis (E1, . . . , Ed)

Ric(Y, Z)(x) := Ric(X 7→ R(X,Y )Z) =d∑i=1

〈R(Ei(x), Y (x))Z(x), Ei(x)〉Tx .

Remark 5.41. We recall that ∂∂y1

, . . . , ∂∂yd

is a basis for Tx at x. In the same waydy1, . . . , dyd is a basis for the dual space (Tx)∗ with

dyi

∣∣∣x

(∂

∂yj

∣∣∣x

)= δij =

1, i = j

0, i 6= j.

In the same way(dyi

∣∣∣x⊗ dyj

∣∣∣x

)( ∂

∂yk

∣∣∣x⊗ ∂

∂yl

∣∣∣x

)= δikδjl =

1, i = k, j = l

0, otherwise.

Remark 5.42. Taking into account (29) and Remark 5.41 we note that the Ricci tensorhas representation

Ric = Rijdyi ⊗ dyj .


It follows that the Ricci tensor is given by the matrix described by

Rjk =d∑i=1

Riijk. (29)

This gives an intuitive access to Ricci curvature in the form of a matrix.

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Papers

The articles associated with this thesis have been removed for copyright reasons. For more details about these see: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-121636

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-121636

a class of infinite dimensional stochastic processes with...

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