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On the density of systems of non-linearspatially homogeneous SPDEsEulalia Nualart aa Institut Galilée , Université Paris 13, 93430, Villetaneuse , FrancePublished online: 28 Feb 2012.

To cite this article: Eulalia Nualart (2013) On the density of systems of non-linear spatiallyhomogeneous SPDEs, Stochastics An International Journal of Probability and Stochastic Processes:formerly Stochastics and Stochastics Reports, 85:1, 48-70, DOI: 10.1080/17442508.2011.653567

To link to this article: http://dx.doi.org/10.1080/17442508.2011.653567

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On the density of systems of non-linear spatially homogeneous SPDEs

Eulalia Nualart*

Institut Galilee, Universite Paris 13, 93430 Villetaneuse, France

(Received 29 November 2009; final version received 23 December 2011)

In this paper, we consider a system of k second-order nonlinear stochastic partialdifferential equations with spatial dimension d $ 1, driven by a q-dimensionalGaussian noise, which is white in time and with some spatially homogeneouscovariance. The case of a single equation and a one-dimensional noise has largely beenstudied in the literature. The first aim of this paper is to give a survey of some of theexisting results. We will start with the existence, uniqueness and Holder’s continuity ofthe solution. For this, the extension of Walsh’s stochastic integral to cover somemeasure-valued integrands will be recalled. We will then recall the results concerningthe existence and smoothness of the density, as well as its strict positivity, which areobtained using techniques of Malliavin calculus. The second aim of this paper is toshow how these results extend to our system of stochastic partial differential equations(SPDEs). In particular, we give sufficient conditions in order to have existence andsmoothness of the density on the set where the columns of the diffusion matrix spanRk.We then prove that the density is strictly positive in a point if the connected componentof the set where the columns of the diffusion matrix span Rk which contains this pointhas a non-void intersection with the support of the law of the solution. We will finallycheck how all these results apply to the case of the stochastic heat equation in any spacedimension and the stochastic wave equation in dimension d [ {1; 2; 3}.

Keywords: spatially homogeneous Gaussian noise; Malliavin calculus; non-linearstochastic partial differential equations; strict positivity of the density

AMS 2000 Subject Classification: 60H15; 60H07

1. Introduction

Consider the system of stochastic partial differential equations

Luiðt; xÞ ¼Xq

j¼1

s ijðuðt; xÞÞ _Wjðt; xÞ þ biðuðt; xÞÞ; t $ 0; x [ Rd; ð1:1Þ

i ¼ 1; . . . ; k, with vanishing initial conditions.HereL is a second-order differential operator,

ands ij; bi : Rk 7! R are globally Lipschitz functions, which are the entries of a k £ qmatrix

s and a k-dimensional vector b. We denote bys 1; . . . ;s q the columns of thematrixs . The

driving perturbation _Wðt; xÞ ¼ ð _W1ðt; xÞ; . . . ; _Wqðt; xÞÞ is a q-dimensional Gaussian noise

which is white in time and with a spatially homogeneous covariance f, that is,

E½ _Wiðt; xÞ _Wjðs; yÞ� ¼ dðt 2 sÞf ðx 2 yÞdij;

ISSN 1744-2508 print/ISSN 1744-2516 online

q 2013 Taylor & Francis

http://dx.doi.org/10.1080/17442508.2011.653567

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*Email: [email protected], http://nualart.es

Stochastics: An International Journal of Probability and Stochastic Processes

Vol. 85, No. 1, February 2013, 48–70

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http://nualart.es

where dð�Þ denotes the Dirac delta function, dij is the Kronecker symbol and f is a positive

continuous function on Rdn{0}.

The basic examples we are interested in are the stochastic wave and heat equations

with vanishing initial conditions, that is, L ¼ ð›2=›t 2Þ2 D and L ¼ ð›=›tÞ2 D, where D

denotes the Laplacian operator in Rd, and the spatial covariance f to be the Riesz kernel,

that is, f ðxÞ ¼ kxk2b

, 0 , b , d.

Let ðF tÞt$0 denote the filtration generated by W, and let T . 0 be fixed. By definition,

the solution to the formal Equation (1.1) is an adapted stochastic process {uðt; xÞ ¼

ðu1ðt; xÞ; . . . ; ukðt; xÞÞ; ðt; xÞ [ ½0; T� £ Rd} such that

uiðt; xÞ ¼Xq

j¼1

ðt

0

ðRd

Gðt 2 s; x 2 yÞs ijðuðs; yÞÞW jðds; dyÞ

þ

ðt

0

ðRd

biðuðt 2 s; x 2 yÞÞGðs; dyÞds;

ð1:2Þ

where G denotes the fundamental solution of the deterministic equation Lu ¼ 0.

Recall that when Gðt; xÞ is a real-valued function, the stochastic integral appearing in

(1.2) is the classical Walsh stochastic integral (see [26]). However, when G is a measure,

Dalang [5] extended Walsh’s stochastic integral using techniques of Fourier analysis, and

covered, for instance, the case of the wave equation in dimension 3. Nualart and Quer-

Sardanyons [19] extend Walsh’s stochastic integral using techniques of stochastic

integration with respect to a cylindrical Wiener process (see [12]), in order to cover some

classes of measure-valued integrands.

In this paper, we are interested in studying the existence, smoothness and strict

positivity of the density of the solution to the system of SPDEs (1.1), on the set where

{s 1; . . . ;s q} span Rk. The case of a single equation has largely been studied in the

literature, so our aim is to give a survey of the known results and explain how they extend

to the case of a system of equations.

One of the motivations of the results of this article is to develop in a further work

potential theory for solutions of systems of the type (1.1) (see [9]). For the moment,

potential theory for systems of nonlinear SPDEs has been studied by Dalang and Nualart

[6] for the wave equation, and by Dalang, Khoshnevisan and Nualart [7,8], for the heat

equation, all driven by space-time white noise. That is, taking in Equation (1.1), d ¼ 1, f

the Dirac delta function and L the fundamental solution of the deterministic wave or heat

equation. In all these works, the existence, smoothness and strict positivity of the density

of the solution to the system of SPDEs are required.

The paper is organized as follows. Section 2 deals with the existence and uniqueness of

the solution to (1.2). For this, we first need to define rigorously the noise W, and explain in

which sense the stochastic integral in (1.2) is understood, recalling the mentioned results in

Refs [5] and [19]. Studying the density of an SPDE is sometimes related to the Holder

continuity properties of the paths of its solution. Thus, we will recall some results

concerning the Holder continuity of the solution to the stochastic heat and wave equations.

This will be done in Section 3. Section 4 will be devoted to the study of the existence and

smoothness of the density of the solution to (1.1). For this, some elements of Malliavin

calculus need to be introduced. Finally, the aim of Section 5 is to show the strict positivity

of the density. This last result turns out to be new in the literature. Its proof extends the

method used by Bally and Pardoux [1] for the case of the stochastic heat equation driven

by a space-time white noise, in our spatially homogeneous situation.

Stochastics: An International Journal of Probability and Stochastic Processes 49

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2. The stochastic integral

The aim of this section is to recall the results concerning the existence and uniqueness of

the solution to our class of SPDEs (1.1). For this, we will first recall in our q-dimensional

situation the extension of Walsh’s stochastic integral given in Ref. [19], which uses Da

Prato and Zabczyk’s stochastic integration theory with respect to cylindrical Wiener

processes. In the recent paper [11], Dalang and Quer-Sardanyons extensively explain how

this extension is related to the pioneer work by Dalang [5].

Fix a time interval ½0; T�. The Gaussian random perturbation W ¼ ðW 1; . . . ;W qÞ

is a q-dimensional zero mean Gaussian family of random variables

W ¼ {W jðwÞ; 1 # j # q;w [ C10 ð½0; T� £ Rd;RÞ}, defined on a complete probability

ðV;F ; PÞ with covariance

E½W iðwÞW jðcÞ� ¼ dij

ðT

0

ðRd

ðRd

wðt; xÞf ðx 2 yÞcðt; yÞdxdydt;

where f : Rd ! Rþ is a continuous function on Rdn{0} which is integrable in a

neighbourhood of 0.

Observe that in order that there exists a Gaussian process with this covariance

functional, it is necessary and sufficient that the covariance functional is non-negative

definite. This implies that f is symmetric, and is equivalent to the fact that it is the Fourier

transform of a non-negative tempered measure m on Rd. That is,

f ðxÞ ¼ FmðxÞ :¼

ðRd

e22pix�jm ðdjÞ;

and for some integer m $ 1, ðRd

mðdjÞ

ð1þ kjk2Þm

, þ1:

Elementary properties of the convolution and Fourier transform show that this

covariance can also be written in terms of the measure m as

E½W iðwÞW jðcÞ� ¼ dij

ðT

0

ðRd

FwðtÞðjÞFcðtÞðjÞmðdjÞdt;

where Fc denotes the complex conjugate of Fc.

Let Hq denote the completion of the Schwartz space SðRd;RqÞ of C1ðRd;RqÞ

functions with rapid decrease, endowed with the inner product

kw;clHq ¼Xq

l¼1

ðRd

ðRd

wlðxÞf ðx 2 yÞclðyÞdx dy ¼Xq

l¼1

ðRd

FwlðjÞFclðjÞmðdjÞ;

f;c [ SðRd;RqÞ. Notice that Hq may contain distributions. Set HqT ¼ L2ð½0; T�;HqÞ.

In particular, HqT is the completion of the space C10 ð½0; T� £ Rd;RqÞ with respect to the

scalar product

kw;clHqT¼Xq

l¼1

ðT

0

ðRd

FwlðtÞðjÞFclðtÞðjÞmðdjÞdt:

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Then the Gaussian family W can be extended to HqT and we will use the same notation

{WðgÞ; g [ HqT}, where WðgÞ ¼

Pqi¼1W

iðgiÞ.

Now, set WtðhÞ ¼Pq

i¼1Wið1½0;t�hiÞ, for any t $ 0, h [ Hq. Then {Wt; t [ ½0; T�} is

a cylindrical Wiener process in the Hilbert space Hq (cf. [12], Section 4.3.1). That is, for

any h [ Hq, {WtðhÞ; t [ ½0; T�} is a Brownian motion with variance tkhk2Hq , and

E½WsðhÞWtðgÞ� ¼ ðs ^ tÞkh; glHq :

Let ðF tÞt$0 denote the s -field generated by the random variables

{WsðhÞ; h [ Hq; 0 # s # t} and the P-null sets. We define the predictable s -field as

the s -field in V £ ½0; T� generated by the sets {ðs; t� £ A; 0 # s , t # T ;A [ F s}. Then

following Ref. [12] (Chapter 4), we can define the stochastic integral of any predictable

process g [ L2ðV £ ½0; T�;HqÞ with respect to the cylindrical Wiener process W as

ðT

0

ðRd

g�dW :¼X1j¼1

ðT

0

kgt; ejlHqdWtðejÞ;

where ðejÞ is an orthonormal basis ofHq. Moreover, the following isometry property holds:

E

ðT

0

ðRd

g�dW

��2

" #¼ E

ðT

0

kgtk2Hqdt

� �:

We next introduce the following condition on the fundamental solution of Lu ¼ 0, G.

(H1): For all t . 0, GðtÞ is a non-negative distribution with rapid decrease, such thatðT

0

ðRd

jFGðtÞðjÞj2mðdjÞdt , þ1: ð2:1Þ

Moreover, G is a non-negative measure of the form Gðt; dxÞdt such that

supt[½0;T�

ðRd

Gðt; dxÞ # CT , þ1: ð2:2Þ

Then Ref. [19] (Lemma 3.2 and Proposition 3.3) shows that under condition (H1),

G belongs to HT , and one can define the stochastic integral of a predictable process

G ¼ Gðt; dxÞ ¼ Zðt; xÞGðt; dxÞ [ L2ðV £ ½0; T�;HqÞ with respect to W, provided that

Z ¼ {Zðt; xÞ; ðt; xÞ [ ½0; T� £ Rd} is a predictable process such that

supðt;xÞ[½0;T�£Rd

E½kZðt; xÞk2� , þ1:

In this case, we denote the stochastic integral as

ðT

0

ðRd

Gðs; yÞ�Wðds; dyÞ ¼

ðT

0

ðRd

Gðs; yÞZðs; yÞ�Wðds; dyÞ:

We next state the existence and uniqueness result of the solution to Equation (1.2).

This is an extension to systems of SPDEs of Ref. [19] (Theorem 4.1; see also [5]

Theorem 13) and can be proved in the same way.


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Theorem 2.1. Under condition (H1), there exists a unique adapted process

{uðt; xÞ ¼ ðu1ðt; xÞ; . . . ; ukðt; xÞÞ; ðt; xÞ [ ½0; T� £ Rd} solution to Equation (1.2), which

is continuous in L2 and satisfies that for all T . 0 and p $ 1,

supðt;xÞ[½0;T�£Rd

E½ uðt; xÞk kp� , þ1: ð2:3Þ

The basic exampleswe are interested in are the stochastic heat andwave equations.More

precisely, it is well known (see for e.g. [5]) that if L is the heat operator inRd, d $ 1, that is,

L ¼ ð›=›tÞ2 D, where D denotes the Laplacian operator in Rd, or if L is the wave operator

in Rd, d [ {1; 2; 3}, that is, L ¼ ð›2=›t 2Þ2 D, condition (H1) is satisfied if and only if

ðRd

mðdjÞ

ð1þ kjk2Þ, þ1: ð2:4Þ

For instance, one can take f to be a Riesz kernel, that is, f ðxÞ ¼ kxk2b

, 0 , b , d.

Then, mðdjÞ ¼ cd;bkjkb2d

dj and (2.4) holds if and only if 0 , b , ð2 ^ dÞ.

3. Holder continuity of the solution

Studying existence, smoothness and strict positivity of the density of the solution to the

system of equations (1.1) will require to know the behaviour of the moments of the

increments of the solution, which in particular implies the Holder continuity of the paths.

Let us introduce the following conditions.

(H2): There exists g1 . 0 such that for all t [ ½0; T�, h [ ½0; T 2 t�, x [ Rd, and

p . 1,

E½kuðt þ h; xÞ2 uðt; xÞkp� # cp;T hg1p;

for some constant cp;T . 0. In particular, for all d . 0 the trajectories of u are

ðg1 2 dÞ-Holder continuous in time

(H3): There exists g2 . 0 such that for all t [ ½0; T�, x; y [ Rd, and p . 1,

E½kuðt; xÞ2 uðt; yÞkp� # cp;Tkx 2 yk

g2p;

for some constant cp;T . 0. In particular, for all d . 0 the trajectories of u are

ðg2 2 dÞ-Holder continuous in space

Using Kolmogorov’s continuity theorem, Sanz-Sole and Sarra [25] prove that if there

exists 1 [ ð0; 1Þ such that

ðRd

mðdjÞ

ð1þ kjk2Þ1

, þ1; ð3:1Þ

then the stochastic heat equation satisfies (H2) for all g1 [ ð0; ð12 1Þ=2Þ, and (H3)

for all g2 [ ð0; 12 1Þ. In particular, if f is the Riesz kernel, that is mðdjÞ ¼ cd;bkjkb2d

dj,

0 , b , ð2 ^ dÞ, then (H2) holds for all g1 [ ð0; ð22 bÞ=4Þ, and (H3) holds for all

g2 [ ð0; ð22 bÞ=2Þ. Observe that in this case (3.1) holds for all 1 . b=2.

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For the stochastic wave equation in dimensions d [ {1; 2}, one obtains that under

condition (3.1), (H2) and (H3) are satisfied for all g1; g2 [ ð0; 12 1Þ, that is,

g1; g2 [ ð0; ð22 bÞ=2Þ for the Riesz kernel case. The two-dimensional case was studied

by Millet and Sanz-Sole [16] (Proposition 1.4). See Ref. [26] for the one-dimensional case.

A different approach based on Sobolev embedding theorem is needed to handle the

stochastic wave equation in dimension 3. This was done by Dalang and Sanz-Sole [10] for

the case where the spatially homogeneous covariance is defined as a product of a Riesz

kernel and a smooth function. In particular, when it is a Riesz kernel, they show that (H2)

and (H3) are satisfied with g1; g2 [ ð0; ð22 bÞ=2Þ.

4. Existence and smoothness of the density

We are now interested in proving existence and smoothness of the density of the solution

to system (1.1) on the set where {s 1; . . . ;s q} span Rk. As is well known, the Malliavin

calculus provides a powerful tool in order to show this kind of results for solutions to

stochastic differential equations (SDEs) and SPDEs. Let us first recall some existing

results for single equations of type (1.1). For the existence and regularity of the density of

the wave equation with d ¼ 1, one refers to the work by Carmona and Nualart [3]. The

wave equation in spatial dimension 2 was studied by Millet and Sanz-Sole [16]. The three-

dimensional case was then handled by Quer-Sardanyons and Sanz-Sole, see [23,24]. The

case of the stochastic heat equation in any dimension can be found in Ref. [14] (the one-

dimensional case was done earlier in Ref. [1]). All these results were unified and

generalized by Nualart and Quer-Sardanyons [19]. They assumed the following condition.

(H4): There exists h . 0 such that for all t [ ½0; 1�,

cth #

ðt0

ðRd

jFGðtÞðjÞj2mðdjÞdt;

for some constant c . 0

Then they show that if s ; b are C1 functions with bounded derivatives of order greater

than or equal to one, and js j $ c . 0, under conditions (H1) and (H4), for all

ðt; xÞ [�0; T� £ Rd, the law of uðt; xÞ has a C1 density.

Observe that condition (H4) is satisfied for the stochastic heat equation for any h $ 1

(see [19] Remark 6.3), and with h ¼ 3 for the stochastic wave equation in dimensions 1; 2and 3 (see [23] (A.3)).

The proof of this result uses tools of Malliavin calculus. One needs to check first that

the random variable uðt; xÞ is smooth in the Malliavin sense. This can be easily generalized

to systems of equations (see Proposition 4.3). Secondly, one needs to show that the

Malliavin matrix has inverse moments of all orders. For this, one uses condition (H4) and

the non-degeneracy condition on s . The extension of this result to systems of equations is

more delicate as ones deals with a matrix s instead of a function, and because of that

reason we need to consider the following additional assumptions on G.

(H5): Conditions (Hi), i ¼ 1; 2; 3 are satisfied. Moreover, the non-negative measure C

defined as kxkg2Gðt; dxÞ satisfies that

ðT

0

ðRd

jFCðtÞðjÞj2mðdjÞdt , þ1:


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Moreover, there exist a1;a2 . 0, 2g1 , a1, 2g2 , a2, such that for all t [ ½0; 1�,ðt0

r 2g1

ðRd

jFGðrÞðjÞj2mðdjÞdr # cta1 ;

for some constant c . 0, andðt0

ðRd

jFCðrÞðjÞj2mðdjÞdr # cta2 ;

for some constant c . 0

(H6): Conditions (H4) and (H5) are satisfied, and a :¼ a1 ^ a2 . h

We will then show the following result.

Theorem 4.1. Assume hypothesis (H6) and that s ; b are C1 functions with bounded

partial derivatives of order greater than or equal to one. Then for all ðt; xÞ [�0; T� £ Rd, the

law of the random vector uðt; xÞ admits a C1 density on the open subset of Rk

S :¼ {y [ Rk : s 1ðyÞ; . . . ;s qðyÞ span Rk}. That is, there exists a function pt;x [C1ðS;RÞ such that for every bounded and continuous function f : Rk 7! R with support

contained in S,

E½f ðuðt; xÞÞ� ¼

ðRk

f ðyÞpt;xðyÞdy:

We next prove an intermediary result that will be needed for the proof of Theorem 4.1.

Lemma 4.2. Assume hypothesis (H5). Then for all t [ ½0; T�, x [ Rd, 1 [ ð0; 1� and

p . 1,

E

ð10

ðs ijðuðt 2 r; *ÞÞ2 s ijðuðt; xÞÞÞGðr; x 2 *Þ�� 2

Hdr

��p� �

# cp;T1ap;

for some constant cp;T . 0, where a :¼ a1 ^ a2 and a1;a2 are the parameters in

hypothesis (H5).

Proof. Set Gðr; dyÞ :¼ ðs ijðuðt 2 r; yÞÞ2 s ijðuðt; xÞÞÞGðr; x 2 dyÞ. Then by Ref. [19]

(Proposition 3.3), Gðr; dyÞ [ L2ðV £ ½0; T�;HÞ, and following the proof of this

proposition, we can write

kGðr; x 2 *Þjj2H ¼

ðRd

ðRd

Gðr; x 2 dyÞf ðy 2 zÞGðr; x 2 dzÞ:

Using Minkowski’s inequality with respect to the finite measure

Gðr; x 2 dyÞGðr; x 2 dzÞf ðy 2 zÞdr, together with the Lipschitz property of s , we get

that for all p . 1,

E

ð10

Gðr; x 2 *Þk k2Hdr

��p� �

# cpj

ð10

ðRd

ðRd

Gðr; x 2 dyÞ

£ E½kuðt 2 r; yÞ2 uðt; xÞkpkuðt 2 r; zÞ2 uðt; xÞk

p�

� �1=pGðr; x 2 dzÞf ðy 2 zÞdrj

p:

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Therefore, appealing to hypotheses (H2) and (H3), we find that the last term is bounded by

cp;T

ð10

ðr g1 þ kx 2 *kg2 ÞGðr; x 2 *Þ

�� 2Hdr

��p

# cp;T

ð10

r 2g1

ðRd

FGðrÞðjÞj j2mðdjÞdr

��p

þ

ð10

ðRd

jFCðrÞðjÞj2mðdjÞdr

��p� �

:

Hence, using hypothesis (H5), we conclude that

E

ð10

Gðr; x 2 *Þk k2Hdr

��p� �

# cp;T ð1a1p þ 1a2pÞ # cp;T1

ap;

where a :¼ a1 ^ a2. A

Next, we recall some elements of Malliavin calculus. Consider the Gaussian family

{WðhÞ; h [ HqT} defined in Section 2, that is, a centred Gaussian process such that

E½WðhÞWðgÞ� ¼ kh; glHq

T. Then, we can use the differential Malliavin calculus based on it

(see for instance [18]). We denote the Malliavin derivative by D ¼ ðD ð1Þ; . . . ;D ðqÞÞ, which

is an operator in L2ðV;HqT Þ. For any m $ 1, the domain of the iterated derivative Dm in

LpðV; ðHqT Þ

^mÞ is denoted by Dm;p, for any p $ 2. We set D1 ¼ >p$1 >m$1 Dm;p. Recall

that for any differentiable random variable F and any r ¼ ðr1; . . . ; rmÞ [ ½0; T�m, DmFðrÞ

is an element of H^m, which will be denoted by Dmr F. We define the Malliavin matrix of

F [ ðD1Þm by gF ¼ ðkDFi;DFjlHqTÞ1#i;j#m.

The next result is the q-dimensional extension of Ref. [19] (Proposition 6.1; see also

[24] Theorem 1). Its proof follows exactly along the same lines working coordinate by

coordinate, and is therefore omitted.

Proposition 4.3. Assume that (H1) holds, and that s ; b are C1 functions with bounded

partial derivatives of order greater than or equal to one. Then, for every

ðt; xÞ [ ð0; T� £ Rd, the random variable uiðt; xÞ belongs to the space D1, for all

i ¼ 1; . . . ; k. Moreover, the derivative Duiðt; xÞ ¼ ðD ð1Þuiðt; xÞ; . . . ;D ðqÞuiðt; xÞÞ is an HqT -

valued process that satisfies the following linear stochastic differential equation, for all

i ¼ 1; . . . ; k:

DðjÞr uiðt; xÞ ¼ s ijðuðr; *ÞÞGðt 2 r; x 2 *Þ

þ

ðt

r

ðRd

Gðt 2 s; x 2 yÞXq

l¼1

DðjÞr ðs ilðuðs; yÞÞW lðds; dyÞ

þ

ðt

r

ðRd

Gðt 2 s; dyÞDðjÞr ðbiðuðs; x 2 yÞÞds;

ð4:1Þ

for all r [ ½0; t�, and is 0 otherwise. Moreover, for all p $ 1, m $ 1 and i ¼ 1; . . . ; k,

it holds that

supðt;xÞ[ð0;T�£Rd

E kDmuiðt; xÞjjp

ðHq

TÞ^m

h i, þ1: ð4:2Þ

Remark 4.4. Recall that the iterated derivative of uiðt; xÞ satisfies also the q-dimensional

extension of equation ([19], (6.27)).


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Recall that the stochastic integral on the right-hand side of (4.1) must be understood by

means of a Hilbert-valued stochastic integral (see [19] Section 3) and the Hilbert-valued

pathwise integral of (4.1) is defined in Ref. [19] (Section 5).

In order to prove Theorem 4.1, we will use the following localized variant of

Malliavin’s absolute continuity theorem.

Theorem 4.5 ([1], Theorem 3.1). Let Sm , Rk, m [ N, m $ 1, be a sequence of open

sets such that �Sm , Smþ1 and let F [ ðD1Þk such that for any q . 1 and m [ N,

E ðdetgFÞ2q1{F[Sm}

, þ1: ð4:3Þ

Then the law of F admits a C1 density on the set S ¼ <mSm.

Proof of Theorem 4.1. For each m [ N, m $ 1, define the open set

Sm ¼ y [ Rk : kjTs ðyÞk2.

1

m;;j [ Rk; kjk ¼ 1

� �: ð4:4Þ

By Proposition 4.3, it suffices to prove that condition (4.3) of Theorem 4.5 holds true in

each Sm. Indeed, observe that S ¼ <mSm ¼ {s 1; . . . ;s q span Rk}.

Let ðt; xÞ [�0; T� £ Rd and m $ 1 be fixed. It suffices to prove that there exists

d0ðmÞ . 0 such that for all 0 , d # d0, and all p . 1,

P ðdetguðt;xÞ , dÞ1{uðt;xÞ[Sm}

� �# Cdlp; ð4:5Þ

for some l . 0, and for some constant C . 0 not depending on d. This implies (4.3),

taking p such that 1 , ðl=qÞp in (4.5).

We write

detguðt;xÞ $ infj[Rk :kjk¼1ðjTguðt;xÞjÞ

�k

: ð4:6Þ

Let j [ Rk with kjk ¼ 1, and fix 1 [ ð0; 1�. The inequality

ka þ bk2Hq $

1

2kak

2Hq 2 jbj

2Hq ; ð4:7Þ

together with (4.1), gives

jTguðt;xÞj $

ðt

t21

Xk

i¼1

Drðuiðt; xÞÞji

��2

Hq

dr $1

2A1 2A2;

where

A1 ¼

ðt

t21

ðjTs ðuðr; *ÞÞÞGðt 2 r; x 2 *Þ�� 2

Hqdr;

A2 ¼

ðt

t21

Xk

i¼1

aiðr; t; x; *Þji

��2

Hq

dr;

aiðr; t; x; *Þ ¼Xq

l¼1

ðt

r

ðRd

Gðt 2 s; x 2 yÞDrðs ilðuðs; yÞÞW lðds; dyÞ

þ

ðt

r

ðRd

Drðbiðuðs; x 2 yÞÞGðt 2 s; yÞdyds:

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Now, assume that uðt; xÞ [ Sm. Then, appealing again to inequality (4.7), we obtain

that A1 $ ð1=2ÞA1;1 2A1;2, where

A1;1 ¼

ðt

t21

ðjTs ðuðt; xÞÞÞGðt 2 r; x 2 *Þ�� 2

Hqdr;

A1;2 ¼

ðt

t21

ðjT ðs ðuðr; *ÞÞ2 s ðuðt; xÞÞÞÞGðt 2 r; x 2 *Þ�� 2

Hqdr:

Note that we have added and subtracted a ‘local’ term to make the ellipticity property

appear (see (4.4)). A similar idea is used in Ref. [16] for the stochastic wave equation in

dimension 2.

Then, using the fact that uðt; xÞ [ Sm, we get that

A1;1 .1

m

ð10

ðRd

jFGðrÞðjÞj2mðdjÞdr ¼:

1

mgð1Þ:

Now we find out upper bounds for the pth moment, p . 1, of the terms A1;2 and A2.

For A1;2, it suffices to apply Lemma 4.2 to get that, for all p . 1,

E supj[Rd :kjk¼1

jA1;2jp

" ## cp;T1

ap:

We next treatA2. Using the Cauchy–Schwarz inequality, for any p . 1, it yields that

E supj[Rd :kjk¼1

jA2jp

" ## cp

Xk

i¼1

E

ðt

t21

kaiðr; t; x; *Þjj2Hqdr

��p� �

# cp

Xk

i¼1

E

ð10

kI iðr; t; x; *Þjj2Hqdr

��p� �

þ E

ð10

kJ iðr; t; x; *Þjj2Hqdr

��p� ��

;

ð4:8Þ

where

I iðr; t; x; *Þ : ¼Xq

l¼1

ðt

t2r

ðRd

Gðt 2 s; x 2 yÞDt2rðs ilðuðs; yÞÞW lðds; dyÞ;

J iðr; t; x; *Þ : ¼

ðt

t2r

ðRd

Gðt 2 s; yÞDt2rðbiðuðs; x 2 yÞÞdyds:

Now, using Holder’s inequality, the boundedness of the coefficients of the derivatives of

s , and Ref. [19] ((3.13) and (5.26)), we get that

E

ð10

I iðr; t; x; *Þk k2Hqdr

��p� �

# cp1p21 sup

ðs;yÞ[ð0;1Þ£Rd

E Dt2�uiðt 2 s; yÞk k2pHq

1

h i

£

ð10

ðRd

jFGðsÞðjÞj2mðdjÞds

� �p

# cpgð1Þ2p:

Moreover, using Holder’s inequality, the boundedness of the partial derivatives of

the coefficients of b, hypothesis (2.2), and Ref. [19] ((5.17) and (5.26)), for the second term

in (4.8) corresponding to the Hilbert-valued pathwise integral, we obtain that

E

ð10

J iðr; t; x; *Þk k2Hqdr

��p� �

# cp1p21 sup

ðs;yÞ[ð0;1Þ£Rd

E Dt2�uiðt 2 s; yÞk k2pHq

1

h i

£

ð10

ðRd

Gðs; yÞdyds

� �# cp1

pgð1Þp:


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Hence, we conclude that, for any p . 1,

E supj[Rd :kjk¼1

jA2jp

" ## cp gð1Þ2p þ 1pgð1Þp

� �:

Appealing to (4.6), we have proved that, on the set {uðt; xÞ [ Sm},

det guðt;xÞ .1

2mgð1Þ2 I

� �k

; ð4:9Þ

where I is a random variable such that for all p . 1, E½jIjp� # ð1ap þ gð1Þ2p þ 1pgð1ÞpÞ.

We now choose 1 ¼ 1ðd;mÞ in such a way that d1=k ¼ ð1=4mÞgð1Þ. By hypothesis (H4)

this implies that 4md1=k $ c1h, that is, 1 # Cd1=hk. Then using (4.9), we conclude that for

all 0 , d # d0 and p . 1,

P ðdet guðt;xÞ , dÞ1{uðt;xÞ[Sm}

� �#

1ap þ gð1Þ2p þ 1pgð1Þp

ðð1=2mÞgð1Þ2 d1=kÞp# CðmÞ dp=k þ dp=hk þ dlp

�;

with l ¼ ða2 hÞ=hk. Recall that h , a from hypothesis (H6). This proves (4.5). A

We end this section with the verification of condition (H6) for the stochastic heat and

wave equations with a Riesz kernel as spatial homogeneous covariance, that is,

f ðxÞ ¼ kxk2b

and mðdjÞ ¼ cd;bkjkd2b

dj, 0 , b , ð2 ^ dÞ.

Consider first the stochastic heat equation in any spatial dimension, that is,

Gðt; xÞ ¼ ð4ptÞ2d=2exp 2kxk

2

4t

!; t $ 0; x [ Rd:

For all t [ ½0; T�, using the change variables ½~y ¼ y=ffiffir

p; ~z ¼ z=

ffiffir

p�, it yields that

ðt

0

ðRd

ðRd

ky 2 zk2b

Gðr; yÞGðr; zÞdydzdr

¼

ðt

0

r2b=2

ðRd

ðRd

k~y 2 ~zk2b

Gð1; ~yÞGð1; ~zÞd~yd~zdr ¼ ct ð22bÞ=2:

Thus, hypothesis (H4) is satisfied for h ¼ ð22 bÞ=2. Furthermore, hypothesis (H5) holds

with a1 ¼ ðð22 bÞ=2Þ þ 2g1 and a2 ¼ ðð22 bÞ=2Þ þ g2. Indeed, using the change of

variables ½~y ¼ y=ffiffir

p; ~z ¼ z=

ffiffir

p�, for all t [ ½0; 1�, we have that

ðt0

r 2g1

ðRd

ðRd

ky 2 zk2b

Gðr; zÞGðr; yÞdz dy dr

¼

ðt0

r 2g12ðb=2Þ

ðRd

ðRd

k~y 2 ~zk2b

Gð1; ~zÞGð1; ~yÞd~z d~y dr ¼ ct ðð22bÞ=2Þþ2g1 ;

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andðt0

ðRd

ðRd

kzkg2kyk

g2ky 2 zk2b

Gðr; zÞGðr; yÞdz dy dr

¼

ðt0

r g22ðb=2Þ

ðRd

ðRd

k~zkg2k~yk

g2k~y 2 ~zk2b

Gð1; ~zÞGð1; ~yÞd~zd~ydr ¼ ct ðð22bÞ=2Þþg2 :

Observe that a ¼ a1 ^ a2 ¼ ðð22 bÞ=2Þ þ ð2g1Þ ^ g2, which is bigger that h, thus

proves (H6).

We next treat the case of the stochastic wave equation with spatial dimension

d [ {1; 2; 3}. Let Gd be the fundamental solution of the deterministic wave equation in Rd

with null initial conditions. In this case, changing variables ½ ~j ¼ rj�, for all t [ ½0; T�,

we get thatðt

0

ðRd

jFGdðrÞðjÞj2kjk

b2ddj dr ¼

ðt

0

ðRd

sin2ð2prkjkÞ

4p2kjk2

kjkb2d

dj dr

¼

ðt

0

r 22b

ðRd

sin2ð2pk ~jkÞ

4p2k ~jk2

k ~jkb2d

d ~j dr

¼ ct 32b:

Therefore, hypothesis (H4) is satisfied for h ¼ 32 b. Notice also that

ðT

0

ðRd

jFCðrÞðjÞj2kjk

b2ddj dr # T 2g2

ðT

0

ðRd

jFGdðrÞðjÞj2kjk

b2ddj dr , þ1:

Moreover, changing variables ½ ~j ¼ rj�, for all t [ ½0; 1�, we have that

ðt0

ðRd

jFCðrÞðjÞj2kjk

b2ddj dr ¼

ðt0

r2b

ðRd

jFCðrÞ~j

r

� �j2k ~jk

b2dd ~j dr

¼ kCð1; *Þk2H

ðt0

r 22bþ2g2dr

¼ ct32bþ2g2 ;

and

ðt0

r 2g1

ðRd

jFGdðrÞðjÞj2kjk

b2ddj dr ¼ ct32bþ2g1 :

Thus, hypotheses (H5) and (H6) hold taking a ¼ 32 bþ 2ðg1 ^ g2Þ . 32 b ¼ h.

5. Strict positivity of the density

The second aim of this article is to show that under the same condition (H6), the density of

the law of the solution to system (1.1) is strictly positive in a point if the connected

component of the set where {s 1; . . . ;s q} span Rk that contains this point has a non-void

intersection with the support of the law of the solution (see Theorem 5.1). For this, we will

first extend in our situation a criterion of strict positiveness of densities proved by Bally

and Pardoux [1], which uses essentially an inverse function type result (see Lemma 5.6)

and Girsanov’s theorem.


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We will then apply this criterion to our system of SPDEs (1.1). In Ref. [1], the authors

apply their criterion to the density of the law of the random vector ðuðt; x1Þ; . . . :; uðt; xkÞÞ,

0 # x1 # · · · # xk # 1, where u denotes the solution to the nonlinear stochastic heat

equation driven by a space-time white noise, by using a localizing argument. Hence, their

situation is a bit different as ours, as we deal with a system of SPDEs and we evaluate the

solution at a single point ðt; xÞ [�0; T� £ Rd . In a similar context, Chaleyat and Sanz-Sole

[4] study the strict positivity of the density of the random vector ðuðt; x1Þ; . . . :; uðt; xkÞÞ,

0 # x1 # · · · # xk # 1, where u is the solution to the stochastic wave equation in two

spatial dimensions, that is, take in Equation (1.1) d ¼ 2, k; q ¼ 1 and L ¼ ð›2=›t 2Þ2 D.

Wewill also apply the criterion of strict positivity to the case of a single equation, which

is the solution to (1.1) with k; q ¼ 1.Wewill obtain that the density is strictly positivity in all

R under the same hypotheses that Nualart and Quer-Sardanyons [19] obtained its existence

and smoothness (see Theorem 5.3), with the addition condition of s being bounded. Recall

that in the recent papers [20] and [21], the same authors obtain lower and upper bounds of

Gaussian type for the density of the solution to single spatially homogeneous SPDEs of class

(1.1) in the case where s is a constant, which in particular shows the strict positivity of the

density in the quasi-linear case. A first step to show bounds for the nonlinear case is done in

the recent pre-print [22], where upper and lower bounds for the density of nonlinear

stochastic heat equations in any space dimension are established.

There are also other many SPDEs for which the strict positivity of its density is studied.

For example, the case of nonlinear hyperbolic SPDEs has been studied by Millet and Sanz-

Sole [15], Fournier [13] considers a Poisson-driven SPDE, and the Cahn-Hilliard

stochastic equation is studied by Cardon-Weber [2].

We next state the main result of this paper.

Theorem 5.1. Assume hypothesis (H6) and that s ; b are C1 functions with bounded

partial derivatives of order greater than or equal to one and s is bounded. Then for all

ðt; xÞ [�0; T� £ Rd, the law of the random vector uðt; xÞ admits a C1 density on S :¼

{y [ Rk : s 1ðyÞ; . . . ;s qðyÞ span Rk} such that pt;xðyÞ . 0 if the connected component

of S which contains y has a non-void intersection with the support of uðt; xÞ.

Remark 5.2. In the case where s is uniformly elliptic, that is, ks ðyÞjk2$ c . 0, for all

j [ Rq and y [ Rk, under the hypotheses of Theorem 5.1, we get that for all

ðt; xÞ [�0; T� £ Rd, the law of the randomvector uðt; xÞ admits a C1 strictly positive density.

The case of a single SPDE of the type (1.1) is as follows.

Theorem 5.3. Assume that s ; b are C1 functions with bounded derivatives of order

greater than or equal to one, and 0 , c # js j # C. Then, under conditions (H1) and (H4),

for all ðt; xÞ [�0; T� £ Rd, the law of uðt; xÞ has a C1 strictly positive density function.

Remark 5.4. Observe that we only require the continuity of the density in order to show the

strict positivity (see Theorem 5.7). Hence the C1 condition on the coefficients can be

weakened in order to show the strict positivity of the density.

5.1 Criterion for the strict positivity of the density

As in Section 4, we will study the strict positivity of the density on the set where the

columns of the k £ q matrix s , s 1; . . . ;s q span Rk. If y [ Rk is in this set, then there

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exists l1ðyÞ ¼: l1; . . . ; lkðyÞ ¼: lk such that s l1ðyÞ; . . . ;s lk ðyÞ span Rk. Thus, it suffices to

make all the calculations using W l1ðyÞ; . . . ;W lkðyÞ and ignoring the other W j. In order to

simplify the notation we will take li ¼ i.

Given T . 0, a predictable process g [ HkT and z [ Rk, we define the process

W ¼ ðW 1; . . . ; WkÞ as

W jð1½0;t�hjÞ ¼ W jð1½0;t�hjÞ þ zj

ðt

0

khjð*Þ; gjðs; *ÞlHds;

for any h [ Hk, j ¼ 1; . . . ; k, t [ ½0; T�.

We set WtðhÞ ¼Pk

j¼1W jð1½0;t�hjÞ. Then, by Ref. [12] (Theorem 10.14),

{Wt; t [ ½0; T�} is a cylindrical Wiener process in Hk on the probability space

ðV;F ; PÞ, where

dP

dPðvÞ ¼ JzðvÞ; v [ V;

where

Jz ¼ exp 2Xk

j¼1

zj

ðT

0

ðRd

g jðs; yÞW jðds; dyÞ21

2

Xk

j¼1

z2j

ðT

0

gjðs; *Þ�� 2

Hds

!: ð5:1Þ

Then, for any predictable process Z [ L2ðV £ ½0; T�;HkÞ and j ¼ 1; . . . ; k, it yields that

ðT

0

ðRd

Zjðs; yÞW jðds; dyÞ ¼

ðT

0

ðRd

Zjðs; yÞW jðds; dyÞ þ zj

ðT

0

kZjðs; *Þ; gjðs; *ÞlHds:

For any ðt; xÞ [ ½0; T� £ Rk, let u zðt; xÞ be the solution to Equation (1.2) with respect to

the cylindrical Wiener process W, that is, for i ¼ 1; . . . ; k,

uzi ðt; xÞ ¼

Xk

j¼1

ðt

0

ðRd

Gðt 2 s; x 2 yÞs ijðuzðs; yÞÞW jðds; dyÞ

þXk

j¼1

zj

ðt

0

kGðt 2 s; x 2 �Þs ijðuzðs; *ÞÞ; gjðs; *ÞlHds

þ

ðt

0

ðRd

biðuzðt 2 s; x 2 yÞÞGðs; dyÞds:

ð5:2Þ

Then, the law of u under P coincides with the law of u z under P, which implies that for all

non-negative continuous and bounded function f : Rk 7! R,

E½f ðuðt; xÞÞ� ¼ E½f ðu zðt; xÞÞJz�: ð5:3Þ

Given a sequence {gn}n$1 of predictable processes inHkT and z [ Rk, let uz

nðt; xÞ be the

solution to Equation (1.2) with respect to the cylindrical Wiener process {Wn

t ; t [ ½0; T�},

where Wn

t ðhÞ ¼Pk

j¼1Wn;jð1½0;t�hjÞ for any h [ Hk, and

Wn;jð1½0;t�hjÞ ¼ W jð1½0;t�hjÞ þ zj

ðt

0

khjð*Þ; gjnðs; *ÞlHds:


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We denote by wznðt; xÞ the k £ k Jacobian matrix of the random function z ! uz

nðt; xÞ,

that is,

wznðt; xÞ ¼ wz

n;i;jðt; xÞ �

i;j:¼

›

›zj

uzn;iðt; xÞ

� �i;j

;

and by cznðt; xÞ its k £ k £ k Hessian matrix, that is,

cznðt; xÞ ¼ cz

n;i;j;mðt; xÞ �

i;j;m:¼

›2

›zj›zm

uzn;iðt; xÞ

� �i;j;m

:

As uznðt; xÞ is k-dimensional, the latter is a k-dimensional vector such that each

i-component is a k £ k matrix that corresponds to the Hessian matrix of the random

variable uzn;iðt; xÞ. Finally, we denote by k�k the norm of a n £ n matrix A defined as

kAk ¼ supj[Rn;kjk¼1

kAjk:

We next proceed to the study of the strict positivity of the density pt;xð�Þ of the law of

uðt; xÞ, where ðt; xÞ [�0; T� £ Rd are fixed. We need to introduce the following condition.

We say that y [ Rk satisfies Ht;xðyÞ if

Ht;xðyÞ: there exists a sequence of predictable processes {gn}n$1 in HkT , and positive

constants c1, c2, r0 and d such that

(i) lim supn!1

P ðkuðt; xÞ2 yk # rÞ> ðdetw0nðt; xÞ $ c1Þ

� �. 0; for all r [�0; r0�:

(ii)limn!1

P supkzk#d

ðkwznðt; xÞk þ kcz

nðt; xÞkÞ # c2

( )¼ 1:

Remark 5.5. Observe that if a point y verifies Ht;xðyÞ then automatically belongs to the

support of the law of uðt; xÞ.

We next provide the main result of this section which is a criterion for the strict

positivity of a density, which was proved in Ref. [1] for the case where HkT is replaced by

L2ð½0; 1�;RkÞ. Our case follows along the same lines as theirs. This criterion uses the

following quantitative version of the classical inverse function theorem. Let Bðx; rÞ denote

the ball of Rk with centre x and radius r . 0.

Lemma 5.6. ([1] Lemma 3.2 and [17] Lemma 4.2.1) Let F : Rk 7! Rk be a C2 mapping

such that for some constants b . 1 and d . 0,

jdetF0ð0Þj $1

band sup

kzk#d

jFðzÞj þ jF0ðzÞj þ jF00ðzÞj� �

# b:

Then there exist constants R [�0; 1� and a . 0, such that F is a diffeomorphism from a

neighbourhood of 0 contained in the ball Bð0;RÞ onto the ball BðFð0Þ;aÞ.

Theorem 5.7. Let ðt; xÞ [�0; T� £ Rd and y [ Rk be such that Ht;xðyÞ holds true. Suppose

that the law of random vector uðt; xÞ has a continuous density pt;x in a neighbourhood

of y. Then pt;xðyÞ . 0. Moreover, if Ht;xðyÞ holds on SuppðPuðt;xÞÞ> S, with

S ¼ {y [ Rk : s 1ðyÞ; . . . ;s qðyÞ span Rk}, then for every connected subset ~S , S

such that SuppðPuðt;xÞÞ> ~S is non-void, pt;x is a strictly positive function on ~S.

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Proof. Let y [ Rk satisfying Ht;xðyÞ. Then, proceeding as Ref. [1] (Theorem 3.3), using

Lemma 5.6 with FðzÞ ¼ uznðt; xÞ, one can show that there exists n sufficiently large such

that for any continuous and bounded function f : Rk 7! Rþ it holds that

E½f ðuðt; xÞÞ� $

ðBðy;a=2Þ

f ðvÞuynðvÞdv;

where a . 0 is the parameter of Lemma 5.6 and uyn is a strictly positive continuous

function on Bðy;a=2Þ. This proves the first statement of the theorem.

We next assume that Ht;xðyÞ holds on SuppðPuðt;xÞÞ> S. It suffices to check that if~S , S is a connected component of S such that SuppðPuðt;xÞÞ> ~S is non-void, then~S , SuppðPuðt;xÞÞ. Indeed, this implies that Ht;xðyÞ holds for all y [ ~S and by the first part

of the theorem one obtains that pt;xðyÞ . 0 and the theorem is proved.

Suppose that ~S , SuppðPuðt;xÞÞ is false. Then one may find x1 [ ~S such that

x1 � SuppðPuðt;xÞÞ. Moreover, since SuppðPuðt;xÞÞ> ~S is non-void one may find

x2 [ SuppðPuðt;xÞÞ> ~S. Now, since ~S is connected, one can find a continuous curve

xðlÞ, l [ ½0; 1� contained in ~S with xð0Þ ¼ x2 and xð1Þ ¼ x1. Take now

l* ¼ sup{l : xðlÞ [ SuppðPuðt;xÞÞ}. Since x1 ¼ xð1Þ � SuppðPuðt;xÞÞ and the complemen-

tary of SuppðPuðt;xÞÞ is an open set, it follows that l* , 1. Then, we have a sequence ln " l*such that xðlnÞ [ SuppðPuðt;xÞÞ and we also know that xðlÞ � SuppðPuðt;xÞÞ for l* , l # 1.

This means that xðl*Þ is on the boundary of SuppðPuðt;xÞÞ, and since this set is closed, we

conclude that xðl*Þ [ SuppðPuðt;xÞÞ. Since xðl*Þ belongs also to ~S we have that

pt;xðxðl*ÞÞ . 0. But this is contradictory with the fact that xðl*Þ is on the boundary of

SuppðPuðt;xÞÞ. A

Remark 5.8. Observe that Theorem 5.7 says that if a specific point y* is in the support of the

law of uðt; xÞ then pt;x is strictly positive on the connected component of S that contains y*.

However, this criterion based on the inverse function theorem does not give information

about the support of the law. Nevertheless, in order to prove that pt;xðyÞ . 0, we do not

have to check that y itself belongs to the support, but only that the connected component of

S which contains y has a non-void intersection with the support, and in particular, if we

have ellipticity everywhere, then S ¼ Rk and so the above condition is automatically

verified (see Remark 5.2).

Remark 5.9. There is a little mistake in page 56, lines 11–13 of Ref. [1]. Indeed, the

support is a closed set and their set {w – 0}d is open, thus they may not intersect, so it is

not always possible to choose a point y in the boundary of the support and such that

wðyÞ – 0. For example, if {w – 0} ¼ {2 1; 1}c and the support is ½21; 1�, in the

boundary of the support one has that w ¼ 0.

5.2 Proof of Theorem 5.1

Let ðt; xÞ [�0; T� £ Rd be fixed. As explained at the beginning of Section 5.1, let y [SuppðPuðt;xÞÞ be fixed such that s 1ðyÞ; . . . ;s kðyÞ span Rk. Consider the sequence of

predictable processes {gn}n$1 in HkT , defined by

gjnðs; zÞ ¼ v21

n 1½t222n;t�ðsÞGðt 2 s; x 2 zÞ; n $ 1; j ¼ 1; . . . ; k; ð5:4Þ


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where

vn :¼

ð22n

0

ðRd

jFGðrÞðjÞj2mðdjÞdr:

We are going to prove that assumptions (i) and (ii) ofHt;xðyÞ are satisfied for this sequence

of predictable processes. Then, the second statement of Theorem 5.7 will give the

conclusion of Theorem 5.1.

We start by some preliminary computations. We write

wzn;i;jðt; xÞ ¼

ðt

0

DðjÞr ðuz

n;iðt; xÞÞ; gjnðr; *Þ

D EHdr; 1 # i; j # k;

which follows since the processes defined by the left-hand side and right-hand side satisfy

the same equation, and there is uniqueness of solution. Then by (5.2) and the stochastic

differential equation satisfied by the derivative (4.1), we obtain that

wzn;i;jðt; xÞ ¼ Az

n;i;jðt; xÞ þ Bzn;i;jðt; xÞ þ Cz

n;i;jðt; xÞ;

where

Azn;i;jðt; xÞ ¼ v21

n

ð22n

0

ks ijðuznðt 2 r; *ÞÞGðr; x 2 *Þ;Gðr; x 2 *ÞlHdr;

Bzn;i;jðt; xÞ : ¼

ðt

0

ðt

r

ðRd

Gðt 2 s; x 2 yÞXk

l;m¼1

›ms ilðuznðs; yÞÞ

;

£ kDðjÞr ðuz

n;mðs; yÞÞ; gjnðr; *ÞlHWn;lðds; dyÞ

!dr

Czn;i;jðt; xÞ : ¼

ðt

0

ðt

r

ðRd

Xk

m¼1

›mbiðuznðs; x 2 yÞÞ

£ kDðjÞr ðuz

n;mðs; x 2 yÞÞ; gjnðr; *ÞlHGðt 2 s; dyÞds

!dr:

Note that gjnðr; *Þ ¼ 1½t222n;t�ðrÞg

jnðr; *Þ, and that Drðu

zn;iðs; yÞÞ ¼ 0 if s , r. Hence, using

these facts and Fubini’s theorem, it yields that

Bzn;i;jðt; xÞ ¼

ðt

t222n

ðRd

Gðt 2 s; x 2 yÞXk

l;m¼1

›ks ilðuznðs; yÞÞ

£

ðs

t222n

kDðjÞr ðuz

n;mðs; yÞÞ; gjnðr; *ÞlHdr

� �Wn;lðds; dyÞ;

Czn;i;jðt; xÞ ¼

ðt

t222n

ðRd

Xk

m¼1

›mbiðuznðs; x 2 yÞÞ

£

ðs

t222n

kDðjÞr ðuz

n;mðs; x 2 yÞÞ; gjnðr; *ÞlHdr

� �Gðt 2 s; dyÞds:

Therefore, we have proved that

wzn;i;jðt; xÞ ¼ Az

n;i;jðt; xÞ þDzn;i;jðt; xÞ þ Ez

n;i;jðt; xÞ þ Czn;i;jðt; xÞ; ð5:5Þ

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where

Dzn;i;jðt; xÞ : ¼

ðt

t222n

ðRd

Gðt 2 s; x 2 yÞXk

l;m¼1

›ms ilðuznðs; yÞÞwz

n;m;jðs; yÞW lðds; dyÞ;

Ezn;i;jðt; xÞ : ¼

Xk

l;m¼1

zl

ðt

t222n

Gðt 2 s; x 2 *Þ›ms ilðuznðs; *ÞÞw

zn;m;jðs; *Þ; gl

nðs; *ÞD E

Hds;

Czn;i;jðt; xÞ ¼

ðt

t222n

ðRd

Xk

m¼1

›mbiðuznðs; x 2 yÞÞwz

n;m;jðs; x 2 yÞGðt 2 s; dyÞds:

Next, we study upper bounds for the p-moments of the four terms on the right-hand

side of wzn;i;jðt; xÞ. For this, we assume that kzk # d for some d . 0.

First observe that since s is bounded,

jAzn;i;jðt; xÞj # K: ð5:6Þ

Appealing to Ref. [19] ((3.11)) and using the fact that the partial derivatives of the

coefficients of s are bounded, we get that for all p . 1,

E Dzn;i;jðt; xÞ

�� ph i# cpvp=2

n

Xk

m¼1

supðs;yÞ[½0;T�£Rd

E wzn;m;jðs; yÞ

�� ph i: ð5:7Þ

Using the Cauchy–Schwarz inequality and the fact that the derivatives of s are

bounded, it yields that for all p . 1,

E Ezn;i;jðt; xÞ

�� ph i# cpd

pXk

m¼1


E wzn;m;jðs; yÞ

�� ph i: ð5:8Þ

We finally use Minkowski’s inequality with respect to the finite measure Gðs; dyÞds,

the boundedness of the partial derivatives of the coefficients of b, and hypothesis (2.2), to

see that for all p . 1,

E Czn;i;jðt; xÞ

�� ph i# cp

Xk

m¼1


E wzn;m;jðs; yÞ

�� ph i ð22n

0

ðRd

Gðs; dyÞds

� �p

# cp;T22npXk

m¼1


E wzn;m;jðs; yÞ

�� ph i:

ð5:9Þ

Now, introducing (5.6)–(5.9) into (5.5), we get that for all p . 1,

Xk

m¼1

E wzn;m;jðt; xÞ

�� ph i# Kp þ cp;T vp=2

n þ dp þ 22np �Xk

m¼1


E jwzn;m;jðs; yÞj

ph i

:

ð5:10Þ

Observe that, proceeding as in the proof of (4.2), one can show that

supkzk#d


E wzn;i;jðs; yÞ

�� ph i, 1; ð5:11Þ


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as we have shown in (5.5) that wzn;i;jðt; xÞ satisfies a linear equation with initial condition

Azn;i;jðt; xÞ, which is bounded. Thus, choosing n large and d small such that

cp;T vp=2n þ dp þ 22np

� �# 1=2, we obtain from (5.10) and (5.11) that

supkzk#d


E wzn;i;jðs; yÞ

�� ph i# Kp: ð5:12Þ

We are now ready to show that (i) and (ii) of Ht;xðyÞ are verified.

Proof of (i). Take z ¼ 0 in (5.5), and use (5.7), (5.9) and (5.12), to get that

w0n;i;jðt; xÞ ¼ A0

n;i;jðt; xÞ þRn;i;jðt; xÞ;

where, for any p . 1,

E Rn;i;jðt; xÞ�� p

# cp;T vp=2n þ 22np

�:

We now write

A0n;i;jðt; xÞ ¼ s ijðuðt; xÞÞ þOn;i;jðt; xÞ;

where

On;i;jðt; xÞ ¼ v21n

ð22n

0

ðs ijðuðt 2 r; *ÞÞ2 s ijðuðt; xÞÞÞGðr; x 2 *Þ;Gðr; x 2 *Þ� �

Hdr:

Applying the Cauchy–Schwarz inequality, it yields that for all p . 1,

E On;i;jðt;xÞ�� p

# v2pn E

ð22n

0

kGðr;x2*Þjj2Hdr

��p� �� 1=2

E

ð22n

0

kGðr;x2*Þjj2Hdr

��p� �� 1=2

;

whereGðr; dyÞ :¼ ðs ijðuðt 2 r; yÞÞ2 s ijðuðt; xÞÞÞGðr; x 2 dyÞ (as in theproofofLemma4.2).

Now, appealing to Lemma 4.2 and hypothesis (H6), we obtain that for all p . 1,

E½jOn;i;jðt; xÞjp� # cp;T2

2nða2hÞpða . hÞ:

Now, as y [ SuppðPuðt;xÞÞ> S, there exists r0 . 0 such that for all 0 , r # r0,

Bðy; rÞ , S; and P{uðt; xÞ [ Bðy; rÞ} . 0:

Moreover, s 1ðyÞ; . . . ;s kðyÞ span Rk. Let s ðyÞ denotes the matrix with columns

s 1ðyÞ; . . . ;s kðyÞ. Hence, for all 0 , r # r0,

P ðkuðt; xÞ2 yk # rÞ> ðdets ðuðt; xÞÞ $ 2c1Þf g . 0;

where

c1 :¼1

2inf

z[Bðy;rÞinf

kjk¼1ks ðzÞjk

2

� �k

:

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Thus, we conclude that

lim supn!1

P kuðt; xÞ2 yk # rÞ> ðdetw0nðt; xÞ $ c1

� �� . 0;

which proves (i).

Proof of (ii). We start proving that there exist c . 0 and d . 0, such that

limn!1

P supkzk#d

kwznðt; xÞk # c

( )¼ 1: ð5:13Þ

Observe that (5.5), together with (5.6)–(5.9), show that

kwznðt; xÞk # K þ sup

kzk#d

kGznðt; xÞk; ð5:14Þ

where for any p . 1,

supkzk#d

E kGznðt; xÞk

p # cp;T vp=2

n þ dp þ 22np �

:

Hence, in order to prove (5.13) one only needs to check the uniformity in z. Let z and z0

such that kzk _ kz0k # d. Then, using (5.5) and similar computations as in (5.7)–(5.9), and

appealing to the Lipschitz property of the derivatives of the coefficients of s and b and

(5.12), together with the Cauchy–Schwarz inequality, and finally choosing n large and d

small, we obtain that


E wznðs; yÞ2 wz0

n ðs; yÞ�� ph i

# cp supðs;yÞ[½0;T�£Rd

E Aznðs; yÞ2Az0


þ Cp;T supðs;yÞ[½0;T�£Rd

E uznðs; yÞ2 uz0

n ðs; yÞ�� 2p� �1=2

:

We now claim that for all p . 1,


E uznðs; yÞ2 uz0


# cp;Tkz 2 z0kp: ð5:15Þ

Indeed, using the Lipschitz property of the coefficients of s and b, together with Ref. [19]

((3.9) and (5.15)), we obtain that

E uznðt;xÞ2 uz0

n ðt;xÞ�� ph i

# cpkz2 z0kp

þ cp;T

ðt

0

supy[Rd

E uznðs;yÞ2 uz0

n ðs;yÞ�� ph ið

Rd

kFGðt2 sÞðjÞk2mðdjÞds

þ cp;T

ðt

0

supy[Rd

E uznðt2 s;yÞ2 uz0

n ðt2 s;yÞ�� ph ið

Rd

Gðs;dyÞds:

Thus, hypotheses (2.2) and (2.1), and Gronwall’s lemma prove (5.15).


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We next use the Lipschitz property of s together with (5.15), to obtain that


E Aznðs; yÞ2Az0


# cp;Tkz 2 z0kp:

Therefore, we have proved that

E wznðt; xÞ2 wz0

n ðt; xÞ�� ph i

# cp;Tkz 2 z0kp;

which, together with (5.14) concludes the proof of (5.13).

The proof of (ii) for cznðt; xÞ follows along the same lines, therefore we only give the

main steps. Recall that

czn;i;j;mðt; xÞ ¼

›2

›zj›zm

uzn;iðt; xÞ:

Then we have that

czn;i;j;mðt; xÞ ¼

ðt

0

ðt

0

DðmÞs DðjÞ

r ðuzn;iðt; xÞÞ; gj

nðr; *Þ^gmn ðs; *Þ

D EH^H

dr ds;

where the H^H-valued process DðmÞs DðjÞ

r ðuzn;iðt; xÞÞ satisfies the following linear stochastic

differential equation:

DðmÞs DðjÞ

r uzn;iðt; xÞ

�¼ Gðt 2 r; x 2 *ÞD

ðmÞs ðs ijðuðr; *ÞÞÞ þ Gðt 2 s; x 2 *ÞD

ðjÞr ðs imðuðs; *ÞÞÞ

þ

ðt

r_s

ðRd

Gðt 2 w; x 2 yÞXk

l¼1

DðmÞs DðjÞ

r ðs ilðuðw; yÞÞÞW lðdw; dyÞ

þ

ðt

r_s

ðRd

Gðt 2 w; dyÞDðmÞs DðjÞ

r ðbiðuðw; x 2 yÞÞÞdw:

Using the chain rule and the stochastic differential equation satisfied by the first derivative,

one can compute the different terms of czn;i;j;mðt; xÞ as we did for wz

n;i;jðt; xÞ, and bound their

pth-moments. Finally, one estimates the pth-moments of the difference cznðt; xÞ2 cz0

n ðt; xÞ

as we did for wznðt; xÞ in order to get the desired result.

5.3 Proof of Theorem 5.3

The existence and smoothness of the density follow from Ref. [19] (Theorem 6.2). Hence,

we only need to prove the strict positivity. For this, one applies Theorem 5.7 as in Theorem

5.1 taking S ¼ R. In this case, in order to prove hypothesis (i), one proceeds as in Theorem

5.1, using hypotheses (H1), to show that for all p . 1,

limn!1

E jw0nðt; xÞ2A0

nðt; xÞjp

¼ 0:

Next using the non-degeneracy assumption on s , one gets that A0nðt; xÞ $ c, which

implies that

w0nðt; xÞ $ c 2 jw0

nðt; xÞ2A0nðt; xÞj:

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Finally, these assertions imply that for all y [ SuppðPuðt;xÞÞ

lim supn!1

P ðjuðt; xÞ2 yj # rÞ> w0nðt; xÞ $

c

2

�n o. 0;

which proves (i). The proof of (ii) follows exactly as in Theorem 5.1.

Acknowledgements

The author would like to thank Professors Vlad Bally, Robert C. Dalang, Marta Sanz-Sole and LluısQuer-Sardanyons for stimulating discussions on the subject, and the anonymous referees whohandled this article by all their comments and corrections that have helped to improve the paper.

References

[1] V. Bally and E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, PotentialAnal. 9 (1998), pp. 27–64.

[2] C. Cardon-Weber, Cahn-Hilliard stochastic equation: Strict positivity of the density,Stochastics Stochastics Rep. 72 (2002), pp. 191–227.

[3] R. Carmona and D. Nualart, Random nonlinear wave equations: Smoothness of the solutions,Probab. Theor. Relat. Fields 79 (1988), pp. 469–508.

[4] M. Chaleyat-Maurel and M. Sanz-Sole, Positivity of the density for the stochastic waveequation in two spatial dimensions, ESAIM Probab. Stat. 7 (2003), pp. 89–114.

[5] R.C. Dalang, Extending martingale measure stochastic integral with applications to spatiallyhomogeneous S.P.D.E’s, Electron. J. Probab. 4 (1999), pp. 1–29.

[6] R.C. Dalang and E. Nualart, Potential theory for hyperbolic SPDEs, Ann. Probab. 32 (2004),pp. 2099–2148.

[7] R.C. Dalang, D. Khoshnevisan, and E. Nualart, Hitting probabilities for systems of non-linearstochastic heat equations with additive noise, ALEA 32 (2008), pp. 2099–2148.

[8] R.C. Dalang, D. Khoshnevisan, and E. Nualart, Hitting probabilities for systems of non-linearstochastic heat equation with multiplicative noise, Probab. Theor. Relat. Fields 32 (2009),pp. 2099–2148.

[9] R.C. Dalang, D. Khoshnevisan, and E. Nualart, Hitting probabilities for systems of non-linearstochastic heat equations in spatial dimension k $ 1, Preprint (2012).

[10] R.C. Dalang and M. Sanz-Sole, Holder-Sobolev regularity of the solution to the stochasticwave equation in dimension 3, Mem. Am. Math. Soc. 199 (2009).

[11] R.C. Dalang and L. Quer-Sardanyons, Stochastic integrals for S.P.D.E’s: A comparison,Expositiones Mathematicae 29 (2011), pp. 67–109.

[12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Vol. 44, Encyclopediaof Mathematics and its Applications, Cambridge University Press, Cambridge, 1992.

[13] N. Fournier, Strict positivity of the density for a Poisson driven S.D.E, Stochastics StochasticsRep. 68 (1999), pp. 1–43.

[14] D. Marquez-Carreras, M. Mellouk, and M. Sarra, On stochastic partial differential equationswith spatially correlated noise: Smoothness of the law, Stochastic Processes Appl. 93 (2001),pp. 269–284.

[15] A. Millet and M. Sanz-Sole, Points of positive density for the solution to a hyperbolic SPDEs,Potential Anal. 7 (1997), pp. 623–659.

[16] A. Millet andM. Sanz-Sole, A stochastic wave equation in two space dimension: Smoothness ofthe law, Ann. Probab. 27 (1999), pp. 803–844.

[17] D. Nualart, Analysis on Wiener Space and Anticipating Stochastic Calculus, Vol. 1690,Ecole d’Ete de Probabilites de Saint-Flour XXV, Lectues Notes in Mathematics, Springer-Verlag, Berlin, 1998, pp. 123–227.

[18] D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer-Verlag, Berlin, 2006.[19] D. Nualart and L. Quer-Sardanyons, Existence and smoothness of the density for spatially

homogeneous SPDEs, Potential Anal. 27 (2007), pp. 281–299.[20] D. Nualart and L. Quer-Sardanyons, Gaussian density estimates for solutions of quasi-linear

stochastic partial differential equations, Stochastic Processes Appl. 119 (2009),pp. 3914–3938.


Dow

nloa

ded

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Uni

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ity o

f A

rizo

na]

at 1

0:04

18

May

201

4

[21] D. Nualart and L. Quer-Sardanyons, Optimal Gaussian density estimates for a class ofstochastic equations with additive noise, Infinite Dimension. Anal. Quantum Probab. Relat.Topics 14 (2011), pp. 25–34.

[22] E. Nualart and L. Quer-Sardanyons, Gaussian estimates for the density of the non-linearstochastic heat equation in any space dimension, Stochastic Processes Appl. 122 (2012),pp. 418–447.

[23] L. Quer-Sardanyons and M. Sanz-Sole, Absolute continuity of the law of the solution to the3-dimensional stochastic wave equation, J. Funct. Anal. 206 (2004), pp. 1–32.

[24] L. Quer-Sardanyons andM. Sanz-Sole, A stochastic wave equation in dimension 3: Smoothnessof the law, Bernoulli 10 (2004), pp. 165–186.

[25] M. Sanz-Sole and M. Sarra, Holder continuity for the stochastic heat equation with spatiallycorrelated noise, Seminar on Stochastic Analysis, Random Fields and Applications,III (Ascona, 1999) Prog. Probab. 52 (2002), pp. 259–268.

[26] J.B. Walsh, An Introduction to Stochastic Partial Differential Equations, Vol. 1180, Ecoled’Ete de Probabilites de Saint-Flour XIV, Lectures Notes in Mathematics, Springer-Verlag,Berlin, 1986, pp. 266–437.

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