Lecturenotes London

8/14/2019 Lecturenotes London

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LMS-EPSRC Short CourseStochastic Partial Differential

EquationsImperial College London, 7-11 July 2008

Applications of Malliavin Calculusto Stochastic Partial Differential

Equations

Marta Sanz-SoleFacultat de MatematiquesUniversitat de Barcelona

Version August 2008

Research supported by the grant MTM 2006-01351 from the Ministerio de Ciencia y Tec-

nologa, Spain


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Contents

1 Introduction 5

2 Integration by parts and absolute continuity of probabilitylaws 72.1 Properties derived from an integration by parts formula . . . . 72.2 Malliavins results . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Stochastic calculus of variations on an abstract Wiener space 143.1 Finite dimensional Gaussian calculus . . . . . . . . . . . . . . 143.2 Infinite dimensional framework . . . . . . . . . . . . . . . . . 19

3.3 The derivative and divergence operators . . . . . . . . . . . . 223.4 Some calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Criteria for Existence and Regularity of Densities 304.1 Existence of density . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Smoothness of the density . . . . . . . . . . . . . . . . . . . . 34

5 Watanabe-Sobolev Differentiability of SPDEs 375.1 A class of linear homogeneous SPDEs . . . . . . . . . . . . . . 375.2 The Malliavin derivative of a SPDE . . . . . . . . . . . . . . . 46

6 Analysis of Non-Degeneracy 556.1 Existence of moments of the Malliavin covariance . . . . . . . 556.2 Some references . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Small perturbations of the density 647.1 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 667.2 An example: the stochastic heat equation . . . . . . . . . . . . 69


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1 Introduction

Nowadays, Malliavin calculus is underpinning important developments instochastic analysis and its applications. In particular, research on SPDEs isbenefiting from the ideas and tools of this calculus. Unexpectedly, this hardmachinery is successfully used in financial engineering for the computationof Greeks, and in numerical approximations of SPDEs. The analysis of thedependence of the Malliavin matrix on its structural parameters is used inproblems of potential theory involving SPDEs, like obtaining the optimalsize of some hitting probabilities. The study of such questions, but also ofsome classical issues like the absolute continuity of measures derived fromprobability laws of SPDEs, is still an underdeveloped field.

These notes are a brief introduction to the basic elements of Malliavin calcu-lus and to some of its applications to SPDEs. They have been prepared for aseries of six lectures at the LMS-EPSRC Short Course on Stochastic PartialDifferential Equations.The first three sections are devoted to introduce the calculus: its motivations,the main operators and rules, and the criteria for existence and smoothnessof densities of probabilities laws. The last three ones deal with applicationsto SPDEs. To be self-contained, we provide some ingredients of the SPDEframework we are using. Then we study differentiability in the Malliavinsense, and non-degeneracy of the Malliavin matrix. The last section is de-voted to sketch a method to analyze the asymptotic behaviour of densitiesof small perturbations of SPDEs. Altogether, this is a short, very short,

journey through a deep and fascinating subject.To close this short presentation, I would like to express my gratitude toProfessor Dan Crisan, the scientific organizer of the course, for a wonderfuland efficient job, to the London Mathematical Society for the financial sup-port, and to the students whose interest and enthusiasm has been a sourceof motivation and satisfaction.

Barcelona, August 2008

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2 Integration by parts and absolute continu-

ity of probability lawsThis lecture is devoted to present the classical sufficient conditions for exis-tence and regularity of density of finite measures on Rn and therefore for thedensities of probability laws. The results go back to Malliavin (see [35], butalso [74], [79] and [46]). To check these conditions, Malliavin developed adifferential calculus on the Wiener space, which in particular allows to provean integration by parts formula. The essentials on this calculus will be givenin the next lecture.

2.1 Properties derived from an integration by partsformula

The integration by parts formula of Malliavin calculus is a simple but ex-tremely useful tool underpinning many of the sometimes unexpected appli-cations of this calculus. To illustrate its role and give a motivation, we startby showing how an abstract integration by parts formula leads to explicitexpressions for the densities and their derivatives.Let us introduce some notation. Multi-indices of dimension r are denotedby= (1, . . . , r) {1, . . . , n}r, and we set|| =ri=1 i. For any differ-entiable real valued functiondefined on Rn, we denote by the partial

derivative||1,...,r. If||= 0, we set = , by convention.Definition 2.1 LetF be aRn-valued random vector andG be an integrablerandom variable defined on some probability space (, F, P). Let be amulti-index. The pairF, Gsatisfies an integration by parts formula of degree|| if there exists a random variableH(F, G)L1()such that

E

()(F)G

= E

(F)H(F, G)

, (2.1)

for any Cb (Rn).The property expressed in (2.1) is recursive in the following sense. Let =(, ), with = (1, . . . , a), = (1, . . . , b). Then

E

()(F)G

= E

()(F)H(F, G)

=E

(F)H(F, H(F, G))

=E

(F)H(F, G)

.

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The interest of this definition in connection with the study of probability

laws can be deduced from the next result.Proposition 2.1 1. Assume that (2.1) holds for= (1, . . . , 1)andG=

1. Then the probability law ofFhas a densityp(x) with respect to theLebesgue measure onRn. Moreover,

p(x) =E

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Hence the law ofFis absolutely continuous and its density is given by (2.2).

SinceH(1,...,1)(F, 1) is assumed to be in L1

(), formula (2.2) implies the con-tinuity ofp, by bounded convergence. This finishes the proof of part 1.The proof of part 2 is done recursively. For the sake of simplicity, we shallonly give the details of the first iteration for the multi-index = (1, . . . , 1).Let f C0 (Rn), (x) =

x1

xn f(y)dy, (x) =

x1

xn(y)dy.

By assumption,

E

f(F)

= E

(F)H(1,...,1)(F, 1)

=E

(F)H(1, ,1)(F, H(1,...,1)(F, 1))

=E

(F)H(2,...,2)(F, 1)

.

Fubinis Theorem yields

E

(F)H(2,...,2)(F, 1)

=E

F1

dy1 Fn

dyn

y1

dz1 yn

dznf(z)

H(2,...,2)(F, 1)

=E

F1

dz1 Fn

dznf(z) F1

z1dy1

Fnzn

dynH(2,...,2)(F, 1)

=Rn

dzf(z)E

ni=1(Fi zi)+H(2,...,2)(F, 1)

.

This shows that the density ofFis given by

p(x) =E

ni=1(Fi xi)+H(2,...,2)(F, 1)

,

using a limit argument, as in the first part of the proof. The functionx ni=1(Fi xi)+ is differentiable, except when xi = Fi for somei = 1, . . . , n, which happens with probability zero, since F is absolutelycontinuous. Therefore by bounded convergence

(1,...,1)p(x) = (1)nE

11[x,)(F)H(2,...,2)(F, 1)).

Remark 2.1 The conclusion in part 2 of the preceding Proposition is quite

easy to understand by formal arguments. Indeed, roughly speaking the func-tion in (2.1) should be such that its derivative is the delta Dirac function0. Since taking primitives makes functions smoother, the higher|| is, thesmoothershould be. Thus, having (2.1) for any multi-indexyields infinite

differentiability forp(x) =E

0(F x)

.

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Remark 2.2 Assume that (2.1) holds for = (1, . . . , 1) and a positive,

integrable random variableG. By considering the measuredQ= GdP, andwith a similar proof as for the first statement of Proposition 2.1, we concludethat the measureQ1Fis absolutely continuous with respect to the Lebesguemeasure and its density p is given by

p(x) =E

11(xF)H(1,...,1)(F, G)

.

2.2 Malliavins results

We now give Malliavins criteria for the existence of density (see [35]). Tobetter understand the assumption, let us explore first the one-dimensionalcase.Consider a finite measureon R. Assume that for every function C0 (R)there exists a positive constants C, not depending on , such that

R

d

C||||.Define

a,b(x) =

0 ifxaxaba

ifa < x < b

1 ifxb,(2.5)

< a < b < +. By approximating a,b by a sequence of functions inC0 (R) we obtain

([a, b])C(b a).Since this holds for any such a < b, it follows that is absolutely continuouswith respect to the Lebesgue measure.Malliavin proved that the same result holds true in dimension n > 1 as isstated in the next proposition

Proposition 2.2 Letbe a finite measure onRn. Assume that for anyi{1, 2, . . . , n} and every function C0 (Rn), there exist positive constants

Ci, not depending on, such thatR

id Ci||||. (2.6)

Then is absolutely continuous with respect to the Lebesgue measure and thedensity belongs to L

nn1 .

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When applying this proposition to the law of a random vector F, we have

the following particular statement:

Proposition 2.3 Assume that for any i {1, 2, . . . , n} and every function C0 (Rn), there exist positive constantsCi, not depending on, such that

|E((i)(F))| Ci||||. (2.7)

Then the law ofF has a density.

In [35], the density obtained in the preceding theorem is proved to be in L1;however in a remark the improvement to L

nn1 is mentioned and a hint for

the proof is provided. We prove Proposition 2.2 following [46] which takesinto account Malliavins remark.

Proof: Consider an approximation of the identity on Rn, for example

(x) = (2)n2 exp

|x|

2

2

.

Consider also functions cM, M 1, belonging toC0 (Rn), 0cM 1, suchthat

cM(x) = 1 if|x| M0 if|x| M+ 1

and with partial derivatives uniformly bounded, independently on M. ThefunctionscM ( ) clearly belong toC0 (Rn) and give an approximationof. Then, by Gagliardo-Nirenberg inequality (see a note at the end of thislecture)

cM ( )L nn1n

i=1

i(cM ( ))1n

L1.

We next prove that he right-hand side of this inequality is bounded. For this,we notice that assumption (2.6) implies that the functional

C0 (Rn)Rn

id

is linear and continuous and therefore it defines a signed measure with finitetotal mass (see for instance [32], page 82). We shall denote by i,i = 1, . . . , n

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this measure. Then,

i(cM ( ))L1Rn

cM(x)Rn

i(x y)(dy) dx

+Rn

|icM(x)|Rn

(x y)(dy) dx

Rn

Rn

(x y)i(dy) dx

+Rn

|icM(x)|Rn

(x y)(dy) dx.

By applying Fubinis theorem, and because of the choice of, it is easy tocheck that each one of the two last terms is bounded by a finite constant,

independent ofM and. As a consequence, the set of functions{cM ( ), M1, >0}is bounded inL nn1 . By using the weak compactness of theunit ball ofL

nn1 (Alouglus theorem), we obtain that has a density and it

belongs to L nn1 .

The next result (see [74]) gives sufficient conditions on ensuring smoothnessof the density with respect to the Lebesgue measure.

Proposition 2.4 Let be a finite measure on Rn. Assume that for any

multi-index and every function C0 (R

n

) there exist positive constantsC not depending on such that

Rn

d C. (2.8)

Then possesses a density which is aC function.

When particularising to the law of a random vector F, condition (2.8)clearly reads

|E(() (F))| C. (2.9)

Remark 2.3 When checking (2.6), (2.8), we have to get rid of the deriva-tivesi, and thus one naturally thinks of an integration by parts procedure.

Some comments:

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1. Letn = 1. The assumption in part 1) of Proposition 2.1 implies (2.6).

However, forn >1, both hypotheses are not comparable. The conclu-sion in the former Proposition gives more information on the densitythan in Proposition 2.4.

2. Letn >1. Assume that (2.1) holds for any multi-index with||= 1.Then, by the recursivity of the integration by parts formula, we obtainthe validity of (2.1) for = (1, , 1).

3. Since the random variableH(F, G) in (2.1) belongs toL1(), the iden-

tity (2.1) with G = 1 clearly implies (2.9). Therefore the assumptionin part 2 of Proposition 2.1 is stronger than in Proposition 2.4 but theconclusion more precise too.

AnnexGagliardo-Nirenberg inequality

Letf C0 (Rn), thenf

Ln

n1

ni=1

if1n

L1.

For a proof, we refer the reader to [73], page 129.

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3 Stochastic calculus of variations on an ab-

stract Wiener spaceThis lecture is devoted to introduce the main ingredients of Malliavin calcu-lus: the derivative, divergence and Ornstein Uhlenbeck operators, and rulesof calculus for them.

3.1 Finite dimensional Gaussian calculus

To start with, we shall consider a very particular situation. Letm be thestandard Gaussian measure on Rm:

m(dx) = (2)m2 exp |x|2

2

dx.

Consider the probability space (Rm, B(Rm), m). Here ndimensional ran-dom vectors are functions F : Rm Rn. We shall denote by Em the expec-tation with respect to the measure m.The purpose is to find sufficient conditions ensuring absolute continuity withrespect to the Lebesgue measure on Rn of the probability law ofF, and thesmoothness of the density. More precisely, we would like to obtain expressionssuch as (2.1). This will be done in a quite sophisticated way, as a prelude tothe methodology we shall apply in the infinite dimensional case. For the sake

of simplicity, we will only deal with multi-indices of order one. Hence, weshall only address the problem of existence of density for the random vectorF. As references of this section we mention [35], [74], [54].

The Ornstein-Uhlenbeck operator

Let (Bt, t 0) be a standard Rm-valued Brownian motion. Consider thelinear stochastic differential equation

dXt(x) =

2dBt Xt(x)dt, (3.1)with initial condition xRm. Using Itos formula, it is immediate to checkthat the solution to (3.1) is given by

Xt(x) = exp(t)x+

2 t0

exp((t s))dBs. (3.2)

The operator semigroup associated with the Markov process solution to (3.1)is defined byPtf(x) =Emf(Xt(x)), for a suitable class of functionsf. Notice

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that the law ofZt(x) =

2

t0exp((t s))dBs is Gaussian, mean zero and

with covariance given by (1exp(2t))I. This fact, together with (3.2),yieldsPtf(x) =

Rm

f(exp(t)x+

1 exp(2t)y)m(dy). (3.3)We are going to identify the class of functionsffor which the right hand-sideof (3.3) makes sense, and we will also compute the infinitesimal generator ofthe semigroup. This is the Ornstein-Uhlenbeck operator in finite dimension.

Lemma 3.1 The semigroup generated by(Xt, t0) satisfies the following:1. (Pt, t0) is a contraction semigroup onLp(Rm; m), for allp1.

2. For anyf C2

b (Rm

) and everyxRm

,

limt0

1

t

Ptf(x) f(x)

= Lmf(x), (3.4)

whereLm= x =mi=1 2xixi mi=1 xixi.3. (Pt, t0) is a symmetric semigroup onL2(Rm; m).

Proof. 1) LetX and Ybe independent random variables with law m. The

law of exp(t)X+

1 exp(2t)Y is alsom. Therefore, (m m)T1 =m, where T(x, y) = exp(

t)x+ 1 exp(2t)y. Then, the definition ofPtfand this remark yields

Rm|Ptf(x)|pm(dx)

Rm

Rm

|f(T(x, y))|pm(dx)m(dy)

=Rm

|f(x)|pm(dx).

2) This follows very easily by applying the Ito formula to the process f(Xt).3) We must prove that for any gL2(Rm; m),

RmPtf(x)g(x)m(dx) =

Rm

f(x)Ptg(x)m(dx),

or equivalently

Em

f(exp(t)X+

1 exp(2t)Y)g(X)

=Em

g(exp(t)X+

1 exp(2t)Y)f(X)

,

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where X and Y are two independent standard Gaussian variables. This

follows easily from the fact that the vector (Z, X), where

Z= exp(t)X+

1 exp(2t)Y,

has a Gaussian distribution and each component has law m.

The adjoint of the differential

We are looking for an operator m which is the adjoint of the gradient inL2(Rm, m). Such an operator must act on functions : R

m Rm, takevalues in the space of real-valued functions defined on Rm, and satisfy theduality relation

Emf, = Em(f m), (3.5)where, denotes the inner product in Rm. Let = (1, . . . , m). Assumefirst that the functions f, i : Rm R, i = 1, . . . , m, are continuouslydifferentiable. An usual integration by parts yields

Emf, =m

i=1

Rm

if(x)i(x)m(dx)

=m

i=1

Rm

f(x)

xii(x) ii(x)

m(dx).

Hence

m=m

i=1

(xii ii). (3.6)

Notice that onC2(Rm),m =Lm.The definition (3.6) yields the next useful formula

m(fg) =f, g f Lmg, (3.7)

for anyf , g smooth enough.

Example 3.1 Let n 1; consider the Hermite polynomial of degree n onR, which is defined by

Hn(x) =(1)n

n! exp

x2

2

dn

dxnexp

x

2

2

.

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The operator1 satisfies

1Hn(x) =xHn(x) Hn(x) =xHn(x) Hn1(x)= (n+ 1)Hn+1(x).

Therefore it increases the order of a Hermite polynomial by one.

An integration by parts formula

Using the operators ,mand Lm, and for random vectorsF = (F1, . . . , F n)regular enough (meaning that all the differentiations performed throughoutthis section make sense), we are going to establish an integration by parts

formula of the type (2.1).We start by introducing the finite dimensionalMalliavin matrix, also termedcovariance matrix, as follows:

A(x) =Fi(x), Fj (x)

1i,jn

.

Notice that by its very definition, A(x) is a symmetric, non-negative definitematrix, for anyxRm. ClearlyA(x) =DF(x)DF(x)T, whereDF(x) is theJacobian matrix atx and the superscriptTmeans the transpose.Let us consider a function C1(Rn), and perform some computationsshowing that (i)(F), i = 1, . . . , n, satisfies a linear system of equations.Indeed, by the chain rule,

(F(x)

, Fl(x)

=

mj=1

nk=1

(k)(F(x))j Fk(x)jF

l(x)

=n

k=1

Fl(x), Fk(x)(k)(F(x))

=

A(x)(T)(F(x))

l, (3.8)

l = 1, . . . , n. Assume that the matrix A(x) is inversible m-almost every-where. Then one gets

(i)(F) =n

l=1

(F(x))

, A1i,l (x)Fl(x)

, (3.9)

for everyi = 1, . . . , n,m-almost everywhere.

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Taking expectations and using (3.7), (3.9) yields

Em

(i)(F)

=n

l=1

Em

(F)

, A1i,lFl

=n

l=1

Em

(F)m(A1i,lFl)

=n

l=1

Em

(F)

A1i,l , Fl A1i,l LmFl

. (3.10)

Hence we can write

Em

i(F)

= Em

(F)Hi(F, 1)

, (3.11)

with

Hi(F, 1) =n

l=1

m(A1i.lFl)

=n

l=1

A1i,l , Fl +A1i,lLmFl

. (3.12)

This is an integration by parts formula, as in Definition 2.1, for multi-indicesof length one.

For multi-indices of length greater than one, things are a little bit moredifficult; essentialy the same ideas would lead to the analogue of formula(2.1) with = (1, , 1) andG= 1.The preceding discussion and Proposition 2.2 yield the following result.

Proposition 3.1 LetFbe continuous differentiable up to the second ordersuch thatFand its partial derivatives up to order two belong to Lp(Rm; m),

for anyp[1, [. Assume that:

(1) The matrixA(x)is invertible for everyx Rm, m-almost everywhere.

(2) det A1 Lp(Rm; m), (det A1) Lr(Rm; m), for some p, r (1, ).

Then the law ofF is absolutely continuous with respect to Lebesgue measureonRn.

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Proof: The assumptions on Fand in (2) show that

Ci:=n

l=1

EmA1i,l, Fl + A1i,l LmFl

is finite. Therefore, one can take expectations on both sides of (3.9). By(3.10), it follows that

|Em(i)(F)| Ci||||.

This finishes the proof of the Proposition.

Remark 3.1 The proof of smoothness properties for the density requires aniteration of the procedure presented in the proof of Proposition 3.1.

3.2 Infinite dimensional framework

This section is devoted to describe an infinite dimensional analogue of theprobability space (Rm, B(Rm), m). We start by introducing a family of Gaus-sian random variables. LetHbe a real separable Hilbert space. Denote by||||Hand , Hthe norm and the inner product onH, respectively. There ex-ist a probability space (, G, ) and a family M= W(h), hHof randomvariables defined on this space, such that the mapping hW(h) is linear,each W(h) is Gaussian, EW(h) = 0 and E

W(h1)W(h2)

=h1, h2H (see

for instance, [63], Chapter 1, Proposition 1.3). Such family is constructedas follows. Let (en, n 1) be a complete orthonormal system in H. Con-sider the canonical probability space (, G, P) associated with a sequence(gn, n 1) of standard independent Gaussian random variables. That is, = RN,G =BN, =N1 where, according to the notations of Chapter1, 1 denotes the standard Gaussian measure on R. For each h H, theseries

n1h, enHgn converges in L2(, G, ) to a random variable that we

denote by W(h). Notice that the setM is a closed Gaussian subspace ofL2() that is isometric to H. In the sequel, we will replace

Gby the -field

generated byM.

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Examples

White Noise

LetH=L2(A, A, m), where (A, A, m) is a separable -finite, atomlessmeasure space. For anyF A with m(F)


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LetH denote the completion of (E, , E). Elements of the GaussianfamilyM= (W(h), h H) satisfy

E(W(h1)W(h2)) =Rd

(dx)

h1 h2

(x),

h1, h2 H.The family

W(11F), F Bb(Rd)

can be rigourously defined by approx-

imating 11Fby a sequence of elements inH. It is called a colored noisewith covariance .

We notice that for =0,

, E=, L2

(Rd

).

White-Correlated Noise

In the theory of SPDEs, stochastic processes are usually indexed by(t, x) R+Rd and the role oft and x is different -time and space,respectively. Sometimes the driving noise of the equation is white intime and in space (see the example termedwhite noisebefore). Anotherimportant class of examples are based on noises white in time andcorrelated in space. We give here the background for this type of noise.

With the same notations and hypotheses as in the preceding example,

we consider functions, D(Rd+1) and defineJ(, ) =

R+

dsRd

(dx)

(x). (3.15)

By the above quoted result in [71], J defines an inner product. SetHT =L2([0, T]; H). Elements of the Gaussian familyM= (W(h), hHT) satisfy

E(W(h1)W(h2)) =R+

dsRd

(dx)

h1(s) h2(s)

(x), (3.16)

h1, h2 HT. We can then consider W(t, A), t[0, [, A Bb(Rd),whereW(t, A) :=W(11[0,t] 11A) is defined by an approximation proce-dure. This family is called a Gaussian noise, white in time and station-ary correlated (or coloured) in space.

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3.3 The derivative and divergence operators

Throughout this section, we consider the probability space ( , G, ), definedin section 3.2 and a Gaussian familyM = (W(h), h H), as has beendescribed before.There are several possibilities to define the Malliavin derivative for randomvectorsF : Rn. Here we shall follow theanalytic approachwhich roughlyspeaking consists of an extension by a limiting procedure of differentiationin Rm.To start with, we consider finite-dimensionalobjects, termed smooth func-tionals. They are random variables of the type

F =f(W(h1), . . . , W (hn)), (3.17)

withh1, . . . , hnHand f : Rn Rregular enough.Different choices of regularity offlead to different classes ofsmoothfunction-als. For example, iff Cp (Rn), the set of infinitely differentiable functionssuch thatfand its partial derivatives of any order have polynomial growth,we denote the corresponding class ofsmoothfunctionals by S; iff Cb (Rn),the set of infinitely differentiable functions such thatfand its partial deriva-tives of any order are bounded, we denote bySb the corresponding class. Iffis a polynomial, then smoothfunctionals are denoted byP. ClearlyP SandSb S.We define the operator D on

S (on

P, on

Sb) with values on the set of

H-valued random variables, by

DF =n

i=1

if

W(h1), . . . , W (hn)

hi. (3.18)

FixhHand setFh =f

W(h1) +h, h1H, . . . , W (hn) +h, hnH

,

>0. Then it is immediate to check thatDF,hH= dd Fh

=0. Therefore,

for smooth functionals, D is a directional derivative. It is also routine toprove that ifF, Gare smooth functionals then,D(F G) =F DG+GDF.

Our next aim is to prove that D is closable as an operator from Lp() toLp(; H), for anyp1. That is, if{Fn, n1} Sis a sequence convergingto zero inLp() and the sequence{DFn, n1}converges toG in Lp(; H),then G = 0. The tool for arguing this is a simple version of an integrationby parts formulaproved in the next lemma.

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Lemma 3.2 For anyF S, hH, we have

EDF,hH

= E

F W(h)

. (3.19)

Proof: Without loss of generality, we shall assume that

F =f

W(h1), . . . , W (hn)

,

whereh1, . . . , hn are orthonormal elements ofHand h1= h. Then

EDF,hH

=

Rn

1f(x)n(dx)

=Rn f(x)x1n(dx) =E

F W(h1)

.

The proof is complete.

Formula (3.19) is a statement about duality between the operator D and aintegralwith respect to W.LetF, G S. Applying formula (3.19) to the smooth functional F G yields

E

GDF,hH

=E

FDG,hH

+E

F GW(h)

. (3.20)

With this result, we can now prove that D is closable. Indeed, consider a

sequence{Fn, n1} Ssatisfying the properties stated above. Let hHandF Sb be such that F W(h) is bounded. Using (3.20), we obtain

E

FG, hH

= limn

E

FDFn, hH

= limn

E

FnDF,hH+ FnF W(h)

= 0.

Indeed, the sequence (Fn, n 1) converges to zero in Lp andDF,hH,F W(h) are bounded. This yields G= 0.

Let D1,p be the closure of the setSwith respect to the seminorm

||F||1,p = E(|F|p) +E(||DF||pH)1p

. (3.21)

The set D1,p is the domain of the operator D in Lp(). Notice that D1,p isdense inLp(). The above procedure can be iterated as follows. Clearly, onecan recursively define the operator Dk, k N, on the setS. This yields an

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Hk-valued random vector. As for D, one proves that Dk is closable. Then

we can introduce the seminorms

||F||k,p =

E(|F|p) +k

j=1

E(||DjF||pHj 1p , (3.22)

p[1, ), and define the sets Dk,p to be the closure ofSwith respect to theseminorm (3.22). Notice that by definition, Dj,q Dk,p forkj andpq.By convention D0,p =Lp() and 0,p = p, the usual norm in Lp().We now introduce thedivergence operator, which corresponds to the infinitedimensional analogue of the operator m defined in (3.6).

For this, we notice that the Malliavin derivative D is an unbounded operatorfromL2() intoL2(; H). Moreover, the domain ofD inL2(), denoted byD

1,2, is dense in L2(). Then, by an standard procedure (see for instance[80]) one can define the adjointofD, that we shall denote by .

Indeed, the domain of the adjoint, denoted by Dom , is the set of randomvectors uL2(; H) such that for any F D1,2,

EDF,uHc||F||2,wherec is a constant depending on u. IfuDom , then u is the elementofL

2

() characterized by the identity

E

F (u)

= EDF,uH

, (3.23)

for all F D1,2.Equation (3.23) expresses the duality between D and . It is called theintegration by parts formula(compare with (3.19)). The analogy betweenandmdefined in (3.6) can be easily established onfinite dimensionalrandomvectors ofL2(; H), as follows.

LetSH be the set of random vectors of the type

u=n

j=1

Fjhj,

whereFj S, hjH,j= 1, . . . , n. Let us prove that uDom .

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Indeed, owing to formula (3.20), for any F S,EDF,uH= n

j=1

E

FjDF,hjH

n

j=1

EFDFj, hjH + EF FjW(hj)

C||F||2.HenceuDom . Moreover, by the same computations,

(u) =n

j=1

FjW(hj) n

j=1

DFj, hjH. (3.24)

Hence, the gradient operator in the finite dimensional case is replaced bythe Malliavin directional derivative, and the coordinate variables xj by therandom coordinatesW(hj).

Remark 3.2 The divergence operator coincides with a stochastic integral in-troduced by Skorohod in [72]. This integral allows for non adapted integrands.It is actually an extension of Itos integral. Readers interested in this topicare suggested to consult the monographs [46] and [47].

3.4 Some calculus

In this section we prove several basic rules of calculus for the two operatorsdefined so far. The first result is a chain rule.

Proposition 3.2 Let: Rm R be a continuously differentiable functionwith bounded partial derivatives. LetF = (F1, . . . , F m) be a random vectorwhose components belong to D1,p for somep1. Then(F)D1,p and

D((F)) =m

i=1

i(F)DFi. (3.25)

The proof of this result is straightforward. First, we assume that F

S; in

this case, formula (3.25) follows by the classical rules of differential calculus.The proof for F D1,p is done by an approximation procedure.The preceding chain rule can be extended to Lipschitz functions. The toolfor this improvement is given in the next Proposition. For its proof, we usethe Wiener chaos decomposition ofL2(, G) (see [22]).

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Proposition 3.3 Let(Fn, n1) be a sequence of random variables inD1,2

converging to F inL2

()and such that

supn

E||DFn||2H


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that n C and that the sequence (n, n 1) converges to uniformly.In additionn is bounded by the Lipschitz constant of.Proposition 3.2 yields,

D(n(F)) =m

i=1

in(F)DFi. (3.28)

Now we apply Proposition 3.3 to the sequence Fn =n(F). It is clear thatlimn n(F) = (F) in L

2(). Moreover, by the boundedness propertyonn, the sequence D(n(F)), n 1, is bounded in L2(; H). Hence(F)D1,2 andD(n(F)), n1 converges in the weak topology ofL2(; H)to D((F)). Since the sequence (

n(F), n

1), is bounded, a.s., there

exists a subsequence that converges to some random bounded vector G inthe weak topology of L2(; H). By passing to the limit as n theequality (3.28), we finish the proof of the Proposition.

Remark 3.3 Let Cp (Rm) and F = (F1, . . . , F m) be a random vectorwhose components belong top[1,)D1,p. Then the conclusion of Proposition3.2 also holds. Moreover, (F) p[1,)D1,p.The chain rule (3.25) can be iterated; we obtain Leibnizs rule for Malliavinsderivatives. For example, if F is one-dimensional (m= 1) then

Dk((F)) =k

l=1

Pl

cl(l)(F)li=1D

|pi|F, (3.29)

wherePl denotes the set of partitions of{1, , k} consisting of l disjointsetsp1, , pl, l= 1, , k,|pi| denotes the cardinal of the setpi andcl arepositive coefficients.

For anyF DomD, hHwe set DhF =DF,hH. The next propositionsprovide important calculus rules.

Proposition 3.5 Letu SH. Then

Dh((u)) =u, hH+(Dhu). (3.30)

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Proof: Fixu =

n

j=1 Fj hj ,Fj S,hjH,j = 1, . . . , n. By virtue of (3.24),we have

Dh((u)) =n

j=1

(DhFj)W(hj) +Fjhj , h D(DhFj), hjH

.

Notice that by (3.24),

(Dhu) =n

j=1

(DhFj)W(hj ) D(DhFj), hjH

. (3.31)

Hence (3.30) holds.

The next result is anisometry propertyfor the integral defined by the operator.

Proposition 3.6 Letu, v D1,2(H). Then

E

(u)(v)

= E(u, vH) +E(tr(Du Dv)), (3.32)

wheretr(DuDv) =

i,j=1 Dej u, eiHDeiv, ejH, with(ei, i1) a complete

orthonormal system inH.Consequently, ifuD1,2(H) thenuDom and

E

(u)2 E(||u||2H) +E(||Du||2HH). (3.33)

Proof: Assume first thatu, v SH. The duality relation between D and yields

E((u)(v)) = E

v, D((u))H

= E

i=1v, eiHDei((u))

.

By virtue of (3.30), this last expression is equal to

E

i=1

v, eiH(u, eiH+ (Deiu)

.

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The duality relation between D andimplies

Ev, eiH(Deiu)

= E

Deiu, Dv, eiHH

=

j=1

EDeiu, ejHej, Dv, eiH

=

j=1

E

Deiu, ejHDej v, eiH

.

This establishes (3.32). Taking u = v and applying Schwarz inequality yield(3.33).The extension to u, v D1,2(H) is done by a limit procedure.

Remark 3.4 Proposition 3.6 can be used to extend the validity of (3.30) tou D2,2(H). Indeed, let un SH be a sequence of processes convergingto u inD2,2(H). Formula (3.30) holds true for un. We can take limits inL2(; H) asn tends to infinity and conclude, because the operatorsD andare closed.

Proposition 3.7 Let F D1,2, u Dom , F u L2(; H). If F (u)DF,uHL2(), then

(F u) =F (u) DF,uH. (3.34)

Proof: Assume first thatF Sandu SH. LetG S. Then by the dualityrelation betweenD and and the calculus rules on the derivatives, we have

E(G(F u)) = E(DG,FuH)=E(u, (D(F G) GDF)H)=E(G(F (u) u,DFH)).

By the definition of the operator , (3.34) holds under the assumptions ofthe proposition.

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4 Criteria for Existence and Regularity of

DensitiesIn lecture 1, we have shown how an integration by parts formula (see Def-inition 2.1) leads to results on densities of probability laws. The questionwe tackle in this lecture is how to derive such a formula. In particular wewill give an expression for the random variable H(F, G). For this, we shallapply the calculus developed in Section 3.We consider here the probability space associated with a Gaussian family(W(h), hH), as has been described in Section 3.2.

4.1 Existence of densityLet us start with a very simple example.

Proposition 4.1 LetFbe a random variable belonging to D1,2. Assume thatthe random variable DF

||DF||2H

belongs to the domain of in L2(; H). Then

the law ofF is absolutely continuous. Moreover, its density is given by

p(x) =E

11(F >x) DF||DF||2H

(4.1)

and therefore it is continuous and bounded.

Proof: We will check that for any C

b

(R),

E(

(F)) =E

(F) DF||DF||2H

. (4.2)

Thus (2.1) holds for G = 1 with H1(F, 1) =

DF||DF||2

H

. Then the results

follow from part 1 of Proposition 2.1.The chain rule of Malliavin calculus yields D((F)) =

(F)DF. Thus,

(F) =

D((F)),

DF

||DF||2H

H

.

Therefore, the integration by parts formula implies

E

(F)

= E

D((F)), DF||DF||2H

H

=E

(F)

DF

||DF||2H

,

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proving (4.2).

Remark 4.1 Notice the analogy between (4.2) and the finite dimensionalformula (3.11).

Remark 4.2 Using the explicit formula (4.1) to particular examples andLp() estimates of the Skorohod integral leads to interesting estimates forthe density (see for instance [47]).

Remark 4.3 In Proposition 4.1 we have established the formula

H1(F, 1) = DF

||DF||2H , (4.3)

whereF : R.For random vectorsF, (n >1), we can obtain similar results by using matrixcalculus, as it is illustrated in the next statement. In the computations,instead ofDFH, we have to deal with the Malliavin matrix, a notion givenin the next definition.

Definition 4.1 LetF : Rn be a random vector with componentsFj D

1,2, j = 1, . . . , n. The Malliavin matrix ofF is then n matrix, denotedby , whose entries are the random variables i,j =

DFi, DFj

H, i, j =

1, . . . , n.

Proposition 4.2 Let F : Rn be a random vector with componentsFj D1,2, j = 1, . . . , n. Assume that

(1) the Malliavin matrix is inversible, a.s.

(2) For every i, j = 1, . . . , n, the random variables (1)i,j DFj belong to

Dom .

Then for any function Cb (Rn),E(i(F)) =E((F)Hi(F, 1)), (4.4)

with

Hi(F, 1) =n

l=1

((1)i,lDFl). (4.5)

Consequently the law ofF is absolutely continuous.

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Proof: Fix Cb (Rn). By virtue of the chain rule, we have(F) D1,2

and

D((F)), DFlH=n

k=1

k(F)DFk, DFlH

=n

k=1

k(F)k,l,

l = 1, . . . , n. Since is inversible a.s., this system of linear equations ink(F), k = 1, . . . , n, can be solved, and

i(F) =n

l=1

D((F)), (1)i,lDFl

H, (4.6)

i= 1, . . . , n, a.s.The assumption (2), the duality formula along with (4.6) yield

nl=1

E

(F)

(1)i,lDFl

=n

l=1

ED((F)), (1)i,lDFlH

=E

i(F)

.

Hence (4.4), (4.5) is proved.Notice that by assumption Hi(F, 1) L2(). Thus Proposition 2.2 part 1)yields the existence of the density.

Remark 4.4 The equalities (4.4), (4.5) give the integration by parts formula(in the sense of Definition 2.1) forndimensional random vectors, for multi-indices of length one.

The assumption of part 2 of Proposition 4.2 may not be easy to check. In the

next Proposition we give a statement which is more suitable for applications.Theorem 4.1 LetF : Rn be a random vector satisfying the followingconditions:

(a) Fj D2,4, for anyj= 1, . . . , n,

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(b) the Malliavin matrix is inversible, a.s.

Then the law ofFhas a density with respect to Lebesgue measure onRn.

Proof: As in the proof of Proposition 4.2, we obtain the system of equations(4.6) for any function Cb . That is,

i(F) =n

l=1

D((F)), (1)i,lDFlH,

i= 1, . . . , n, a.s.We would like to take expectations on both sides of this expression. However,assumption (a) does not ensure the integrability of1. We overcome this

problem by localising (4.6), as follows.For any natural number N 1, we define the set

CN=

L(Rn,Rn) :|||| N, | det | 1N

.

Then we consider a nonnegative function N C0 (L(Rn,Rn)) satisfying(i)N() = 1, ifCN,(ii)N() = 0, ifCN+1.

From (4.6), it follows that

E

N()i(F)

=n

l=1

ED((F)), N()DFl(1)i,lH

(4.7)

The random variable N()DFl(1)i,l belongs to D

1,2(H), by assumption(a). Consequently N()DF

l(1)i,l Dom (see Proposition 3.6). Hence,by the duality identity,

E

N()i(F)=

nl=1

ED((F)), N()DFl(1)i,lH

E n

l=1

N()DFl(1)i,l

||||.Let PNbe the finite measure on (, G) absolutely continuous with respectto P with density given by N(). Then, by Proposition 2.2, PN F1 is

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absolutely continuous with respect to Lebesgue measure. Therefore, for any

B B(Rn

) with Lebesgue measure equal to zero, we haveF1(B)

N()dP = 0.

Let N . Assumption (b) implies that limN N() = 1. Hence, bybounded convergence, we obtain P(F1(B)) = 0. This finishes the proof ofthe Proposition.

Remark 4.5 The existence of density for the probability law of a randomvectorFcan be obtained under weaker assumptions than in Theorem 4.1 (or

Proposition 4.2). Indeed, Bouleau and Hirsch proved a better result usingother techniques in the more general setting of Dirichlet forms. For the sakeof completenes we give one of their statements, the most similar to Theorem4.1, and refer the reader to [8] for complete information.

Proposition 4.3 Let F : Rn be a random vector satisfying the fol-lowing conditions:

(a) Fj D1,2, for anyj= 1, . . . , n,(b) the Malliavin matrix is inversible, a.s.

Then the law ofFhas a density with respect to the Lebesgue measure onR

n

.

4.2 Smoothness of the density

As we have seen in the first lecture, in order to obtain regularity propertiesof the density, we need an integration by parts formula for multi-indices oforder greater than one. In practice, this can be obtained recursively. In thenext proposition we give the details of such a procedure.

An integration by parts formula

Proposition 4.4 LetF : Rn

be a random vector such thatFj

D

for anyj = 1, . . . , n. Assume that

det 1 p[1,)Lp(). (4.8)Then:

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(1) det 1 D and1 D(Rm Rm).(2) LetG D. For any multi-index {1, . . . , n}r, r 1, there exists

a random variable H(F, G) D such that for any function Cb (Rn),

E

()(F)G

= E

(F)H(F, G)

. (4.9)

The random variablesH(F, G) can be defined recursively as follows:

If||= 1, = i, then

Hi(F, G) =n

l=1

(G(1)i,lDFl), (4.10)

and in general, for= (1, . . . , r1, r),

H(F, G) =Hr(F, H(1,...,r1)(F, G)). (4.11)

Proof: Consider the sequence of random variables

YN= (det + 1N

)1, N1

. Fix an arbitrary p[1, [. Assumption (4.8) clearly yields

limN

YN= det 1

inLp().We now prove the following facts:

(a)YN D, for anyN 1,(b) (DkYN, N 1) is a Cauchy sequence in Lp(; Hk), for any natural

number k .

Since the operator Dk is closed, the claim (1) will follow.Consider the function N(x) = (x+

1N

)1, x 0. Notice that N Cb .Then Remark 3.3 yields recursively (a). Indeed, det D.Let us now prove (b). The sequence of derivatives

(n)

N (det ), N 1 isCauchy in Lp(), for any p [1, ). This can be proved using (4.8) andbounded convergence. The result now follows by expressing the differenceDkYN DkYM, N, M 1, by means of Leibnizs rule (see (3.29)) and usingthat det D.

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Once we have proved that det 1 D, we trivially obtain1 D(Rm R

m

), by a direct computation of the inverse of a matrix and using thatFj D.The proof of (4.9)(4.11) is done by induction on the order r of the multi-index . Let r = 1. Consider the identity (4.6), multiply both sides by Gand take expectations. We obtain (4.9) and (4.10).Assume that (4.9) holds for multi-indices of order r 1. Fix =(1, . . . , r1, r). Then,

E

()(F)G

= E

(1,...,r1)((r)(F))G

=E

(r)(F)H(1,...,r1)(F, G)

=E

(F)Hr(F, H(1,...,r1)(F, G)

.

The proof is complete.

A criterion for smooth densities

As a consequence of the preceding proposition and part 2 of Proposition 2.1we have a criterion on smoothness of density, as follows.

Theorem 4.2 LetF : Rn be a random vector satisfying the assump-tions

(a) Fj D, for anyj = 1, . . . , n,(b) the Malliavin matrix is invertible a.s. and

det 1 p[1,)Lp().

Then the law of F has an infinitely differentiable density with respect toLebesgue measure onRn.

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5 Watanabe-Sobolev Differentiability of

SPDEs

5.1 A class of linear homogeneous SPDEs

LetL be a second order differential operator acting on real functions definedon [0, [Rd. Examples ofL where the results of this lecture can be appliedgather the heat operator and the wave operator. With some minor modi-fications, the damped wave operator and some class of parabolic operatorswith time and space dependent coefficients could also be covered. We areinterested in SPDEs of the following type

Lu(t, x) = (u(t, x)) W(t, x) +b (u(t, x)) , (5.1)

t]0, T],x Rd, with suitable initial conditions. This is a Cauchy problem,with finite time horizon T > 0, driven by the differential operator L, andwith a stochastic input given by W(t, x). For the sake of simplicity we shallassume that the initial conditions vanish.

Hypotheses on W

We assume that (W(), D(Rd+1)) is a Gaussian process, zero mean, andnon-degenerate covariance function given by E(W(1)W(2)) = J(1, 2),where the functional J is defined in (3.15). By setting =

F1, the

covariance can be written as

E(W(1)W(2)) =R+

dsRd

(d)F1(s)()F2(s)(),

(see (3.16)).From this process, we obtain the Gaussian family (W(h), h HT) (see Sec-tion 3.2).

A cylindrical Wiener process derived from (W(h), h HT)The process (Wt, t[0, T]) defined by

Wt =

j=1

ejj(t),

where (ej, j 1) is a CONS ofH and j , j 1, a sequence of independentstandard Wiener processes, defines a cylindrical Wiener process onH (see

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[18], Proposition 4.11, page 96, for a definition of this object). In particular,

Wt(g) :=Wt, gH satisfiesE(Wt(g1)Ws(g2)) = (s t)g1, g2H.

The relationship between (Wt, t[0, T]) and (W(h), h HT) can be estab-lished as follows. Consider h HTof the particular type h= 11[0,t]g, g H.Then, the respective laws of the stochastic processes W(11[0,t]g), t[0, T] andWt, gH, t[0, T] are the same.Indeed, by linearity,

W(11[0,t]g) =

j=1

g, ej

HW(11[0,t]ej ),

By the definition of (W(h), h HT), the family (W(11[0,t]ej), t[0, T], j1)is a sequence of independent standard Brownian motions.On the other hand,

Wt, gH=

j=1

g, ejHj(t).

This finishes the proof of the statement.In connection with the process (Wt, t [0, T]), we consider the filtration(Ft, t [0, T]), whereFt is the -field generated by the random variablesW

s(g), 0

s

t, g

H. It will be termed thenatural filtration associated

withW.

Hypotheses on L

We shall denote by the fundamental solution of Lu = 0, and we shallassume(HL) is a deterministic function oft taking values in the space of non-negative measures with rapid decrease (as a distribution), satisfying

T0

dtRd

(d) |F(t)()|2


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Examples

1 Heat operator: L= t d,d1.The fundamental solution of this operator possesses the following property:for anyt0, Rd,

C1t

1 + ||2 t0

ds|F(s)()|2 C2 t+ 11 + ||2 , (5.4)

for some positive constants Ci,i = 1, 2.Consequently (5.2) holds if and only if

Rd

(d)

1 + ||2 1), we have1 exp(42t||2)

42||2 1

22||2 C

1 + ||2 .

On the other hand, on (|| 1), we use the property 1 ex x, x 0,and we obtain

1 exp(42t||2)42

|

|2

Ct1 +

|

|2

.

This yields the upper bound in (5.4).Moreover, the inequality 1 ex x1+x , valid for any x0, implies t

0ds|F(t)()|2 C t

1 + 42t||2 .

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Assume that 42t||2 1. Then 1 + 42t||2 82t||2; i f 42t||2 1then 1 + 4

2

t||2

< 2 and therefore, 1

1+42t||2 1

2(1+||2) . Hence, we obtainthe lower bound in (5.4) and now the equivalence between (5.2) and (5.5) isobvious.Condition (5.3) is clearly satisfied.

2 Wave operator: L= 2tt d,d1.For anyt0, Rd, it holds that

c1(t t3) 11 + ||2

t0

ds|F(s)()|2 c2(t+t3) 11 + ||2 , (5.6)

for some positive constants ci, i = 1, 2. Thus, (5.2) is equivalent to (5.5).Let us prove (5.6). It is well known (see for instance [75]) that

F(t)() =sin(2t||)2|| .

Therefore

|F(t)()|2 122(1 + ||2)11(||1)+t

211(||1)

C 1 +t2

1 + ||2 .

This yields the upper bound in (5.6).

Assume that 2t|| 1. Then sin(4t||)

2t|| and consequently, t0

dssin2(2s||)

(2||)2 C t

1 + ||2 2t0

sin2(u||)du

=C t

1 + ||2 (2 sin(4t||)

2t|| )

C t1 + ||2 .

Next we assume that 2t|| 1 and we notice that for r [0, 1], sin2 rr2

sin2 1. Thus, t

0ds

sin2(2s||)(2||)2 Csin

2 1 2t0

s2ds

C t3

1 + ||2 .

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This finishes the proof of the lower bound in (5.6).

Ford3, condition (5.3) holds true. In fact,

(t,dx) =

1211{|x|


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t]0, T], x Rd, is required to be predictable and to belong to the spaceL2

( [0, T]; H).We address this question following the approach of [53] with a few changes,in particular we allow more general covariances (see Lemma 3.2 and Propo-sition 3.3 in [53]).

Lemma 5.1 Assume that satisfies (HL), then HT and

2HT = T0

dtRd

(d)|F(t)()|2.

Proof: Let be a non-negative function inC

(Rd) with support containedin the unit ball and such that

Rd

(x)dx= 1. Set n(x) =nd(nx), n1.Define n(t) = n(t). It is well known that n 0 inS(Rd) andn(t) S(Rd). Moreover, for any Rd,|Fn(t)()| |F(t)()|.By virtue of (5.2), (n, n1) HT, and it is Cauchy sequence. Indeed,

n mHT = T0

dtRd

(d)|F(t)()|2|F(n() m(t))|2

and sinceF(n() m()) converges pointwise to zero as n, m , wehave

limn,m

n

mHT

= 0,

by bounded convergence. Consequently, (n, n1) converges in HTand thelimit is . Finally, by using again bounded convergence,

2HT = limn n2HT

= T0

dtRd

(d)|F(t)()|2.

The next Proposition gives a large class of examples for which the stochasticconvolution againstWcan be defined.

Proposition 5.1 Assume that satisfies (HL). LetZ ={Z(t, x), (t, x)[0, T] Rd} be a predictable process, bounded inL2. Set

CZ := sup(t,x)[0,T]Rd

E|Z(t, x)|2

.

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Then, z(t,dx) :=Z(t, x)(t,dx)is a predictable process belonging to L2( [0, T]; H) and

Ez2HT

CZ

T0

dtRd

(d)|F(t)()|2.

Hence, the stochastic integralT

0

Rd

zdW is well-defined as anL2()-valuedrandom variable and

T0

Rd

zdW

2

L2()

=Ez2HT

CZ

T

0dt

Rd

(d)|F

(t)()|2. (5.8)

Proof: By decomposing the process Z into its positive and negative part,it suffices to consider non-negative processes Z. Since (t) is a temperedmeasure, so is z(t). Hence we can consider the sequence ofS(Rd)-valuedfunctions zn(t) = nz(t), n 1, where n is the approximation of theidentity defined in the proof of the preceding lemma.Using Fubinis theorem and the boundedness property ofZwe obtain

E(

zn

2HT

) =E T

0

dt R

d(dx) [zn(t)

zn(t)] (x)

CZ T0

dtRd

(dx)Rd

dzn(t, d(z))n(t, d(x z))

=CZ

T0

dtRd

(dx)n(t) n(t)

(x)

=CZ

T0

dtRd

(d)|Fn(t)()|2

CZ T0

dtRd

(d)|F(t)()|2


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then, using similar arguments as in the preceding lemma, we can prove that

(zn, n 1) converges in L2

([0, T]; H) to z, finishing the proof of theproposition.For the proof of (5.9), we proceed as follows. Firstly, to simplify the expre-sions, we write zn,m instead ofzn zm, and n,m forn m.

E(zn,m2HT) =E T

0dt

Rd

(dx)Rd

dyzn,m(t, x y)zn,m(t, y)

=E T

0dt

Rd

(dx)Rd

dy

Rddzn,m(x y z)z(t, z)

Rd dzn,m(y z)z(t, z)

=E T

0dt

Rd

Rd

Z(t, z)Z(t, z)(t,dz)(t,dz)

Rd

(dx)[n,m(. z) n,m(.+z]

=E T

0dt

Rd

Rd


Rd

(d)e2i(zz)|Fn,m()|2

. (5.10)

Then, since the Fourier transform of a convolution is the product of theFourier transform of the corresponding factors, using Fubinis theorem thislast expression is equal to

E

T0

dtRd

(d)|F(Z(t)(t))()|2|Fn,m()|2

.

Hence (5.9) is established.Finally, (5.8) is obtained by the isometry property of the stochastic convolu-tion, combined with the estimate of the integrand proved before.

Remark 5.1 Assume that the processZ is bounded away from zero, that itinf(t,x)[0,T]Rd |Z(t, x)| c0 > 0. Then

E(zn2HT)c20 T0

dtRd

(d)|F(t)()|2|Fn()|2. (5.11)

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Indeed, arguing as in (5.10), we see that

E(zn2HT) =E

T

0dt

Rd

Rd


Rd

(dx)[n(. z) n(.+z]

c20 T0

dtRd

Rd

(t,dz)(t,dz)

Rd

(dx)[n(. z) n(.+z]

=c20

T0

dtRd

Rd

(t,dz)(t,dz)

Rd (d)e2i(zz)

|Fn()|2

=c20

T0

dtRd

(d)|F(t)()|2|Fn()|2.

Remark 5.2 Assume that the coefficient in (5.7) has linear growth andthat the processu satisfies the conditions given at the beginning of the section.ThenZ(s, y) :=(u(s, y))satisfies the assumption of Proposition 5.1 and thestochastic integral (stochastic convolution) in (5.7) is well-defined.

A result on existence and uniqueness of solution

Theorem 5.1 Assume that, b: R R are Lipschitz functions and thatsatisfies (HL). Then there exists a unique mild solution to Equation (5.1).Such a solution is a random field indexed by(t, x)[0, T] Rd, continuousinL2(), and for anyp[1, [,

sup(t,x)[0,T]Rd

E(|u(t, x)|p)


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for n 1. We refer the reader to Theorem 13 of [16] for the details ofthe proof of the convergence of the Picard sequence and the extensions ofGronwalls lemma suitable thereof (see also Theorem 6.2 and Lemma 6.2 in[69]).

5.2 The Malliavin derivative of a SPDE

Consider a SPDE in its mild formulation (see 5.7). We would like to studyits differentiability in the Watanabe-Sobolev sense. There are two aspects ofthe problem:

(A) to prove differentiability,

(B) to give an equation satisfied by the Malliavin derivative.

A useful tool to prove differentiability of Wiener functionals is provided bythe next result, which is an immediate consequence from the fact that theNth order Malliavin derivative is a closed operator defined on Lp() withvalues in Lp(; HN), for any p[1, [. In our context H:=HT. A resultof the same vein has been presented in Proposition 3.3.

Lemma 5.2 Let(Fn, n1) be a sequence of random variables belonging toD

N,p. Assume that:

(a) there exists a random variableFsuch thatFn converges toF inLp

(),asn tends to,(b) the sequence(DNFn, n1) converges inLp(; HNT ), asn tends to,

ThenF belongs to DN,p andDNF =Lp(; HNT ) limn DNFn.We shall apply this lemma to F := u(t, x), the solution of Equation (5.7).Therefore, we have to find out an approximating sequence of the SPDE sat-isfying the assumptions (a) and (b) above. A possible candidate is providedin the proof of the existence and uniqueness of solution: the Picard approx-imations defined in (5.13). The verification of condition (a) for the sequence

(un(t, x), n0), for fixed (t, x)[0, T] Rd is part of the proof of Theorem5.1As regards condition (b), we will avoid too many technicalities by focussingon the first order derivative and taking p = 2. For this we need the functions and b to be of classC1.

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A possible strategy might consists in proving recursively thatun(t, x) belongs

toD

1,2

, then applying rules of Malliavin calculus (for instance, an extensionof (3.30)) we will obtain

Du0(t, x) = 0

Dun(t, x) = (t , x )(un1(, ))+

t0

Rd

(t s, x y)(un1(s, y))Dun1(s, y)W(ds, dy)

+ t0

dsRd

(s,dy)b(un1(t s, x y))Dun1(t s, x y),(5.14)

n1.A natural candidate for the limit of this sequence is the process satisfying(5.16).At this point some comments are pertinent:

1. The Malliavin derivative is a random vector with values inHT. There-fore, Equations (5.14) and (5.16) correspond to the mild formulationof a Hilbert-valued SPDE.

2. The notation (t, x)(un1(, )) aims to show up the dependenceon the time variable (written with a dot) and on the space variable(written with a star). By Proposition 5.1 and Remark 5.2 such a termis in L2( [0, T]; H).

3. The stochastic convolution term in (5.14) is not covered by the previousdiscussion, since the process (un1(s, y))Dun1(s, y) takes values onHT. A sketch of the required extension is given in the next paragraphs.

Stochastic convolution with Hilbert-valued integrands

Let K be a separable real Hilbert space with inner-product and norm denotedby, K and K, respectively. LetK={K(s, z), (s, z)[0, T] Rd} beaK-valued predictable process satisfying

CK := sup(s,z)[0,T]Rd

E||K(s, z)||2K


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5.1, zj (t,dx) = Kj(t, x)(t,dx) is a predictable process and belongs to

L2

( [0, T]; H), and then K(t, x)(t,dx) is also a predictable processand belongs to L2( [0, T]; H K). TheK-valued stochastic convolutionT0

Rd

(t, x)K(t, x)W(dt,dx) is defined as T0

Rd

(t, x)Kj (t, x)W(dt,dx), j0

and satisfies

E

T0

Rd

(t, x)K(t, x)W(dt,dx)

2

K

= EK2HK

CK T

0 dtRd (d)|F(t)()|

2

. (5.15)

Going back to the application of Lemma 5.2, we might guess as limit of thesequence (5.14) aHT-valued process (Du(t, x), (t, x)[0, T] Rd) satisfyingthe equation

Du(t, x) = (t , x )(u(, ))+

t0

Rd

(t s, x y)(u(s, y))Du(s, y)W(ds, dy)

+ t0

dsRd

(s,dy)b(u(t s, x y))Du(t s, x y). (5.16)

Yet another result on existence and uniqueness of solution

Theorem 5.1 is not general enough to cover SPDEs like (5.16). In this sectionwe set up a suitable framework for this (actually to deal with Malliavinderivatives of any order). For more details we refer the reader to [69], Chapter6.LetK1,K be two separable Hilbert spaces. If there is no reason for mis-understanding we will use the same notation,|| ||,, , for the norms andinner products in these two spaces, respectively.Consider two mappings

, b:K1 K Ksatisfying the next two conditions for some positive constant C:

(c1)

supxK1

||(x, y) (x, y)|| + ||b(x, y) b(x, y)||

C||y y||,

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(c2) there exists q[1, ) such that

||(x, 0)|| + ||b(x, 0)|| C(1 + ||x||q),

x K1, y, y K.Notice that (c1) and (c2) clearly imply

(c3)||(x, y)|| + ||b(x, y)|| C(1 + ||x||q + ||y||).

Let V =

V(t, x), (t, x) [0, T] Rd

be a predictableK1-valued processsuch that

sup(t,x)[0,T]Rd

E||V(t, x)

||p


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Main result

We will now apply Lemma 5.2 to prove that for any fixed (t, x)[0, T]Rd,u(t, x)D1,2. The next results provide a verification of conditions (a) and (b)of the Lemma. We shall assume that the functions andb are differentiablewith bounded derivatives.

Lemma 5.3 The sequence of random variables

un(t, x), n0

defined re-

cursively in (5.13) is a subset ofD1,2.In addition,

supn0

sup(t,x)[0,T]Rd

EDun(t, x)2HT


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Finally, for the third termB3,n(t, x) we use Schwarzs inequality with respect

to the finite measure (s,dz)ds. Then, the assumptions on b and yield

E(B3,n(t, x)2HT C t0

ds sup(,z)[0,s]Rd

E||Dun1(, z)||2HT

.

Therefore,

sup(s,z)[0,t]Rd

E||Dun(s, z)||2HT

C1 + t

0ds sup

(,z)[0,s]RdE||Du

n1(, z)||2HT(J(t s) + 1).

Then, by Gronwalls Lemma (see Lemma 6.2 in [69]), we finish the proof.

Lemma 5.4 Under the standing hypotheses, the sequenceDun(t, x), n0,converges inL2(; HT), uniformly in (t, x) [0, T] Rd, to theHT-valuedstochastic processes

U(t, x), (t, x)[0, T] Rd

solution of the equation

U(t, x) =H(t, x)

+ t0

Rd

(t s, x z)U(s, z)(u(s, z))W(ds, dz)

+ t0

dsRd

(s,dz)U(t s, x z)b(u(t s, x z)), (5.21)

withH(t, x) =(u(, ))(t , x ).

Proof : We must prove

sup(t,x)[0,T]Rd

EDun(t, x) U(t, x)2

HT

0, (5.22)

as n tends to infinity.

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Set

In,NZ (t, x) = (t , x )(un1(, )) (u(, )),In (t, x) =

t0

Rd

(t s, x z)(un1(s, z))Dun1(s, z)W(ds,dz)

t0

Rd

(t s, x z)(u(s, z))U(s, z)W(ds, dz),

Inb(t, x) = t0

dsRd

(s,dz)

b(un1(t s, x z))Dun1(t s, x z) b(u(t s, x z))U(t s, x z)

.

The Lipschitz property of yields

E(||In,NZ (t, x)||2HT)C sup(t,x)[0,T]Rd

E(|un1(t, x) u(t, x)|2) t

0 ds

Rd

(d)|F(s)()|2

C sup(t,x)[0,T]Rd

E(|un1(t, x) u(t, x)|2).

Hence,lim

n sup

(t,x)[0,T]RdE(||In,NZ (t, x)||2HT) = 0. (5.23)

Consider the decomposition

E(

In (t, x)2HT

)

C(D1,n(t, x) +D2,n(t, x),

where

D1,n(t, x) =E t0

Rd

(t s, x z)[(un1(s, z)) (u(s, z))]Dun1(s, z)W(ds, dz)2HT

,

D2,n(t, x) =E t0

Rd

(t s, x z)(u(s, z))[Dun1(s, z) U(s, z)]W(ds, dz)2HT

.

The isometry property of the stochastic integral, Cauchy-Schwarzs inequalityand the properties of yield

D1,n(t, x)C sup(s,y)[0,T]Rd

E(|un1(s, y) u(s, y)|4)E(Dun1(s, y)4HT)

12

t0

dsRd

(d)|F(s)()|2.

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Owing to and Lemma 5.3 we conclude that

limn

sup(t,x)[0,T]Rd

D1,n(t, x) = 0.

Similarly,

D2,n(t, x)C t0

ds sup(,y)[0,s]Rd

E(Dun1(, y) U(, y)2HT)J(t s).(5.24)

For the pathwise integral term, we have

E(

Inb(t, x)

2HT

)

C(b1,n(t, x) +b2,n(t, x)),

with

b1,n(t, x) =E|| t0

dsRd

(s,dz)[b(un1(t s, x z)) b(u(t s, x z))] Dun1(t s, x z)||2HT

,

b2,n(t, x) =E t0

dsRd

(s,dz)b(u(t s, x z)) [Dun1(t s, x z) U(t s, x z)]2HT

.

By the properties of the deterministic integral of Hilbert-valued processes,

the assumptions on band Cauchy-Schwarzs inequality we obtain

b1,n(t, x) t0

dsRd

(s,dz)E|b(un1(t s, x z)) b(u(t s, x z))|2

Dun1(t s, x z)2HT

sup(s,y)[0,T]Rd

E|un1(s, y) u(s, y)|4 EDun1(s, y)4HT

1/2 t0

ds(s,dz).

Thus,lim

n

sup(t,x)[0,T]Rd

b1,n(t, x) = 0.

Similar arguments yield

b2,n(t, x)C t0

ds sup(,y)[0,s]Rd

E(Dun1(, y) U(, y)2HT).

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Therefore we have obtained that

sup(s,x)[0,t]Rd

E(||Dun(s, x) U(s, x)||2HT)

Cn+C t0

ds sup(,x)[0,s]Rd

E(||Dun1(, x) U(, x)||2HT)(J(t s) + 1),

with limn Cn = 0. Thus applying a version of Gronwalls lemma (seeLemma 6.2 in [69]) we complete the proof of (5.22).

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6 Analysis of Non-Degeneracy

In comparison with SDEs, the application of the criteria for existence andsmoothness of density for Gaussian functionals (see for instance Proposition4.3) and Theorem 4.2) to SPDEs is not a well developed topic. Most of theresults for SPDEs are proved under ellipticity conditions. In this lecture,we shall discuss the non-degeneracy of the Malliavin matrix for the classof SPDEs studied in the preceding lecture, in a very simple situation: indimension one and assuming ellipticity.

6.1 Existence of moments of the Malliavin covariance

Throughout this section, we fix (t, x)]0, T] Rd

and consider the randomvariableu(t, x) obtained as a solution of (5.7). Hence we are in the frameworkof Section 5 and therefore, we are assuming in particular that satisfieshypotheses (HL).Following Definition 4.1, the Malliavin matrix is the random variableDu(t, x)HT. In this section, we want to study the property

EDu(t, x)pHT

0, a.s.Clearly, having (6.1) for somep >0 is a sufficient condition for this propertyto hold.

The classical connection between moments and distribution func-tions

Lemma 6.1 Fix p]0, ]. The property (6.1) holds if and only if thereexists0 > 0, depending onp, such that

0

0

(1+p)P(||

Du(t, x)||2HT

< )d )d.

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In fact, this follows easily from Fubinis theorem.

Apply this formula to Y :=||Du(t, x)||2p

HT . We obtain

E(||Du(t, x)||2pHT) =m1+m2,

with

m1= 00

P(||Du(t, x)||2pHT > )d,

m2=

0P(||Du(t, x)||2pHT > )d.

Clearly,m1

0. The change of variable = p implies

m2 =

0P(||Du(t, x)||2pHT > )d

=

0P(||Du(t, x)||2HT <

1p )d

=p 1p

0

0(1+p)P(||Du(t, x)||2HT < )d.

This finishes the proof.

Moments of low order

Knowing the size in of the term P(||Du(t, x)||2HT < ) will help us to verifythe integrability of(1+p)P(||Du(t, x)||2HT < ) at zero, and a posterioritoestablish the validity of (6.1). The next proposition gives a result in thisdirection.

Proposition 6.1 We assume that

(1) there exists0 > 0 such thatinf{|(z)|, z R} 0,

(2) there exist such that for anyt(0, 1),

C1t

t0

dsRd

(d)|F(s)()|2, (6.3)

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Then for any]0, 1[,

PDu(t, x)2HT <

C1 1 . (6.4)

Consequently, (6.1) holds for anyp 0 such thatt 0. From (5.16), the definition ofHT, andthe triangular inequality, we clearly have

Du(t, x)2HT t

tdsDs,u(t, x)2H

12

t

tds(t s, x )(u(s, ))2H I(t, x; ),

where

I(t, x; ) = t

tds

ts

Rd

(t r, x z)(u(r, z))Ds,u(r, z)W(dr, dz)

+ t

sdr

Rd

(t r,dz)b(u(r, x z))Ds,u(r, x z)2H.

Set

M1() = 0

dsRd

(d)|F(s)()|2,

M2() = 0

dsRd

(s, y)dy.

Notice that by (5.3),M2()C.Our aim is to prove

tt

ds(t s, x )(u(s, ))2H20M1(), (6.5)

E(I(t, x; ))CM1() (M1() +M2()) . (6.6)

For any]0, 1[, we can choose := ()> 0 such thatM1() = 420

. Notice

that by (6.3) this is possible. Then, < 420

1

1 , and assuming that (6.5),

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(6.6) hold true, we have

PDu(t, x)2HT <

P

t

tdsDs,u(t, x)2H <

P

I(t, x; )20

2M1()

C1E(I(t, x; ()))C1

2 +1+

1

C1 1 .

This is (6.4).

Proof of (6.5):By a change of variables t

tds(t s, x )(u(s, ))2H =

0

ds(s, x )(u(t s, ))2H.

Then, the inequality (5.11) applied to Z(s, y) =|(u(t s, y))| and T := yields (6.5). Indeed, for this choice ofZand T,

0ds(s, x )(u(t s, ))2H = limnE(zn

2H

)0M1().

Proof of (6.6):

We shall give a bound for the mathematical expectation of each one of theterms

I1(t, x; ) = 0

ds

t

ts

Rd

(t r, x z)(u(r, z))Dts,u(r, z)W(dr,dz)2

H,

I2(t, x; ) = 0

ds t

tsdr

Rd

(t r,dz)b(u(r, x z))Dts,u(r, x z)2

H.

Since is bounded, the inequality (5.15) yields

E(I1(t, x; ))C sup(s,y)[0,]Rd EDt,u(t s, y)2

H

M1().

For the pathwise integral, it is easy to prove that

E(I2(t, x; ))C sup(s,y)[0,]Rd

EDt,u(t s, y)2H

M2().

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Since

sup(s,y)[0,]Rd EDt,u(t s, y)2HCM1(),

(see for instance [61] or Lemma 8.2 in [69]), we get

E(I1(t, x; ))CM1()2,E(I2(t, x; ))CM1()M2().

This finishes the proof of (6.7) and therefore of (6.4).The statement about the validity of (6.1) for the given range ofp is a conse-quence of Lemma 6.1.

An example: the stochastic wave equation in dimensiond3Proposition 6.1 can be applied for instance to the stochastic wave equation.Indeed, let be the fundamental solution of L = 0 with L = 2tt d,d= 1, 2, 3. Assume that the measure satisfies

0 0 such thatinf{|(z)|, z R} 0,(3) satisfies the assumptions (HL),

(4) there exist >0, such that for anyt(0, 1),

C1t

t0

dsRd

(d)|F(s)()|2

Then, for any fixed (t, x)]0, T] Rd, the random variable u(t, x) belongsto D1,2, and for any p < 1 1

, E(Du(t, x)pHT)


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At this point, we can apply Chebychevs inequality to obtain

P

I(t, x; )202

M1() CqE(I(t, x; ()))q ,

for anyq >1.Using Lq()- estimates for the Hilbert-valued stochastic convolutions (seefor instance [69], Theorem 6.1), and for pathwise integrals as well, yield

E(I(t, x; ()))q C()q1

M1(())2q +M1(())

q(())q

.

By the choice of(), this implies

PDu(t, x)2HT <

C q1 +[q q ].

Sinceqcan be chosen arbitrarily large, we obtain (6.1) for any p0.Regularity of the density

Proceeding recursively, it is possible to extend the results in Section 5.2and prove that under suitable assumptions, the solution of (5.7) is infinitelydifferentiable in the Watanabe-Sobolev sense (see [69], Chapter 7). Then,owing to the results discussed in the preceding paragraphs, applying Theorem4.2 yields the following:

Theorem 6.2 Consider the stochastic process{u(t, x), (t, x)[0, T] Rd}solution of (5.7). We assume:

(1) the functions , b belong toC

and have bounded derivatives of anyorder,

(2) there exists0> 0 such thatinf{|(z)|, z R} 0,(3) satisfies the assumptions (HL),

(4) there exist >0, such that for anyt(0, 1),

C1t

t0

dsRd

(d)|F(s)()|2

Then, for any fixed(t, x)]0, T] Rd, the random variableu(t, x)belongs toD

, and for anyp >0,E(Du(t, x)p

HT)


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6.2 Some references

To end this lecture, we mention some references on existence and smoothnessof density of probability laws, as a guide for the reader to have a furtherinsight into the subject.The first application of Malliavin calculus to SPDEs may be found in [51]; itconcerns the hyperbolic equation on Rn

2

stX(s, t) =A(X(s, t)) Ws,t+A0(X(s, t)), (6.8)

withs, t]0, 1] and initial condition X(s, t) = 0 ifs t= 0. HereA: Rn

R

d

R

n, A0 : Rn

R

n,

are smooth functions, andW ad-dimensional Brownian sheet on [0, 1]2, thatis, W = (Ws,t = (W

1s,t, . . . W

ds,t), (s, t) [0, 1]2), with independent Gaussian

components, zero mean and covariance function given by

E(Wis1,t1Wi

s2,t2) = (s1 s2)(t1 t2),

i= 1, . . . , d.In dimension n = 1, this equation is transformed into the standard waveequation after a rotation of forty-five degrees. Otherwise, (6.8) is an extensionto a two-parameter space of the Ito equation. The existence and smoothness

of density for the probability law of the solution to (6.8) at a fixed timeparameter (s, t), with st = 0 has been proved under a specific type ofHormanders condition on the vector fields Ai, i= 1, . . . , d, which does notcoincide with Hormanders condition for diffusions. An extension to a nonrestricted Hormanders condition, that is, including the vector field A0, hasbeen done in [52].The one-dimensional wave equation perturbed by space-time white noise, asan initial value problem but also as a boundary value problem, has beenstudied in [13]. The degeneracy conditions on the free terms of the equationare different from those in Theorem 6.2.Existence for the density of equation (4.1) in dimension one, with L= and

space-time white noise, has been first studied in [58]. The authors considera Dirichlet boundary value problem on [0, 1], with initial conditionu(0, x) =u0(x). The required non-degeneracy condition reads as follows:

(u0(y))= 0, for some y]0, 1[. (6.9)

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The same equation has been analyzed in [3]. In this reference, the authors

consider the random vector (u(t, x1), . . . , u(t, xm)) obtained by looking at thesolution of the equation at time t= 0 and different points x1, . . . , xm]0, 1[.Under assumption (2) of Theorem 6.2, they obtain the smoothness of thedensity.Recently in [45], this result has been improved. The authors prove that form= 1, the assumption 6.9 yields the smoothness of the density as well.The first application of Malliavin calculus to SPDEs with correlated noiseappears in [43], and a first attempt for an unified approach of stochastic heatand wave equations is done in [37]. Hormanders type conditions in a generalcontext of Volterra equations have been given in [64].

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7 Small perturbations of the density

Consider the SPDE (5.1) that we write in its mild form, as in (5.7). Wereplace W(t, x) by W(t, x), with ]0, 1[ and we are interested in the be-haviour, as 0 of the solution of the modified equation, that we willdenote by u. At the moment, this is a very vague plan; roughly speaking,one would like to know the effect of small noise on a deterministic evolutionequation. Several questions may be addressed. For instance, denoting by

the probability law of the solution u, we may want to prove a large deviationprincipleon spaces where the solution lives.We recall that a family (, ]0, 1[) of probability measures on a Polishspace E is said to satisfy a large deviation principlewith rate functional I

ifI : E [0, ] is a lower semicontinous function such that the level sets{I(x)a} are compact, and for any Borel set B E,inf

xBI(x)lim inf

02 log((B))limsup

02 log((B)) inf

xBI(x).

In many of the applications that have motivated the theory of large devi-ations, as 0, degenerates to a delta Dirac measure at zero. Hence,typically a large deviation principle provides the rate of convergence and anaccurate description of the degeneracy.Suppose that the measures live in Rd and have a density with respect tothe Lebesgue measure. A natural question is whether from a large deviation

principle one could obtain a precise lower and upper bound for the density.This question has been addressed by several authors in the context of dif-fusion processes (Azencott, Ben Arous and Leandre, Varadhan, to mentionsome of them), and the result is known as thelogarithmic estimatesand alsoas theVaradhan estimates. We recall this result since it is the inspiration forextensions to SPDEs.Consider the family of stochastic differential equations on Rn (in theStratonovich formulation)

Xt =x+ t0

A(Xs) dBs+ t0

A0(Xs)ds,

t [0, 1], > 0, where A : Rn Rd Rn, A0 : Rn Rn and B is ad-dimensional Brownian motion. For each h in the Cameron-Martin spaceH associated with B, we consider the ordinary (deterministic) equation

Sht =x+ t0

A(Shs )hsds+ t0

A0(Shs )ds,

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t[0, 1], termed the skeletonofX. Fory Rn, setd2(y) = inf{h2H; Sh1 =y}d2R(y) = inf{h2H; Sh1 =y, det Sh

1>0},

where Sh1

denotes the n-dimensional matrix whose en-

tries areD(Sh1 )i, D(Sh1 )j, i, j = 1, . . . , n. Here D stands for the Frechetdifferential operator on Banach spaces and we assume that the random fieldsA1, . . . , Ad (the components ofA) and A0 are smooth enough. Notice thatwe use the same notation for the deterministic matrix (D(Sh1 )i, D(Sh1 )j)i,jthan for the Malliavin matrix. In this context, the former is often termedthe deterministic Malliavin matrix. The quantitiesd2(y), d2R(y) are related

with the energy needed by a system described by the skeleton to leave theinitial condition x Rn and reach y Rn at time t = 1.This is the result for SDEs.

Theorem 7.1 LetA : Rn RdRn,A0: Rn Rn be infinite differentiablefunctions with bounded derivatives of any order. Assume:

(HM) There existsk0 1 such that the vector space spanned by the vectorfields

[Aj1, [Ajk2 , [. . . [Ajk , Aj0]] ], 0kk0,where j0

{1, 2, . . . , d

} and ji

{0, 1, 2, . . . , d

} if 1

i

k at the point

x Rn has dimensionn.Then, the random variableX1 has a smooth densityp, and

d2R(y)liminf0 22 logp(y)limsup

022 logp(y) d2(y). (7.1)

In addition, if inf{det Sh1

;for h such that Sh1 =y}> 0, thend2(y) =d2R(y)and consequently,

lim0

22 logp(y) =d2(y). (7.2)

The assumption (HM) in this theorem is termed Hormanders unrestricted

assumption; the notation [, ] refers to the Lie brackets.The proof of this theorem given in [11] admits an extension to the generalframework of an abstract Wiener space. This fact has been noticed andapplied in [33], and then written in [47]; since then, it has been applied toseveral examples of SPDEs. In this lecture we shall give this general result

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and then some hints on its application to an example of stochastic heat

equation.Throughout this section,{W(h), h H} is a Gaussian family, as has beendefined in Section 3.2. We will consider non-degenerate random vectors F,which in this context means that F D and

det 1F p[1,[Lp(),

where Fdenotes the Malliavin matrix ofF. Notice that by Theorem 4.2,non-degenerate random vectors Fhave an infinitely differentiable density.

7.1 General results

Lower bound

Proposition 7.1 Let {F, ]0, 1]} be a family of non-degenerate n-dimensional random vectors and let Cp (H;Rn) (the space ofC func-tions with polynomial growth) be such that for eachhH, the limit

lim0

1

F

+

h

(h)

= Z(h) (7.3)

exists in the topology ofD and defines an-dimensional random vector with

absolute continuous distribution.Then, setting

d2R(y) = inf{h2H : (h) =y, det (h) > 0},

y Rn, we haved2R(y)liminf0

2 logp(y). (7.4)

Proof: Lety Rn be such thatd2R(y)0 there existshHsuch that (h) =y, det (h) >0 andh2H d2R(y) +. For any functionf C

0 (R

n), we can write

E(f(F)) = exp

h

2H

22

E

f

F

+

h

exp

W(h)

, (7.5)

by Girsanovs theorem.

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Consider a smooth approximation of 11[,] given by a function C,01, such that (t) = 0, ift /[2, 2], (t) = 1 ift[, ]. Thenusing (7.5), and assuming that fis a positive function, we have

E(f(F))exph

2H+ 4

22

E

f

F

+

h

(W(h))

.

We now apply this inequality to a sequence fn, n 1, of smooth approxi-mations of the delta Dirac function at y. Passing to the limit and takinglogarithms, we obtain

22 logp(y)

(

h

2H+ 4) + 2

2 log Ey F

+h

(W(h)) .

Hence, to complete the proof we need checking that

lim0

2 log E

y

F

+

h

(W(h))

= 0. (7.6)

Sincey = (h), we clearly have

E

y

F

+

h

(W(h))

= mE

0

F

+ h

(h)

(W(h))

.

The expression

E

0

F

+ h

(h)

(W(h))

tends to the density ofZ(h) at zero, as 0, as can be proved using theintegration by parts formula (4.9)(4.11). Hence (7.6) holds true and thisends the proof of the Proposition.

Upper boundStating the upper bound for the logarithm of the density needs more de-manding assumptions. Among others, the family{F, ]0, 1]}must satisfya large deviation principle, and there should be a control of the norm of theinverse of the Malliavin matrix in terms of powers of.

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Proposition 7.2 Let {F, ]0, 1]} be a family of non-degenerate n-dimensional random vectors and let C

p (H;R

n

) be such that:1. sup]0,1] Fk,p 0 and N(p) ]1, [ such that

(F)1pN(p), for each integerp,3.{F, ]0, 1]}satisfies a large deviation principle onRn with rate func-

tionI(y), y Rn.Then,

limsup0

22 logp(y)

I(y). (7.7)

Proof: It is an application of the integration by parts formula (4.9)(4.11).Indeed, fixy Rn and a smooth function C0 (Rn), 01 such thatis equal to one in a nighbourhood ofy . Then we can write

p(y) =E((F)y(F

)) .

Applying Holders inequality with p, q]1, [ with 1p + 1q = 1, we obtain

E((F)y(F)) =E

11{Fy}H1,...,1(F

, (F))

E(|H1,...,1(F, (F))|)=E

|H1,...,1(F, (F))|11{Fsupp }

)

(P{F supp }) 1q H1,...,1(F, (F))p.By theLp estimates of the Skorohod integral (see for instance [79], or Propo-sition 3.2.2 in [47]), there exist real numbers greater than one, p, a, b, a, band

H1,...,1(F, (F))pC(F)1pFa,b(F)a,b.The assumptions (1) (2) below ensure

limsup0 22 log H1,...,1(F, (F))p= 0,

while (3) implies

limsup0

22 log P{F supp} inf(I(y), ysupp)

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and consequently,

lim sup0

22 logp(y) 1q

inf(I(y), ysupp).

Set I(supp) = inf(I(y), y supp). For any > 0, there exists q > 1such that 1

qI(supp)I(supp) . Then, by taking a sequence of smooth

functions n (with the same properties as ) such that suppn decreases to{y}, we see that

limsup0

22 logp(y) I(y) +,

and since is arbitrary, we obtain (7.7).This ends the proof.

7.2 An example: the stochastic heat equation

In this section, we consider the SPDE

u(t, x) = t0

R

G(t s, x y)(u(s, y))W(ds,dy)

+ t0

R

G(t s, x y)b(u(s, y))dsdy, (7.8)

t [0, T], where G(t, x) = (2t) 12exp |x|22t and W is space-time whitenoise. In the framework of Section 5 this corresponds to a stochastic heatequation in dimension d = 1 and to a spatial covariation measure given by(dy) =0(y).The companion perturbed family we would like to study is

u(t, x) = t0

R

G(t s, x y)(u(s, y))W(ds, dy)

+ t0

R

G(t s, x y)b(u(s, y))dsdy, (7.9)

]0, 1[.Our purpose is to apply Propositions 7.1 and 7.2 and therefore to obtainlogarithm estimates for the density.Throughout this section we will assume the following condition, althoughsome of the results hold under weaker assumptions.

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(C)The functions , b areC with bounded derivatives.The Hilbert spaceHwe should take into account here is the Cameron-Martinspace associated withW. It consists of the set of functionsh: [0, T]R Rabsolutely continuous with respect to the product Lebesgue measure dtdx

and such thath2H :=T

0

R

|ht,x|2dtdx12 0.Then the family (u, ]0, 1[) is non-degenerate and therefore the randomvariable u(t, x) possesses aC density p.This result has been proved in [3] (see also sections 5 and 6).

2 Large deviation principle

Large deviation principles for the family (u, ]0, 1[) in the topology ofHolder continuous functions have been established under different type ofassumptions by Sowers ([66]) and Chenal and Millet. Here, for the sake ofsimplicity we shall consider the topology of uniform convergence on compactsets and denote by

C([0, T]

R) the set of continuous functions defined on

[0, T]R with respect to this topology. It is known that (u, ]0, 1[) satisfiesa large deviation principle onC([0, T] R) with rate function

I(f) = inf

h2H2

; hH, h =f

.

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This is a functional large deviation principle; the contraction principle (a

transfer principle of large deviations through continuous functionals) givesrise to the following statement:Fix (t, x)]0, T] R. Assuming (C), u(t, x) satisfies a large deviationprinciple on R with rate function

I(y) = inf

h2H2

; hH, h(t, x) =y

, y R. (7.11)

A result for the stochastic heat equation

For the family defined in (7.9), we have the following theorem ([40], Theorem2.1)

Theorem 7.2 Assume that the functions, b satisfy(C) and also (ND).Fix (t, x)]0, T]R and let I : R R be defined in (7.11). Then thedensities(pt,x, ]0, 1[) of(u(t,x,]0, 1[), satisfy

lim0

2 logpt,x(y) =I(y). (7.12).

Proof: We shall consider the abstract Wiener space associated with the space-time white noise W and check that F := u(t, x), ]0, 1[, satisfy theassumptions of Propositions 7.1, 7.2. We give some hints for this in thesequel.

Proposition 7.2, Assumption 1.

Following the results of Section 5, we already know that u(t, x) D, thatis,u(t, x)k,p


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for anyp[1, [. Then we will haveE

||p2p sup]0,1[

E|Q|pC2p,

and we will obtain the desired conclusion with N(p) = 2.Let us give some hints for the proof of (7.13). Remember that =Du(t, x)2H. The Hvalued stochastic process Du(t, x) satisfies an equa-tion similar to (5.16), where is replaced by (and consequently

replaced by ). By uniqueness of solution, it holds that Du(t, x) =(u(., ))Y(t, x), where Y(t, x) is a Hvalued stochastic process solutionto the equation

Y(t, x) =G(t , x ) + t0

Rd

G(t s, x y)(u(s, y))Y(s, y)W(ds,dy)+

t0

Rd

G(t s, x y)b(u(s, y))Y(s, y)dsdy.

ThenQcorresponds to(u(., ))Y(t, x)2H, which essentially behaves likeDu(t, x)2H.Proposition 7.2, Assumption 3.

See the result under the heading large deviation principle.

Assumptions of Proposition 7.1.

The non-degeneracy of the family u

(t, x), ]0, 1[ has already been dis-cussed. Now we will discuss the existence of limit (7.3), which is a hypoth-esis on existence of directional derivative with respect to . Let us proceedformally.For anyhH, setZ,h(t, x)() =u(t, x)( + h ). By uniqueness of solutionZ,h(t, x) is given by

Z,h(t, x) = t0

Rd

G(t s, x y)

(Z,h(s, y)W(ds,dy)

+

(Z,h(s, y)hs,y+b(Z,h(s, y))

dsdy

.

It is now clear that the candidate for (h) in Proposition 7.1 should beZ0,h(t, x), and by uniqueness of solution Z0,h(t, x) = h(t, x). Hence, wehave to check that the mapping ]0, 1[Z,h(t, x) is diffe

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