Some Applications of Malliavin Calculus to SPDE and Convergence of Densities

Fei Lu

Submitted to the graduate degree program in the Department of Mathematics and the Graduate Faculty of the University of Kansas in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy.

Professor Yaozhong Hu, Chairperson

Professor David Nualart, Chairperson

Professor Tyrone Duncan Committee members

Professor Jin Feng

Professor Ted Juhl

Professor Xuemin Tu

Date Defended: February 11, 2013 The Dissertation Committee for Fei Lu certiﬁes that this is the approved version of the following dissertation:

Some Applications of Malliavin Calculus to SPDE and Convergence of Densities

Professor Yaozhong Hu, Chairperson

Professor David Nualart, Chairperson

Professor Tyrone Duncan

Professor Jin Feng

Professor Ted Juhl

Professor Xuemin Tu

Date approved: February 11, 2013

ii Abstract

Some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation theory are studied in this dissertation. In Chapter3, a Feynman-Kac formula is established for a stochastic heat equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion

1 with Hurst parameter H < 2 . To establish such a formula, we introduce and study a nonlinear stochastic integral of the Gaussian noise. The existence of the Feynman-Kac integral then follows from the exponential integrability of this nonlinear stochastic integral. Then, techniques from Malliavin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic heat equation. In Chapter4, the density formula in Malliavin calculus is used to study the joint Holder¨ continuity of the solution to a nonlinear SPDE arising from the study of one dimensional super-processes. Dawson, Vaillancourt and Wang [Ann. Inst. Henri. Poincare´ Probab. Stat., 36 (2000) 167-180] proved that the solution of this SPDE gives the density of the branching particles in a random environment. The time-space joint continuity of the density process was left as an open problem. Li, Wang, Xiong and Zhou [Probab. Theory Related Fields 153 (2012), no. 3-4, 441–469] proved that this solution is joint

1 1 Holder¨ continuous with exponent up to 10 in time and up to 2 in space. Using our new 1 method of Malliavin calculus, we improve their result and obtain the optimal exponent 4 in time.

iii In Chapter5, we study the convergence of densities of a sequence of random variables to a normal density. The random variables considered are nonlinear functionals of a Gaussian process, in particular, the multiple integrals. They are assumed to be nondegenerate so that their probability densities exist. The tool we use is the Malliavin calculus, in particular, the density formula, the integration by parts formula and the Stein’s method. Applications to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck is also considered. In Chapter6, we apply an upper bound estimate from small deviation theory to prove the non-degeneracy of some functional of fractional Brownian motion.

iv Acknowledgements

I owe my gratitude to all those people who have made this dissertation possible and because of whom my graduate experience has been one that I will cherish forever. My deepest gratitude is to my advisors Professor Yaozhong Hu and Professor David Nualart. They patiently taught me how to do research and help me develop a career plan. It is their guidance, caring and support that made my accomplishments possible. Also, I am deeply indebted to Yaozhong and his wife Jun Fu for taking care of me like parents. I am grateful to Professor Jin Feng, Professor Xuemin Tu and Professor Tyrone Dun- can for their help and suggestions in the past years. I would like to thank Professor Shigeru Iwata and Professor Ted Juhl for serving on my committee. I gratefully acknowledge Professor Mat Johnson, Professor Weishi Liu, Professor Al- bert Sheu, Professor Milena Stanislavova, Professor Atanas Stefanov, Professor Zsolt Ta- lata and Professor Rodolfo Torres for their teaching, valuable advice and support. Many thanks are also due to Professor Margaret Bayer and Professor Jack Porter for their su- pervision on my teaching. I am indebted to Professor Xuemin Tu for introducing me to the research area of stochastic computations and for recommending me a great job opportunity. This dissertation can not be made easily without the efforts of many staffs in the department of mathematics. Especially, I am thankful to Kerrie Brecheisen, Debbie Garcia, Gloria Prothe, Samantha Reinblatt and Lori Springs for their help in many ways.

v I would like to thank all my friends who provided so much help and encouragement. Specially, I would like to thank Jarod Hart, Daniel Harnett, Jingyu Huang, Pedro Lei, Xianping Li, Seungly Oh, Jian Song, Xiaoming Song and Mingji Zhang for numer- ous mathematical discussions and lectures on related topics that helped me improve my knowledge. Finally, I would like to thank my family, especially my wife Wenjun Ma, for their love, encouragement and support.

vi Contents

Abstract iii

Acknowledgementsv

1 Introduction1

2 Preliminaries8

2.1 Isonormal Gaussian process and multiple integrals...... 8 2.2 Malliavin operators...... 10 2.3 A Density Formula...... 13

3 Feynman-Kac Formula 15

3.1 Introduction...... 15 3.2 Preliminaries...... 19 3.3 Nonlinear stochastic integral...... 22 3.4 Feynman-Kac integral...... 35 3.5 Validation of the Feynman-Kac Formula...... 39 3.6 Skorohod type equation and Chaos expansion...... 46 3.7 Appendix...... 48

4 Joint continuity of the solution to a SPDE 53

4.1 Introduction...... 53

vii 4.2 Moment estimates...... 57 4.3 Holder¨ continuity in spatial variable...... 66 4.4 Holder¨ continuity in time variable...... 69

5 Convergence of densities 76

5.1 Introduction...... 76 5.2 Stein’s method of normal approximation...... 81 5.3 Density formulae...... 83 5.3.1 Density formulae...... 84 5.3.2 Derivatives of the density...... 88 5.4 Random variables in the qth Wiener chaos...... 93 5.4.1 Uniform estimates of densities...... 93 5.4.2 Uniform estimation of derivatives of densities...... 98 5.5 Random vectors in Wiener chaos...... 104 5.5.1 Main result...... 104 5.5.2 Sobolev norms of the inverse of the Malliavin matrix...... 114 5.5.3 Technical estimates...... 119 5.6 Uniform estimates for densities of general random variables...... 125 5.6.1 Convergence of densities with quantitative bounds...... 126 5.6.2 Compactness argument...... 129 5.7 Applications...... 131 5.7.1 Random variables in the second Wiener chaos...... 132 5.7.2 Parameter estimation in Ornstein-Uhlenbeck processes..... 136 5.8 Appendix...... 141

6 Non-degeneracy of random variables 152

6.1 Introduction...... 152

viii 6.2 An estimate on the path variance of fBm...... 155 6.3 Proof of the main theorem...... 157

Bibliography 165

ix Chapter 1

Introduction

In this dissertation, we study some applications of Malliavin calculus to stochastic partial differential equations (SPDEs) and to normal approximation. It is a collection of my joint works with my advisors, Yaozhong Hu and David Nualart. The Malliavin calculus, also known as the stochastic calculus of variations, is an inﬁnite-dimensional differential calculus on the Wiener space. One can distinguish two parts in the Malliavin calculus. First is the theory of differential operators deﬁned on suitable Sobolev spaces of Wiener functionals. A crucial fact in this theory is the integration by parts formula, which relates the derivative operator on the Wiener space and the Skorohod extended stochastic integral. The second part of this theory establishes general criteria, such as the density formulae, in terms of existence of negative moments of “Malliavin covariance matrix” for a given random vector to possess a density or, a smooth density. This gives a probabilistic proof and extensions to the Hormander’s¨ theorem about the existence and smoothness of the density for a solution of a stochastic differential equation. In the applications of Malliavin calculus to SPDE, we shall use the Malliavin differential operators to study existence and representation of solutions in Chapter3, and use

1 the general criteria for density to study the regularity of probability laws of solutions in Chapter4. Through the Gaussian integration by parts formula, Malliavin calculus can be combined with Stein’s method (a general method to obtain bounds on the distance between two probability distributions with respect to a probability metric) to study normal approximation theory. Their interaction has led to some remarkable new results involving central and non-central limit theorems for functionals of infinite dimensional Gaussian fields. While these central limit results study the convergence in distribution, we shall use the density formulae to study the convergences of densities of the random variables in Chapter5. As mentioned above, a crucial (and sufficient) condition for a random vector to possess a density is the existence of negative moments of its “Malliavin covariance matrix”. In Chapter6 we shall address this problem for some functionals of fractional Brownian motion. In the following we give a brief introduction to Chapter3–6. Chapter3 is taken from [13], in which we derive a Feynman–Kac formula for a SPDE driven by Gaussian noise which is, with respect to time, a fractional Brownian motion (fBm) with Hurst parameter H < 1/2. More precisely, we consider the stochastic heat

d equation (SHE) on R

  ∂u 1 ∂W d  ∂t = 2 ∆u + u ∂t (t,x), t ≥ 0, x ∈ R ,  u(0,x) = u0(x),

d where u0 is a bounded measurable function and W = {W(t,x), t ≥ 0,x ∈ R } is a fBm 1 1 of Hurst parameter H ∈ 4 , 2 in time and it has a spatial covariance Q(x,y), which is locally γ-Holder¨ continuous, with γ > 2 − 4H. We shall show that the solution to this

2 SHE is given by Z t B x x u(t,x) = E u0(Bt )exp W(ds,Bt−s) , 0

x d where B = {Bt = Bt + x,t ≥ 0,x ∈ R } is a d -dimensional Brownian motion starting at d x ∈ R , independent of W. In 1949, Kac [18] proved the famous Feynman-Kac formula representing the solution of a heat equation by the expectation of a stochastic process, and hence established a link between PDE and Probability. Since then, stochastic heat equation (a heat equation with random potentials) has been extensively studied. However, the case of random potentials deﬁned by means of fractional Brownian sheet (fBs) has been investigated

1 only recently by Hu, Nualart and Song [17], where the time Hurst index H > 2 , the spatial Hurst indices are relatively large and space-time derivative are considered. Due to singularity of the fractional Brownian sheet noise, a Feynman-Kac formula can not

1 be written if H ≤ 2 . In order to study the case of fractional noise potential with time 1 Hurst index H < 2 , we let the noise to be regular in space variables. Then we construct a solution to the Cauchy problem via a generalized Feynman-Kac formula. The difﬁculty lies in the lack of fractional smoothness of the fBm in time. To overcome the difﬁculty, we compensate it with regularity in space by introducing a nonlinear stochastic integral

R t 1 0 W(ds,φ(s)) for all φ which is Holder¨ continuous of order α > γ (1 − 2H). The main tool in the proof is fractional calculus. We then show the exponential integrability of the nonlinear stochastic integral. The study of nonlinear stochastic integral is of independent interest and might be used in other related problems. The second effort is to show that the Feynman–Kac formula provides a solution to the SPDE. The approach of approximation with techniques from Malliavin calculus is used. The solution is in the weak sense, and the integral with respect to the noise is in the Stratonovich sense. The Feynman-Kac expression allows us a rather complete analysis

3 of the statistical properties of the solution. We prove that the solution is Lp integrable for

1 1 all p ≥ 1. We also prove that the solution is Holder¨ continuous of order (H − 2 + 4 γ) a.s. if u0 is Lipschitz and bounded. Chapter4 is devoted to study the joint H older¨ continuity of the solution to the following nonlinear SPDE arising from the study of one dimensional superprocesses:

Z t Z t Z Xt(x) = µ(x) + ∆Xu(x)dr − ∇x (h(y − x)Xu (x))W (du,dy) 0 0 R Z t p V (du,dx) + Xu (x) , 0 dx where V and W are two independent Brownian sheets on R+ × R. Dawson, Vaillancourt and Wang [7]) proved that the solution Xt(x) gives the density of the branching particles in the random environment W. The time-space joint continuity of Xt(x) was left as an open problem. Recently, Li et al [25] proved that this solution is almost surely joint

1 1 Holder¨ continuous with exponent up to 10 in time and up to 2 in space. The methods they used are fractional integration by parts technique and Krylov’s Lp theory (cf. Krylov [20]). Comparing to the Holder¨ continuity for the stochastic heat equation which has the Holder¨ continuity of 1/4 in time, it is conjectured that the optimal exponent of Holder¨ continuity of Xt(x) should also be 1/4. By introducing the techniques from Malliavin calculus, we shall give an afﬁrmative answer to this conjecture. We also give a short proof of the Holder¨ continuity with ex-

1 ponent 2 in space using our method. We anticipate that our method would be potentially useful for the study of other interactive branching diffusions. This chapter is taken from [12]. In Chapter5, we study the convergence of densities of a sequence of random variables to a normal density. The motivation is the following. The central limit theorem (CLT)

4 gives the convergence of the distribution functions of a sequence of random variables. A natural question to ask is, what can we say about the probability densities if they exist? Inspired by the following recent developments of CLT for multiple integrals (see the deﬁnition of the multiple integral in Section 2.1), we shall give a partial answer to this question.

Consider a sequence of random variables Fn = Iq( fn) ( in the q–th Wiener chaos with

2 q ≥ 2 and EFn = 1. Nualart and Peccati [41] and Nualart and Ortiz-Latorre [40] have proved that Fn converges in distribution to the normal law N(0,1) as n → ∞ if and only if one of the following three equivalent conditions holds:

4 (i) limn→∞ E[Fn ] = 3,

(ii) For all 1 ≤ r ≤ q − 1, limn→∞ k fn ⊗r fnkH⊗2(q−r) = 0,

2 2 (iii) kDFnkH → q in L (Ω) as n → ∞. where DFn is the Malliavin derivative of Fn (see for deﬁnition in the following sections). The ﬁrst two conditions were proved in [41] using the method of moments and comu- lants, which are called “the fourth moment theorem”, and (iii) was proved in [40] using characteristic function. Later, two new proofs of the fourth moment theorem were given by Nourdin and Peccati [34] by bringing together Stein’s method and Malliavin calculus and by Nourdin [32] using free Brownian motion. Multidimensional versions of this characterization were studied by Peccati and Tudor [47] and Nualart and Ortiz-Latorre [40]. Different extensions and applications of these results can be found in the extensive literature, among them we mention Hu and Nualart [14] and Peccati and Taqqu [45, 46]. Subsequently, quantitative bounds to the fourth moment theorem is provided by Nourdin and Peccati [34] (see also Nourdin and Peccati [35]). Bringing together Stein’s

5 method with Malliavin calculus, they proved a bound for the total variation distance,

q 4 dTV (Fn,N) := sup |P(Fn ∈ B) − P(N ∈ B)| ≤ 2 |EFn − 3|. B∈B(R)

The aim of this chapter is to study the convergence of the densities of Fn = Iq( fn).

It is well-known in Malliavin calculus that each Fn has a density fFn if its Malliavin derivative kDFnkH has negative moments (we say a random vaiable is non-degenerate if its Malliavin derivative has negative moments). Assuming further that

−6 M := supE[kDFnkH ] < ∞, n

we shall prove that fFn ∈ C(R) for each n and there exists a constant Cq,M depending only q and M such that q 4 sup| fFn (x) − φ(x)| ≤ Cq,M |EFn − 3|, x∈R where we denote φ(x) the density of N(0,1). Assuming higher order of negative moments, we also show the uniform convergence of the derivatives of the densities. The convergence of densities for random vectors has also been studied. Applications to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Ulenbeck is also considered. The main ingredients in proof are the density formulae in Malliavin calculus and the combination of Stein’s method and Malliavin calculus. In the application of multivariate Stein’s method, we gave estimates on the solution to the multivariate Stein’s equation with non-smooth unbounded test functions. Our result extends the relatively few current results for non-smooth test functions (see e.g. [4, 50]) for multivariate Stein’s method. As we have seen in the above chapters, non-degeneracy of a random variable plays a fondamental role when we apply density formula. Chapter6 is a short note on non-

6 degeneracy of the following functional of fractional Brownian motion BH:

Z 1 Z 1 H H 2p Bt − Bt0 0 F = 0 q dtdt , 0 0 |t −t | where q ≥ 0 and the integer p satisﬁes (2p − 2)H > q − 1.

1 1 2 In the case of H = 2 , B is a Brownian motion, and the random variable F is the Sobolev norm on the Wiener space considered by Airault and Malliavin in [1]. This norm plays a central role in the construction of surface measures on the Wiener space. Fang [9] showed that F is non-degenerate. Then it follows from the well-known density formula in Malliavin calculus that the law of F has a smooth density. In Chapter6, using an upper bound estimates in small deviation theory, we shall show that F is non-degenerate for all H ∈ (0,1).

7 Chapter 2

Preliminaries

We introduce some basic elements of Gaussian analysis and Malliavin calculus, for which we refer to [39, 35] for further details.

2.1 Isonormal Gaussian process and multiple integrals

Let H be a real separable Hilbert space (with its inner product and norm denoted by

⊗q q h·,·iH and k·kH, respectively). For any integer q ≥ 1, let H (H ) be the qth tensor product (symmetric tensor product) of H. Let X = {X(h),h ∈ H} be an isonormal Gaussian process associated with the Hilbert space H, deﬁned on a complete probability space (Ω,F ,P). That is, X is a centered Gaussian family of random variables such that

E[X(h)X(g)] = hh,giH for all h,g ∈ H.

For every integer q ≥ 0, the qth Wiener chaos (denoted by Hq) of X is the closed lin- 2 ear subspace of L (Ω) generated by the random variables Hq(X(h)) : h ∈ H,khkH = 1 , where Hq is the qth Hermite polynomial recursively deﬁned by H0(x) = 1, H1(x) = x and

Hq+1(x) = xHq(x) − qHq−1(x), q ≥ 1. (2.1)

8 ⊗q For every integer q ≥ 1, the mapping Iq(h ) = Hq(X(h)) , where khkH = 1, can be q √ extended to a linear isometry between H (equipped with norm q!k·kH⊗q ) and Hq 2 (equipped with L (Ω) norm). For q = 0, H0 = R, and I0 is the identity map. It is well-known (Wiener chaos expansion) that L2(Ω) can be decomposed into the

2 inﬁnite orthogonal sum of the spaces Hq. That is, any random variable F ∈ L (Ω) has the following chaotic expansion:

∞ F = ∑ Iq( fq), (2.2) q=0

q where f0 = E[F], and fq ∈ H ,q ≥ 1, are uniquely determined by F. For every q ≥ 0 we denote by Jq the orthogonal projection on the qth Wiener chaos Hq, so Iq( fq) = JqF. q p Let {en,n ≥ 1} be a complete orthonormal basis of H. Given f ∈ H and g ∈ H , for r = 0,..., p ∧ q the r–th contraction of f and g is the element of H⊗(p+q−2r) deﬁned by ∞

f ⊗r g = ∑ h f ,ei1 ⊗ ··· ⊗ eir iH⊗r ⊗ hg,ei1 ⊗ ··· ⊗ eir iH⊗r . (2.3) i1,...,ir=1

Notice that f ⊗r g is not necessarily symmetric. We denote by f ⊗e rg its symmetrization.

Moreover, f ⊗0 g = f ⊗ g, and for p = q, f ⊗q g = h f ,giH⊗q . For the product of two multiple integrals we have the multiplication formula

p∧q pq Ip( f )Iq(g) = ∑ r! Ip+q−2r( f ⊗r g). (2.4) r=0 r r

In the particular case H = L2(A,A , µ), where (A,A ) is a measurable space and µ is a σ–ﬁnite and nonatomic measure, one has that H⊗q = L2(Aq,A ⊗q, µ⊗q) and H q is the space of symmetric and square-integrable functions on Aq. Moreover, for every

q f ∈ H , Iq( f ) coincides with the qth multiple Wiener–Itoˆ integral of f with respect to

9 X, and (2.3) can be written as

Z f ⊗r g t1,...,tp+q−2r = f t1,...,tq−r,s1,...,sr (2.5) Ar ×g t1+q−r,...,tp+q−r,s1,...,sr dµ(s1)...dµ(sr).

2.2 Malliavin operators

We introduce some basic facts on Malliavin calculus with respect to the Gaussian process

X. Let S denote the class of smooth random variables of the form F = f (X(h1),...,X(hn)), ∞ n where h1,...,hn are in H, n ≥ 1, and f ∈ Cp (R ), the set of smooth functions f such that f itself and all its partial derivatives have at most polynomial growth. Given F = f (X(h1),...,X(hn)) in S , its Malliavin derivative DF is the H-valued random variable given by n ∂ f DF = ∑ (X(h1),...,X(hn))hi. i=1 ∂xi

The derivative operator D is a closable and unbounded operator on L2(Ω) taking values in L2(Ω;H). By iteration one can deﬁne higher order derivatives DkF ∈ L2(Ω;H k). For

k,p any integer k ≥ 0 and any p ≥ 1 and we denote by D the closure of S with respect to the norm k·kk,p given by:

k p i p kFkk,p = ∑ E( D F H⊗i ). i=0

∞,p For k = 0 we simply write kFk0,p = kFkp. For any p ≥ 1 and k ≥ 0, we set D = k,p k,∞ k,p ∩k≥0D and D = ∩p≥1D . We denote by δ (the divergence operator) the adjoint operator of D, which is an unbounded operator from a domain in L2(Ω;H) to L2(Ω). An element u ∈ L2(Ω;H)

10 belongs to the domain of δ if and only if it veriﬁes

q 2 E[hDF,uiH] ≤ cu E[F ]

1,2 for any F ∈ D , where cu is a constant depending only on u. In particular, if u ∈ Dom δ, then δ(u) is characterized by the following duality relationship

E(δ(u)F) = E(hDF,uiH) (2.6)

, q q for any F ∈ D1 2. This formula extends to the multiple integral δ , that is, for u ∈ Dom δ q, and F ∈ D 2 we have q q E(δ (u)F) = E(hD F,uiH⊗q ).

We can factor out a scalar random variable in the divergence in the following sense.

, Let F ∈ D1 2 and u ∈ Dom δ such that Fu ∈ L2(Ω;H). Then Fu ∈ Dom δ and

δ (Fu) = Fδ(u) − hDF,uiH , (2.7) provided the right hand side is square integrable. The operators δ and D have the following commutation relationship

Dδ(u) = u + δ(Du) (2.8)

, for any u ∈ D2 2(H) (see [39, page 37]). The following version of Meyer’s inequality (see [39, Proposition 1.5.7]) will be used frequently in this paper. Let V be a real separable Hilbert space. We can also introduce

k,p Sobolev spaces D (V) of V-valued random variables for p ≥ 1 and integer k ≥ 1. Then, k,p k− ,p for any p > 1 and k ≥ 1, the operator δ is continuous from D (V ⊗ H) into D 1 (V).

11 That is,

kδ(u)kk−1,p ≤ Ck,p kukk,p . (2.9)

∞ The operator L deﬁned on the Wiener chaos expansion as L = ∑q=0(−q)Jq is the inﬁnitesimal generator of the Ornstein–Uhlenbeck semigroup. Its domain in L2(Ω) is

( ∞ ) 2 2 2 2,2 Dom L = F ∈ L (Ω) : ∑ q JqF 2 < ∞ = D . q=1

The relation between the operators D, δ and L is explained in the following formula (see [39, Proposition 1.4.3]). For F ∈ L2(Ω), F ∈ Dom L if and only if F ∈ Dom(δD) (i.e.,

, F ∈ D1 2 and DF ∈ Dom δ), and in this case

δDF = −LF. (2.10)

2 −1 ∞ −1 −1 For any F ∈ L (Ω), we deﬁne L F = −∑q=1 q Jq(F). The operator L is called the pseudo-inverse of L. Indeed, for any F ∈ L2(Ω), we have that L−1F ∈ Dom L, and

LL−1F = F − E[F].

We list here some properties of multiple integrals which will be used in Section 4.

q q Fix q ≥ 1 and let f ∈ H . We have Iq( f ) = δ ( f ) and DIq( f ) = qIq−1( f ), and hence ∞,2 Iq( f ) ∈ D . The multiple integral Iq( f ) satisﬁes hypercontractivity property:

Iq( f ) p ≤ Cq,p Iq( f ) 2 for any p ∈ [2,∞). (2.11)

∞ This easily implies that Iq( f ) ∈ D and for any 1 ≤ k ≤ q and p ≥ 2,

kIq( f )kk,p ≤ Cq,k,p Iq( f ) 2 .

12 q As a consequence, for any F ∈ ⊕l=1Hl, we have

kFkk,p ≤ Cq,k,p kFk2 . (2.12)

For any random variable F in the chaos of order q ≥ 2, we have (see [35], Equation (5.2.7))

1 q − 1 VarkDFk2 ≤ E[F4] − (E[F2])2 ≤ (q − 1)VarkDFk2 . (2.13) q2 H 3q H

2 ∞ In the case where H is L (A,A , µ), for a integrable random variable F = ∑q=0 Iq( fq) ∈ , D1 2, its derivative can be represented as an element in of L2 (A × Ω) given by

∞ DtF = ∑ qIq( fq(·,t)). q=1

2.3 A Density Formula

The following lemma gives an explicit expression of the probability density of a random variable.

1,2 DF Lemma 2.1. Let F be a random variable in the space D , and suppose that 2 kDFkH belongs to the domain of the operator δ in L2 (Ω). Then the law of F has a continuous and bounded density given by

" !# DF p(x) = E 1{F>x}δ 2 . kDFkH

From Eδ(u) = 0 for any u ∈ Dom(δ) and the Holder¨ inequality it follows that

13 ,q Lemma 2.2. Let F be a random variable and let u ∈ D1 (H) with q > 1. Then for the 1 1 conjugate pair p and q (i.e. p + q = 1),

1 p E 1{F>x}δ (u) ≤ (P(|F| > |x|)) kδ (u)kLq(Ω) . (2.14)

14 Chapter 3

Feynman-Kac Formula for the Heat Equation Driven by fractional Noise with Hurst parameter H < 1/2

In this chapter, a Feynman-Kac formula is established for a stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. The existence of the Feynman-Kac integral then follows from the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with techniques from Malli- avin calculus is used to show that the Feynman-Kac integral is the weak solution to the stochastic partial differential equation.

3.1 Introduction

d Consider the stochastic heat equation on R

  ∂u 1 ∂W d  ∂t = 2 ∆u + u ∂t (t,x), t ≥ 0, x ∈ R , (3.1)  u(0,x) = u0(x),

15 d where u0 is a bounded measurable function and W = {W(t,x), t ≥ 0,x ∈ R } is a frac- 1 1 tional Brownian motion of Hurst parameter H ∈ 4 , 2 in time and it has a spatial covariance Q(x,y), which is locally γ-Holder¨ continuous (see Section 3.2 for precise meaning of this condition), with γ > 2 − 4H. We shall show that the solution to (3.1) is given by

Z t B x x u(t,x) = E u0(Bt )exp W(ds,Bt−s) , (3.2) 0

x d where B = {Bt = Bt + x,t ≥ 0,x ∈ R } is a d–dimensional Brownian motion starting at d x ∈ R , independent of W. This is a generalization of the well-known Feyman–Kac formula to the case of a ran-

∂W R t x dom potential of the form ∂t (t,x). Notice that the integral 0 W(ds,Bt−s) is a nonlinear stochastic integral with respect to the fractional noise W. This type of Feynman-Kac formula was mentioned as a conjecture by Mocioalca and Viens in [29]. There exists an extensive literature devoted to Feynman-Kac formulae for stochastic partial differential equations. Different versions of the Feynman-Kac formula have been established for a variety of random potentials. See, for instance, a Feynman-Kac formula for anticipating SPDE proved by Ocone and Pardoux [43]. Ouerdiane and Silva [44] give a generalized Feynman-Kac formula with a convolution potential by introducing a generalized function space. Feynman-Kac formulae for Levy´ processes are presented by Nualart and Schoutens [42]. However, only recently a Feynman-Kac formula has been established by Hu et al. [17] for random potentials associated with the fractional Brownian motion. The authors consider the following stochastic heat equation driven by fractional noise

  ∂u 1 ∂ d+1W d  ∂t = 2 ∆u + u ∂t∂x ···∂x (t,x), t ≥ 0, x ∈ R , 1 d (3.3)  u(0,x) = u0(x),

16 d where W = {W(t,x), t ≥ 0,x ∈ R } is fractional Brownian sheet with Hurst parameter 1 (H0,H1,...,Hd). They show ([17], Theorem 4.3) that if H1,...,Hd ∈ ( 2 ,1), and 2H0 +

H1 + ··· + Hd > d + 1, then the solution u(t,x) to the above stochastic heat equation is given by Z t Z B x x u(t,x) = E f (Bt )exp δ(Bt−r − y)W(dr,dy) , (3.4) 0 Rd

x d where B = {Bt = Bt + x,t ≥ 0,x ∈ R } is a d–dimensional Brownian motion starting at d x ∈ R , independent of W. The condition 2H0 + H1 + ··· + Hd > d + 1 is shown to be sharp in that framework. Since the Hi, i = 1,...,d cannot take value greater or equal to 1 1, this condition implies that H0 > 2 .

H0 H0 We remark that if B = {Bt ,t ≥ 0} is a fractional Brownian motion with Hurst

1 R T H0 parameter H0 > 2 , then the stochastic integral 0 f (t)dBt is well deﬁned for a suitable R t R x class of distributions f , and in this sense the above integral d (B − y)W(dr,dy) 0 R δ t−r 1 is well deﬁned for any trajectory of the Brownian motion B. If H0 < 2 , this is no longer true and we can integrate only functions satisfying some regularity conditions. For this reason, it is not possible to write a Feynman-Kac formula for the Equation (3.3) with

1 H0 < 2 . 1 Notice that for d = 1 and H0 = H1 = 2 (space-time white noise) a Feynman-Kac formula can not be written for Equation (3.3), but this equation has a unique mild solution when the stochastic integral is interpreted in the Itoˆ sense. A renormalized Feynman-Kac formula with Wick exponential has been obtained in this case by Bertinin and Cancrini [2]. More generally, if the product appearing in (3.3) is replaced by Wick product, Hu and Nualart [15] showed that a formal solution can be obtained using chaos expansions.

1 In this chapter, we are concerned with the case H0 < 2 , but we use a random potential ∂W of the form ∂t (t,x). One of the main obstacles to overcome is to deﬁne the stochastic R t x integral 0 W(ds,Bt−s). We start with the construction of a general nonlinear stochastic

17 R t 1 integral 0 W(ds,φs) where φ is a Holder¨ continuous function of order α > γ (1 − 2H). It turns out that the irregularity in time of W(t,x) is compensated by the above Holder¨ continuity of φ through the covariance in space, with an appropriate application of the R t fractional integration by parts technique. Let us point out that 0 W(ds,φs) is well-deﬁned 1 1 for all Holder¨ continuous function φ with α > γ ( 2 − H), and we consider here only 1 the case α > γ (1 − 2H) because this condition is required when we show that u(t,x) 1 is a weak solution to (3.1). Furthermore, the condition α > γ (1 − 2H) also allows us R t to obtain an explicit formula for the variance of 0 W(ds,φs). Contrary to [17], it is R t x rather simpler to show that 0 W(ds,Bt−s) is exponentially integrable. A by-product is that u(t,x) deﬁned by (3.2) is almost surely Holder¨ continuous of order which can be

1 γ arbitrarily close to H − 2 + 4 from below. Let us also mention recent work on stochastic integral [11] and [19] with general Gaussian processes which can be applied to the case

1 H < 2 . Another main effort is to show that u(t,x) deﬁned by (3.2) is a solution to (3.1) in a weak sense (see Deﬁnition 3.12). As in [17], this is done by using an approximation scheme together with techniques of Malliavin calculus. Let us point out that in the def- R t inition of 0 W(ds,φs) one can use a one-side approximation, but it is necessary to use 1 γ symmetric approximations (as well as the condition H > 2 − 4 ) to show the convergence of the trace term (3.59). We also discuss the corresponding Skorohod-type equation, which corresponds to taking the Wick product in [15]. We show that a unique mild solution exists for H ∈

1 γ 1 ( 2 − 4 , 2 ).

This chapter is organized as follows. Section 3.2 contains some preliminaries on the fractional noise W and some results on fractional calculus which is needed in the paper. We also list all the assumptions that we make for the noise W in this section.

18 In Section 3.3, we study the nonlinear stochastic integral appeared in Equation (3.2) by using smooth approximation and we derive some basic properties of this integral. Section 3.4 veriﬁes the integrability and Holder¨ continuity of u(t,x). Section 3.5 is devoted to show that u(t,x) is a solution to (3.1) in a weak sense. Section 3.6 gives a solution to the Skorohod type equation. The last section is the Appendix with some technical results.

3.2 Preliminaries

1 1 2H 2H 2H Fix H ∈ (0, 2 ) and denote by RH(t,s) = 2 t + s − |t − s| the covariance function of the fractional Brownian motion of Hurst parameter H. Suppose that W = {W(t,x),t ≥

d 0,x ∈ R } is a mean zero Gaussian random ﬁeld, deﬁned on a probability space (Ω,F ,P), whose covariance function is given by

E(W(t,x)W(s,y)) = RH(t,s)Q(x,y), where Q(x,y) satisﬁes the following properties for some M < 2 and γ ∈ (0,1]:

(Q1) Q is locally bounded: there exists a constant C0 > 0 such that for any K > 0

M Q(x,y) ≤ C0 (1 + K)

d for any x,y ∈ R such that |x|,|y| ≤ K.

(Q2) Q is locally γ-Holder¨ continuous: there exists a constant C1 > 0 such that for any K > 0 M γ γ |Q(x,y) − Q(u,v)| ≤ C1 (1 + K) |x − u| + |y − v| ,

d for any x,y,u,v ∈ R such that |x|,|y|, |u|, |v| ≤ K.

19 Denote by E the vector space of all step functions on [0,T]. On this vector space E we introduce the following scalar product

h1[0,t],1[0,s]iH0 = RH(t,s).

α Let H0 be the closure of E with respect to the above scalar product. Denote by C ([a,b]) the set of all functions which is Holder¨ continuous of order α, and denote by ||·||α the α 1 α-Holder¨ norm. It is well known that C ([0,T]) ⊂ H0 for α > 2 − H. Let H be the Hilbert space deﬁned by the completion of the linear span of indicator

d functions 1[0,t]×[0,x], t ∈ [0,T], x ∈ R under the scalar product

h1[0,t]×[0,x],1[0,s]×[0,y]iH = RH(t,s)Q(x,y).

In the above formula, if xi < 0 we assume by convention that 1[0,xi] = −1[−xi,0]. The mapping W : 1[0,t]×[0,x] → W(t,x) can be extended to a linear isometry between H and the Gaussian space spanned by W. Then, {W(h), h ∈ H} is an isonormal Gaussian process.

We recall the following notations of Malliavin caluclus. Let S be the space of ran-

∞ n dom variables F of the form: F = f (W(ϕ1),...,W(ϕn)), where ϕi ∈ H, f ∈ C (R ), f and all its partial derivatives have polynomial growth. The Malliavin derivative DF of

F ∈ S is an H–valued random variable given by

n ∂ f DF = ∑ (W(ϕ1),...,W(ϕn))ϕi. i=1 ∂xi

20 The divergence operator δ is the adjoint of the derivative operator D, determined by the duality relationship

1,2 E(δ(u)F) = E(hDF,uiH), for any F ∈ D .

, δ(u) is also called the Skorohod integral of u. For any random variable F ∈ D1 2 and φ ∈ H,

FW (φ) = δ(Fφ) + hDF,φiH . (3.5)

Since we deal with the case of Hurst parameter H ∈ (0,1/2), we shall use intensively the fractional calculus. We recall some basic deﬁnitions and properties. For a detailed account, we refer to [51].

Let a,b ∈ R, a < b. Let f ∈ L1 (a,b) and α > 0. The left and right-sided fractional integral of f of order α are deﬁned for x ∈ (a,b), respectively, as

Z x α 1 α−1 Ia+ f (x) = (x − y) f (y)dy Γ(α) a and Z b α 1 α−1 Ib− f (x) = (y − x) f (y)dy. Γ(α) x

α p α p p α α Let Ia+ (L )(resp. Ib− (L )) the image of L (a,b) by the operator Ia+(resp. Ib−). α p α p If f ∈ Ia+ (L )(resp. Ib− (L )) and 0 < α < 1 then the left and right-sided fractional derivatives are deﬁned by

Z x ! α 1 f (x) f (x) − f (y) Da+ f (x) = α + α +1 dy , (3.6) Γ(1 − α) (x − a) a (x − y)α and α Z b ! α (−1) f (x) f (x) − f (y) Db− f (x) = α + α +1 dy (3.7) Γ(1 − α) (b − x) x (y − x)α

21 for all x ∈ (a,b) (the convergence of the integrals at the singularity y = x holds point-wise for almost all x ∈ (a,b) if p = 1 and moreover in Lp-sense if 1 < p < ∞).

1 1 It is easy to check that if f ∈ Ia+(b−) L ,

α 1−α α 1−α Da+Da+ f = D f , Db−Db− f = D f (3.8) and Z b Z b α α α (−1) Da+ f (x)g(x)dx = f (x)Db−g(x)dx (3.9) a a

α p α q 1 1 provided that 0 ≤ α ≤ 1, f ∈ Ia+ (L ) and g ∈ Ib− (L ) with p ≥ 1,q ≥ 1, p + q ≤ 1+α. It is clear that Dα f exists for all f ∈ Cβ ([a,b]) if α < β. The following proposition was proved in [55].

Proposition 3.1. Suppose that f ∈ Cλ ([a,b]) and g ∈ Cµ ([a,b]) with λ + µ > 1. Let R b λ > α and µ > 1 − α. Then the Riemann Stieltjes integral a f dg exists and it can be expressed as Z b Z b α α 1−α f dg = (−1) Da+ f (t)Db− gb− (t)dt, (3.10) a a where gb− (t) = g(t) − g(b).

3.3 Nonlinear stochastic integral

In this section, we introduce the nonlinear stochastic integral that appears in the Feynman- Kac formula (3.2) and obtain some properties of this integral which are useful in the following sections. The main idea to deﬁne this integral is to use an appropriate approximation scheme. In order to introduce our approximation, we need to extend the fractional

d Brownian ﬁeld to t < 0. This can be done by deﬁning W = {W(t,x),t ∈ R,x ∈ R } as a

22 mean zero Gaussian process with the following covariance

1 E [W (t,x)W (s,y)] = |t|2H + |s|2H − |t − s|2H Q(x,y) . 2

For any ε > 0, we introduce the following approximation of W(t,x):

Z t W ε (t,x) = W˙ ε (s,x)ds, (3.11) 0

˙ ε 1 where W (s,x) = 2ε (W(s + ε,x) −W(s − ε,x)).

Deﬁnition 3.2. Given a continuous function φ on [0,T], deﬁne

Z t Z t ε W(ds,φs) = lim W˙ (s,φs)ds, 0 ε→0 0 if the limit exits in L2(Ω).

Now we want to ﬁnd conditions on φ such that the above limit exists in L2(Ω). To R t ˙ ε this end, we set Iε (φ) = 0 W (s,φs)ds and compute E (Iε (φ)Iδ (φ)) for ε,δ > 0. Denote

1 V 2H (r) = |r + ε − δ|2H − |r + ε + δ|2H − |r − ε − δ|2H + |r − ε + δ|2H . ε,δ 4εδ

Using the fact that Q(x,y) = Q(y,x), we have

2H Making the substitution r = θ − η and using the notation Vε,δ , we can write

Z t Z θ 2H E (Iε (φ)Iδ (φ)) = Q(φθ ,φθ−r)Vε,δ (r)drdθ. (3.12) 0 0

23 We need the following two technical lemmas.

Lemma 3.3. For any bounded function ψ : [0,T] → R, we have

Z t Z s Z t 2H 2H−1 2H ψ (s) Vε,δ (r)drds − 2H ψ (s)s ds ≤ 4||ψ||∞ (ε + δ) . (3.13) 0 0 0

R s 2H R t R s 2H 00 Proof. Let g(s) := 0 |r| dr and fε,δ (t) := 0 ψ (s) 0 Vε,δ (r)drds. Note that g exists everywhere except at 0 and g00 (r) = 2Hsign(r)|r|2H−1 for r 6= 0. Then,

1 Z t fε,δ (t) = ψ (s)[g(s + ε − δ) − g(s + ε + δ) − g(s − ε − δ) + g(s − ε + δ)]ds 4εδ 0 1 Z 1 Z 1 Z t = ψ (s)g00 (s + ηε − ξδ)dsdξdη 4 −1 −1 0 1 Z 1 Z 1 Z t = ψ (s)g00 (s − ∆)dsdξdη, 4 −1 −1 0 where ∆ = ξδ − ηε. Case i): If ∆ ≤ 0, we have

Z t 00 2H−1 ψ (s) g (s − ∆) − 2Hs ds 0 Z t 2H−1 2H−1 ≤ 2H ||ψ||∞ s − (s − ∆) ds (3.14) 0 2H 2H 2H 2H = ||ψ||∞ t − (t − ∆) + (−∆) ≤ 2||ψ||∞ |∆| .

Case ii): If ∆ > 0, we assume that ∆ < t (the case ∆ ≥ t follows easily). Then

Z t Z ∆ Z t ψ (s)g00 (s − ∆)ds = −2H ψ (s)(∆ − s)2H−1 ds + 2H ψ (s)(s − ∆)2H−1 ds. 0 0 ∆

Therefore, Z t 00 2H−1 1 2 ψ (s) g (s − ∆) − 2Hs ds ≤ F∆ + F∆ , (3.15) 0

24 where Z ∆ 1 h 2H−1 2H−1i 2H F∆ := 2H ψ (s) (∆ − s) + s ds ≤ 2kψk∞|∆| (3.16) 0 and

Z t 2 h 2H−1 2H−1i F∆ := 2H ψ (s) (s − ∆) − s ds ∆ Z t h 2H−1 2H−1i 2H ≤ 2Hkψk∞ (s − ∆) − s ds ≤ 2kψk∞|∆| . (3.17) ∆

Then (3.13) follows from (3.14)–(3.17).

Lemma 3.4. Let ψ ∈ C([0,T]2) with ψ (0,s) = 0, and ψ (·,s) ∈ Cα ([0,T]) for any s ∈

[0,T]. Assume α + 2H > 1 and sups∈[0,T] ||ψ (·,s)||α < ∞. Then for any 1 − 2H < γ < α and t ≤ T we have

Z t Z s h 2H 2H−2i ψ (r,s) Vε,δ (r) − 2H (2H − 1)r drds 0 0 2H+γ−1 ≤ C sup ||ψ (·,s)||α (ε + δ) , (3.18) s∈[0,T] where the constant C depends on H, γ, α and T, but it is independent of δ,ε and ψ.

Proof. Along the proof, we denote by C a generic constant which depends on H, γ, α and T. Set h(r) := |r|2H. Then h0(r) exists everywhere except at 0 and h0 (r) = 2Hsign(r)|r|2H−1 if r 6= 0. Using (3.8) and (3.10) we have

Z t Z s 2H fε,δ (t) := ψ (r,s)Vεδ (r)drds 0 0 1 Z 1 Z t Z s ∂ = ψ (r,s) [h(r + ε − ξδ) − h(r − ε − ξδ)]drdsdξ 4ε −1 0 0 ∂r Z 1 Z t Z s α0 1 α0 1−α0 = (−1) D0+ψ (r,s)Ds− [h(r + ε − ξδ) − h(r − ε − ξδ)]drdsdξ 4ε −1 0 0 Z 1 Z 1 Z t Z s α0 1 α0 1−α0 0 = (−1) D0+ψ (r,s)Ds− h (r + ηε − ξδ)drdsdξdη, 4 −1 −1 0 0

25 where γ < α0 < α. On the other hand, we also have

Z t Z s Z t Z s 2H−2 α0 α0 1−α0 0 2H(2H − 1) ψ(r,s)r dr = (−1) D0+ψ (r,s)Ds− h (r)drds. 0 0 0 0

Thus,

Z t Z s h 2H 2H−2i Iε,δ := ψ (r,s) Vε,δ (r) − 2H (2H − 1)r drds 0 0 Z 1 Z 1 Z t Z s 1 α0 1−α0 0 1−α0 0 ≤ D0+ψ (r,s) Ds− h (r + ηε − ξδ) − Ds− h (r) drdsdξdη. 4 −1 −1 0 0

Denote ∆ = ξδ − ηε and

Z t Z s α0 h 1−α0 0 1−α0 0 i f∆ (t) := D0+ψ (r,s) Ds− h (r − ∆) − Ds− h (r) drds. (3.19) 0 0

Then we may write 1 Z 1 Z 1 Iε,δ ≤ f∆(t)dξdη. (3.20) 4 −1 −1

Hence, in order to prove (3.18) it sufﬁces to prove

2H+γ−1 f∆ (t) ≤ C sup ||ψ (·,s)||α |∆| . (3.21) s∈[0,T]

By (3.6), we have

r 0 1 ψ (r,s) Z ψ (r,s) − ψ (u,s) Dα ψ (r,s) = + α0 du 0+ 0 α0 α0+1 Γ(1 − α ) r 0 (r − u)

≤ C sup ||ψ (·,s)||α . (3.22) s∈[0,T]

Therefore, 1 2 f∆ (t) ≤ C sup ||ψ (·,s)||α F∆ + F∆ , (3.23) s∈[0,T]

26 where

Z t Z s |h0 (r − ∆) − h0 (r)| F1 = drds, ∆ − 0 0 0 (s − r)1 α Z t Z s Z s |h0 (r − ∆) − h0 (u − ∆) − h0(r) + h0(u)| F2 = dudrds. ∆ − 0 0 0 r (u − r)2 α

As in the proof of Lemma 3.3, we consider the two cases separately: ∆ ≤ 0 and ∆ > 0.

Case i): If ∆ ≤ 0, we can write

0 0 h (r − ∆) − h (r) 0 Z 1 ≤ C(s − r)α −1|∆| (r − ξ∆)2H−2dξ 1−α0 (s − r) 0 0 ≤ C(s − r)α −1r−γ |∆|2H+γ−1 , which implies

1 2H+γ−1 F∆ ≤ C|∆| . (3.24)

For 0 < r < u, we have

0 0 0 0 h (r − ∆) − h (u − ∆) − h (r) + h (u) Z 1 Z 1 = C|∆| (r − ξ∆ + θ (u − r))2H−3 dθdξ (r − u) 0 0 ≤ Cr2H−1−β1−β2 (u − r)β1 |∆|β2

0 for any β1,β2 > 0 such that β1 + β2 < 2H. If α + β1 > 1, we obtain

Z s |h0(r − ∆) − h0(u − ∆) − h0(r) + h(u)| 0 du r (u − r)2−α 0 ≤ Cr2H−1−β1−β2 (s − r)α +β1−1|∆|β2 ,

27 which implies, taking β2 = 2H + γ − 1,

2 2H+γ−1 F∆ ≤ C|∆| . (3.25)

Substituting (3.24) and (3.25) into (3.23), we get (3.21).

Case ii): Now let ∆ > 0. We assume that ∆ < t (the case t ≤ ∆ is simpler and omitted).

1 Let us ﬁrst consider the term F∆ . Deﬁne the sets

D11 = {0 < r < s < ∆}, D12 = {0 < r < ∆ < s < t}, D13 = {∆ < r < s < t}.

Then

1 11 12 13 F∆ = F∆ + F∆ + F∆ , where Z |h0 (r − ∆) − h0 (r)| F1i = drds, i = 1,2,3. ∆ 1−α0 D1i (s − r)

It is easy to see that

Z ∆ Z s 11 h 2H−1 2H−1i α0−1 2H+α0 F∆ ≤ C (∆ − r) + r (s − r) drds ≤ C∆ (3.26) 0 0 and Z t Z ∆ 12 h 2H−1 2H−1i α0−1 2H F∆ ≤ C (∆ − r) + r (s − r) drds ≤ C∆ . (3.27) ∆ 0

13 As for F∆ , we have

Z t Z s |h0 (r − ∆) − h0(r)| Z t−∆ Z u |h0 (v) − h0(v + ∆)| F13 = drds = dvdu. ∆ 1−α − 0 ∆ ∆ (s − r) 0 0 (u − v)1 α

Using the estimate

|h0(v) − h0(v + ∆)| ≤ Cv2H−β−1∆β

28 for all 0 < β < 2H, we obtain

13 β F∆ ≤ C ∆ . (3.28)

Thus, (3.26)–(3.28) yield

1 β F∆ ≤ C∆ , for all 0 < β < 2H . (3.29)

2 Now we study the second term F∆ . Denote

D21 = {0 < r < u < s < ∆ < t}, D22 = {0 < r < u < ∆ < s < t},

D23 = {0 < r < ∆ < u < s < t}, D24 = {0 < ∆ < r < u < s < t}.

Then

2 21 22 23 24 F∆ = F∆ + F∆ + F∆ + F∆ , where for i = 1,2,3,4,

Z |h0 (r − ∆) − h0 (u − ∆) − h0(r) + h0(u)| F2i = dudrds. ∆ 2−α0 D2i (u − r)

21 Consider ﬁrst the term F∆ . We can write

1 0 0 2H−1 2H−1 h (r − ∆) − h (u − ∆) = (∆ − u) − (∆ − r) 2H Z 1 ≤ C (u − r) (∆ − u + θ (u − r))2H−2 dθ 0 ≤ C (u − r)1−β (∆ − u)2H+β−2 ,

29 where 1 − 2H < β < α0. Similarly, we have

0 0 2H+ −2 1− h (r) − h (u) ≤ Cr β (u − r) β .

As a consequence,

Z ∆ Z s Z s 21 α0−β−1 2H+β−2 F∆ ≤ C (u − r) (∆ − u) dudrds 0 0 r Z ∆ Z ∆ Z ∆ 0 ≤ C (u − r)α −β−1 (∆ − u)2H+β−2 dudrds (3.30) 0 0 r 0 ≤ C∆2H+α .

In a similar way we can prove that

Z t Z ∆ Z ∆ 22 α0−β−1 2H+β−2 2H+α0−1 F∆ ≤ C (u − r) (∆ − u) dudrds ≤ C∆ . (3.31) ∆ 0 r

23 For F∆ , notice that when r < ∆ < u,

0 0 0 0 2H−1 2H−1 2H−1 2H−1 h (r − ∆) − h (u − ∆) − h (r) + h (u) = (∆ − r) + (u − ∆) + r + u and

0 0 (u − r)α −2 = (u − ∆ + ∆ − r)α −2

0 0 ≤ (u − ∆)−β (∆ − r)α +β−2 ∧ (u − ∆)−β−2H+1 (∆ − r)2H+α +β−3 , where we can take any β ∈ (0,1) satisfying 2H + β + α0 > 2. Then,

Z 23 h 2H−1 2H−1 2H−1 2H−1i α0−2 F∆ ≤ C (∆ − r) + (u − ∆) + r + u (u − r) dudrds D23 Z h 0 0 i ≤ C (∆ − r)2H+α +β−3 (u − ∆)−β + r2H−1 (u − ∆)−β (∆ − r)α +β−2 dudrds D23

30 0 ≤ C∆2H+α +β−2 .

Taking β = 1 + γ − α0, we obtain

23 2H+α0−1 F∆ ≤ C∆ . (3.32)

24 Finally we consider the last term F∆ . Making the substitutions x = r − ∆, y = u − ∆ we can write

Z |h0 (r − ∆) − h0 (u − ∆) − h0(r) + h0(u)| F24 = dudrds ∆ 2−α0 D24 (u − r) Z t Z s−∆ Z s−∆ |h0 (x) − h0 (y) − h0(x + ∆) + h0(y + ∆)| = dydxds. − 0 ∆ 0 x (y − x)2 α

Note that for 0 < x < y and ∆ > 0,

0 0 0 0 h (x) − h (y) − h (x + ∆) + h (y + ∆)

= x2H−1 − y2H−1 − (x + ∆)2H−1 + (y + ∆)2H−1 Z 1 Z 1 = C (x + θ(y − x) + θ˜∆)2H−3dθdθ˜ 0 0 ≤ Cx2H+β1+β2−3(y − x)1−β1 ∆1−β2 , where

0 0 < β1,β2 < 1, 2H + β1 + β2 > 2, and β1 < α .

Taking β2 = 2 − 2H − γ we get

24 2H+γ−1 F∆ ≤ C ∆ . (3.33)

31 From (3.30)–(3.33), we see that

2 2H+γ−1 F∆ ≤ C∆ . (3.34)

This completes the proof of the lemma.

Theorem 3.5. Suppose that φ ∈ Cα ([0,T]) with γα > 1 − 2H on [0,T]. Then, the non- R t linear stochastic integral 0 W(ds,φs) exists and

Z t 2 Z t 2H−1 E W(ds,φs) = 2H θ Q(φθ ,φθ )dθ 0 0 Z t Z θ 2H−2 +2H (2H − 1) r (Q(φθ ,φθ−r) − Q(φθ ,φθ ))drdθ. (3.35) 0 0

1−2H 0 Furthermore, for any γ < α < α, we have

Z t Z t 2! ε sup E W˙ (s,φs)ds − W(ds,φs) 0≤t≤T 0 0 M γ 2H+γα0−1 ≤ C (1 + kφk∞) 1 + ||φ||α ε , (3.36)

0 where the constant C depends on H, T, γ, α, α and the constants C0 and C1 appearing in (Q1) and (Q2).

Proof. We can write (3.12) as

Z t Z θ 2H E (Iε (φ)Iδ (φ)) = (Q(φθ ,φθ−r) − Q(φθ ,φθ ))Vε,δ (r)drdθ 0 0 Z t Z θ 2H + Q(φθ ,φθ )Vε,δ (r)drdθ . (3.37) 0 0

32 Due to the local boundedness of Q (see (Q1)) and applying Lemma 3.3 to ψ (θ) =

Q(φθ ,φθ ), we see that the second integral converges to

Z t Z θ Z t 2H 2H−1 lim Q(φθ ,φθ )Vε,δ (r)drdθ = 2H Q(φθ ,φθ )θ dθ . ε,δ→0 0 0 0

On the other hand, using the local Holder¨ continuity of Q (see (Q2)) and applying

Lemma 3.4, to ψ (r,θ) = Q(φθ ,φθ−r)−Q(φθ ,φθ ), we see that the ﬁrst integral converges to

Z t Z θ 2H lim (Q(φθ ,φθ−r) − Q(φθ ,φθ ))Vε,δ (r)drdθ ε,δ→0 0 0 Z t Z θ 2H−2 = 2H (2H − 1) (Q(φθ ,φθ−r) − Q(φθ ,φθ ))r drdθ . 0 0

2 This implies that {Iεn (φ),n ≥ 1} is a Cauchy sequence in L (Ω) for any sequence εn ↓ 0. 2 R t As a consequence, limε→0 Iε (φ) exists in L (Ω) and is denoted by I(φ) := 0 W(ds,φs). Letting ε,δ → 0 in (3.37), we obtain (3.35). From (3.37), Lemma 3.3 and Lemma 3.4, we have for any α0 < α,

2 M γ 2H+γα0−1 E (Iε (φ)Iδ (φ)) − E(I (φ)) ≤ C (1 + kφk∞) (1 + kφkα )(ε + δ) . (3.38)

2 In Equation (3.38), let δ → 0 and notice that Iδ (φ) → I(φ) in L (Ω). Then

2 M γ 2H+γα0−1 E (Iε (φ)I(φ)) − E(I (φ)) ≤ C (1 + kφk∞) (1 + kφkα )ε .

On the other hand, if we let ε = δ in (3.38), we obtain

2 2 M γ 2H+γα0−1 EIε (φ) − E(I (φ)) ≤ C (1 + kφk∞) (1 + kφkα )ε .

33 Thus we have

2 2 2 2 E |Iε (φ) − I(φ)| = E Iε (φ) − E I (φ) − 2 E (Iε (φ)I(φ)) − E I (φ) .

Applying the triangular inequality, we obtain (3.36).

The following proposition can be proved in the same way as (3.35).

Proposition 3.6. Suppose φ,ψ ∈ Cα ([0,T]) with αγ > 1 − 2H. Then

Z t Z t Z t 2H−1 E W(dr,φr) W(dr,ψr) = 2H θ Q(φθ ,ψθ )dθ 0 0 0 Z t Z θ 2H−2 +H (2H − 1) r (Q(φθ ,ψθ−r) − Q(φθ ,ψθ ))drdθ (3.39) 0 0 Z t Z θ 2H−2 +H (2H − 1) r (Q(φθ−r,ψθ ) − Q(φθ ,ψθ ))drdθ. 0 0

The following proposition provides the Holder¨ continuity of the indeﬁnite integral.

Proposition 3.7. Suppose φ ∈ Cα ([0,T]) with αγ > 1−2H. Then for all 0 ≤ s < t ≤ T,

Z t Z s 2 M 2H E W(dr,φr) − W(dr,φr) ≤ C (1 + kφk∞) (t − s) , (3.40) 0 0

where the constant C depends on H, T, γ, α and the constants C0 and C1 appearing R t in (Q1) and (Q2). As a consequence, the process Xt = 0 W(dr,φr) is almost surely (H − δ)-Holder¨ continuous for any δ > 0.

Proof. We shall ﬁrst show that

Z t Z s 2 ε ε M 2H E W˙ (r,φr)dr − W˙ (r,φr)dr ≤ C (1 + kφk∞) (t − s) . (3.41) 0 0

34 We can write

Z t Z s 2 Z t 2 ε ε ε E W (dr,φr) − W (dr,φr) = E W (dr,φr) 0 0 s 1 Z t Z t = E [(W(θ + ε,φθ ) −W(θ − ε,φθ ))(W(η + ε,φη ) −W(η − ε,φη ))]dθdη 4ε2 s s Z t Z t 1 2H 2H 2H = Q(φθ ,φη )[|η − θ| − |η − θ − 2ε| − |η − θ + 2ε| ]dθdη 8ε2 s s Z t−s Z t−s 1 2H 2H 2H = 2 Q(φs+θ ,φs+η )[|η − θ| − |η − θ − 2ε| − |η − θ + 2ε| ]dθdη 8ε 0 0 Z t−s Z θ 1 h 2H 2H 2Hi = 2 Q(φs+θ ,φs+θ−r) 2r − |r + 2ε| − |r − 2ε| drdθ. 4ε 0 0

The inequality (3.41) follows from the assumption (Q1) and the inequality (3.65) obtained in the Appendix. Finally, the inequality (3.40) follows from (3.41), Proposition 3.5 and the Fatou’s lemma.

3.4 Feynman-Kac integral

In this section, we show that the random field u(t,x) given by (3.2) is well-defined and study its Holder¨ continuity. Since the Brownian motion Bt has Holder¨ continuous tra- 1 jectories of order δ for any δ ∈ 0, 2 , by Lemma 3.5 the nonlinear stochastic integral R t x 1 γ 0 W(ds,Bt−s) can be defined for any H > 2 − 4 . The following theorem shows that it is exponentially integrable and hence u(t,x) is well-defined. |B − B | Set ||B|| = sup |B | and ||B|| = sup t s for δ ∈ 0, 1 . ∞,T s δ,T δ 2 0≤s≤T 0≤s

1 γ d Theorem 3.8. Let H > 2 − 4 and let u0 be bounded. For any t ∈ [0,T] and x ∈ R , the R t x random variable 0 W(ds,Bt−s) is exponentially integrable and the random ﬁeld u(t,x) given by (3.2) is in Lp(Ω) for any p ≥ 1.

35 Proof. Suppose ﬁrst that p = 1. By (3.40) with s = 0 and the Fernique’s theorem we have

Z t W B W x E |u(t,x)| ≤ ||u0||∞ E E [exp W(ds,Bt−s)] 0 2H M B Ct (1+||B||∞,T ) ≤ ||u0||∞ E [e ] < ∞.

The Lp integrability of u(t,x) follows from Jensen’s inequality

Z t W p B W x E |u(t,x)| ≤ ||u0||∞ E E exp p W(dr,Bt−r) 0 B M 2H ≤ ||u0||∞ E [exp Cp(1 + ||B||∞,T ) T ] < ∞. (3.42)

To show the Holder¨ continuity of u(·,x), we need the following lemma.

Lemma 3.9. Assume that u0 is Lipschitz continuous. Then for 0 ≤ s < t ≤ T and for any 1 α < 2H − 1 + 2 γ,

Z s Z s 2 W x x M γ α E W(dr,Bt−r) − W(dr,Bs−r) ≤ C(1 + ||B||∞,T ) kBkδ,T (t − s) , 0 0

where the constant C depends on H, T, γ and the constant C1 appearing in (Q2).

1 Proof. Suppose δ ∈ 0, 2 . For 0 ≤ u < v < s ≤ T, denote

x x x x x x x x ∆Q(s,t,u,v) := Q(Bt−u,Bt−v) − Q(Bt−u,Bt−u) − Q(Bt−u,Bs−v) + Q(Bt−u,Bs−u).

Note that (Q2) implies

M γ γδ |∆Q(s,t,u,v)| ≤ 2C1(1 + ||B||∞,T ) kBkδ,T (t − s) ,

36 and

M γ γδ |∆Q(s,t,u,v)| ≤ 2C1(1 + ||B||∞,T ) kBkδ,T |u − v| , which imply that for any β ∈ (0,1),

M γ βγδ (1−β)γδ |∆Q(s,t,u,v)| ≤ 2C1(1 + ||B||∞,T ) kBkδ,T (t − s) |u − v| .

Applying (3.39) and using Q(x,y) = Q(y,x), we get

Z s Z s 2 W x x E W(dr,Bt−r) − W(dr,Bs−r) 0 0 Z s Z θ = 2H (2H − 1) r2H−2 [∆Q(s,t,θ,θ − r) + ∆Q(t,s,θ,θ − r)]drdθ 0 0 Z s 2H−1 x x x x x x +2H θ Q(Bt−θ ,Bt−θ ) − 2Q(Bt−θ ,Bs−θ ) + Q(Bs−θ ,Bs−θ ) dθ 0 M γ βγδ ≤ C(1 + ||B||∞,T ) kBkδ,T (t − s) , for any β such that (1 − β)γδ > 1 − 2H, i.e. βγδ < 2H − 1 + γδ. Taking β and δ such that βγδ = α, we get the lemma.

d Theorem 3.10. Suppose u0 is Lipschitz continuous and bounded. Then for each x ∈ R ,

H1 1 1 u(·,x) ∈ C ([0,T]) for any H1 ∈ 0,H − 2 + 4 γ .

Proof. For 0 ≤ s < t ≤ T, from the Minkowski’s inequality it follows that

EW [|u(t,x) − u(s,x)|p] " 1 #p R t x R s x p p B W x 0 W(dr,Bt−r) x 0 W(dr,Bs−r) ≤ E E u0(Bt )e − u0(Bs)e

" 1 #p R t x R s x p p B W 0 W(dr,Bt−r) 0 W(dr,Bs−r) ≤ C ||u0||∞ E E e − e (3.43)

" 1 #p R s x p p B W x x 0 W(dr,Bs−r) +C E E (u0(Bt ) − u0(Bs))e .

37 Since u0 is Lipschitz continuous, using (3.42) and Holder’s¨ inequality we have

" 1 #p R s x p p p B W x x 0 W(dr,Bs−r) E E |u0(Bt ) − u0(Bs)|e ≤ C(t − s) 2 . (3.44)

1 W R t x R s x 2p 2 For the ﬁrst term in (3.43), denoting E (exp 0 W(dr,Bt−r) + exp 0 W(dr,Bs−r)) a b a b by K, using the formula that |e − e | ≤ (e + e )|a − b| for a,b ∈ R and Holder’s¨ inequality we get

Z t Z s p W x x E exp W(dr,Bt−r) − exp W(dr,Bs−r) 0 0 1 " Z t Z s 2p# 2 W x x ≤ K E W(dr,Bt−r) − W(dr,Bs−r) . (3.45) 0 0

Applying Lemma 3.7 and Lemma 3.9, we obtain

Z t Z s 2 Z t 2 W x x W x E W(dr,Bt−r) − W(dr,Bs−r) ≤ 2E W(dr,Bt−r) 0 0 s Z s Z s 2 W x x +2E W(dr,Bt−r) − W(dr,Bs−r) 0 0 M γ 2H1 ≤ C(1 + ||B||∞,T ) kBkδ,T (t − s) . (3.46)

R t x R s x Noting that conditional to B, 0 W(dr,Bt−r) − 0 W(dr,Bs−r) is Gaussian, and using (3.45), (3.46) and (3.42) we get

" 1 #p Z t Z s p p B W x x E E exp W(dr,Bt−r) − exp W(dr,Bs−r) 0 0  1 p Z t Z s 2! 2 B W x x ≤ C E E W(dr,Bt−r) − W(dr,Bs−r)  (3.47) 0 0

≤ C(t − s)pH1 .

38 From (3.43), (3.44) and (3.47), we can see that for any p ≥ 1,

EW [|u(t,x) − u(s,x)|p] ≤ C(t − s)pH1 . (3.48)

Now Kolmogorov’s continuity criterion implies the theorem.

3.5 Validation of the Feynman-Kac Formula

In the last section, we have proved that u(t,x) given by (3.2) is well-deﬁned. In this section, we shall show that u(t,x) is a weak solution to Equation (3.1). To give the exact meaning about what we mean by a weak solution, we follow the idea of [15] and [17]. First, we need a deﬁnition of the Stratonovich integral.

d Deﬁnition 3.11. Given a random ﬁeld v = {v(t,x), t ≥ 0,x ∈ R } such that for all t > 0 R t R d |v(s,x)|dxds < a.s., the Stratonovich integral 0 R ∞

Z t Z v(s,x)W(ds,x)dx 0 Rd is deﬁned as the following limit in probability if it exists

Z t Z lim v(s,x)W˙ ε (s,x)dsdx ε→0 0 Rd where W ε (t,x) is introduced in (3.11).

The precise meaning of the weak solution to equation (3.1) is given below.

d Deﬁnition 3.12. A random ﬁeld u = {u(t,x),t ≥ 0,x ∈ R } is a weak solution to Equa- ∞ d tion (3.1) if for any ϕ ∈ C0 R , we have

Z Z t Z (u(t,x) − u0(x))ϕ(x)dx = u(s,x)∆ϕ(x)dxds Rd 0 Rd

39 Z t Z + u(s,x)ϕ(x)W(ds,x)dx (3.49) 0 Rd almost surely, for all t ≥ 0, where the last term is a Stratonovich stochastic integral in the sense of Deﬁnition 3.11.

The following theorem justiﬁes the Feynman-Kac formula (3.2).

1 1 Theorem 3.13. Suppose H > 2 − 4 γ and u0 is a bounded measurable function. Let ∞ d u(t,x) be the random field defined in (3.2). Then for any ϕ ∈ C0 R , u(t,x)ϕ(x) is Stratonovich integrable and u(t,x) is a weak solution to Equation (3.1) in the sense of Definition 3.12.

Proof. We prove this theorem by a limit argument. We divide the proof into three steps. Step 1. Let uε (t,x) be the unique solution to the following equation:

 ε ∂u ε  1 ε ε ∂W d  = 2 4u + u ∂t (t,x), t > 0,x ∈ R , ∂t (3.50)  ε u (0,x) = u0(x).

Since W ε (t,x) is differentiable, the classical Feynman-Kac formula holds for the solution to this equation, that is,

R t ˙ ε x ε B x 0 W (s,Bt−s)ds u (t,x) := E [u0(Bt )e ].

The fact that uε (t,x) is well-deﬁned follows from (3.41) and Fernique’s theorem. In fact, we have (c.f. the argument in the proof of Lemma 3.8)

Z t W ε p B W ˙ ε x E |u (t,x)| ≤ ||u0||∞ E E exp p W (r,Bt−r)dr 0 B M 2H ≤ ||u0||∞ E [exp Cp(1 + ||B||∞,T ) t ] < ∞. (3.51)

40 Introduce the following notations

1 gε (r,z) := 1 (r)1 (z), s,x 2ε [s−ε, s+ε] [0,x] B g (r,z) := 1 (r)1 x (z), s,x [0,s] [0,Bs−r] Z s ε,B 1 gs,x (r,z) := 1[θ−ε, θ+ε](r)1[0,Bx ](z)dθ . 0 ε s−θ

ε B ε,B From the results of Section 3, we see that gs,x, gs,x, gs,x ∈ H (H is introduced in Section 2), and we can write

1 W˙ ε (s,x) = W 1 (r)1 (z) = W gε , 2ε [s−ε, s+ε] [0,x] s,x

Z s Z s x B ˙ ε x ε,B W dθ,Bs−θ = W gs,x , W θ,Bs−θ dθ = W gs,x . 0 0

Set

ε ε ue (s,x) := u (s,x) − u(s,x).

Step 2. We prove the following claim:

, d uε (s,x) → u(s,x) in D1 2 as ε ↓ 0, uniformly on any compact subset of [0,T] × R , d that is, for any compact K ⊆ R

W h ε 2 ε 2 i sup E |ue (s,x)| + kDue (s,x)kH → 0 as ε ↓ 0. (3.52) s∈[0,T],x∈K

1 Since u0 is bounded, without loss of generality, we may assume u0 ≡ 1. Let B and B2 be two independent Brownian motions, both independent of W. Using the inequality

a b a b ε,B B |e − e | ≤ (e + e )|a − b|,Holder¨ inequality and the fact that W(gt,x ) and W(gt,x) are Gaussian conditioning to B, we have

2 ε,B B EW (uε (t,x) − u(t,x)) = EW [EB(eW(gt,x ) − eW(gt,x))]2

41 ε,B B 2 B W W(gt,x ) W(gt,x) ≤ E E e − e 1 1 ε,B B 4 2 ,B 4 2 B W W(gt,x ) W(gt,x) W ε B ≤ E E e + e E W(gt,x ) −W(gt,x) 1 h ε,B B i 2 ,B 2 B W 4W(gt,x ) 4W(gt,x) B W ε B ≤ C E E e + e E E W(gt,x ) −W(gt,x) .

Note that (3.42) and (3.51) imply

ε,B B EBEW epW(gt,x ) + epW(gt,x) < ∞ (3.53) for any p ≥ 1. On the other hand, applying Theorem 3.5, we have

2 B W ε,B B sup E E W(gt,x ) −W(gt,x) → 0 as ε ↓ 0. (3.54) 0≤t≤T,x∈K

Then it follows that as ε ↓ 0

W ε 2 W ε 2 sup E |ue (t,x)| = sup E (u (t,x) − u(t,x)) → 0. 0≤t≤T,x∈K 0≤t≤T,x∈K

For the Malliavin derivatives, we have

ε B ε,B ε,B Du (s,x) = E exp W gs,x gs,x ,

B B B Du(s,x) = E exp W gs,x gs,x .

Then

W ε 2 E ||Du (s,x) − Du(s,x)||H 2 = EW EB W gε,Bgε,B − W gB gB exp s,x s,x exp s,x s,x H h 2 i ≤ EW EB W gε,B gε,B − gB 2 exp 2 s,x s,x s,x H

42 h 2 2 i + EW EB W gε,B − W gB gB . 2 exp s,x exp s,x s,x H

2 ε,B B 2 W ε,B B Note that ||gt,x − gt,x||H = E W(gt,x ) −W(gt,x) . Then it follows again from (3.53) and (3.54) that as ε ↓ 0

W ε 2 sup E ||Du (s,x) − Du(s,x)||H → 0. 0≤t≤T,x∈K

R t R ε ε Step 3. From Equation (3.50) and (3.52), it follows that d u (s,x) (x)W˙ (s,x)dsdx 0 R ϕ converges in L2 to some random variable as ε ↓ 0. Hence, if

Z t Z ε ε Vε := (u (s,x) − u(s,x))ϕ(x)W˙ (s,x)dsdx. (3.55) 0 Rd converges to zero in L2, then

Z t Z Z t Z lim u(s,x)ϕ(x)W˙ ε (s,x)dsdx = lim uε (s,x)ϕ(x)W˙ ε (s,x)dsdx, ε→0 0 Rd ε→0 0 Rd that is, u(s,x)ϕ(x) is Stratonovich integrable and u(s,x) is a weak solution to Equation

2 (3.1). Thus, it remains to show that Vε converges to zero in L . 2 ε ε In order to show the convergence to zero of (3.55) in L , ﬁrst we write ue (s,x)W(gs,x) as the sum of a divergence integral and a trace term (see (3.5))

uε (s,x)W(gε ) = (uε (s,x)gε ) − Duε (s,x),gε . e s,x δ e s,x e s,x H

Then we have

Z t Z ε ε Vε = ue (s,x)ϕ(x)W(gs,x)dsdx 0 Rd Z t Z = (uε (s,x)gε ) − Duε (s,x),gε (x)dsdx δ e s,x e s,x H ϕ 0 Rd

43 Z t Z = ( ε ) − Duε (s,x),gε (x)dsdx = V 1 −V 2, δ ψ e s,x H ϕ : ε ε 0 Rd where Z t Z ε ε ε ψ (r,z) = ue (s,x)gs,x(r,z)ϕ(x)dsdx. 0 Rd

1 2 For the term Vε , using the estimates on L norm of the Skorohod integral (see (1.47) in [39]), we obtain

h 1 2i h ε 2 i h ε 2 i E Vε ≤ E ||ψ ||H + E ||Dψ ||H⊗H . (3.56)

Denoting supp(ϕ) the support of ϕ, we have

h ε 2 i E ||ψ ||H Z t Z t Z Z = E uε (s ,x )uε (s ,x ) gε ,gε ϕ(x )ϕ(x )ds ds dx dx e 1 1 e 2 2 s1,x1 s2,x2 H 1 2 1 2 1 2 0 0 Rd Rd Z t Z t Z Z ≤ M gε ,gε ϕ(x )ϕ(x )ds ds dx dx 1 s1,x1 s2,x2 H 1 2 1 2 1 2 0 0 Rd Rd Z Z W ε ε = M1 E [W (t,x1)W (t,x2)]ϕ(x1)ϕ(x2)dx1dx2, Rd Rd

ε 2 where M1 := sups∈[0,T],x∈supp(ϕ) E |ue (s,x)| . Note that

Z Z W ε ε lim E [W (t,x1)W (t,x2)]ϕ(x1)ϕ(x2)dx1dx2 ε→0 Rd Rd Z Z W = E [W (t,x1)W (t,x2)]ϕ(x1)ϕ(x2)dx1dx2 (3.57) Rd Rd Z Z 2H = t Q(x1,x2)ϕ(x1)ϕ(x2)dx1dx2 < ∞. Rd Rd

h ε 2 i Thus by (3.52), we get E ||ψ ||H → 0 as ε ↓ 0. ε 2 On the other hand, setting M2 := sups∈[0,T],x∈supp(ϕ) E ||Due (s,x)||H , we have

h ε 2 i E ||Dψ ||H⊗H

44 Z t Z t Z Z = E Duε (s ,x ) ⊗ gε ,Duε (s ,x ) ⊗ gε ϕ(x )ϕ(x )d~sd~x e 1 1 s1,x1 e 2 2 s2,x2 H 1 2 0 0 Rd Rd Z t Z t Z Z = E hDuε (s ,x ),Duε (s ,x )i gε ,gε ϕ(x )ϕ(x )d~sd~x e 1 1 e 2 2 H s1,x1 s2,x2 H 1 2 0 0 Rd Rd Z t Z t Z Z ≤ M gε ,gε ϕ(x )ϕ(x )ds ds dx dx . 2 s1,x1 s2,x2 H 1 2 1 2 1 2 0 0 Rd Rd

h ε 2 i Then (3.52) and (3.57) imply that E ||Dψ ||H⊗H converges to zero as ε ↓ 0. Finally, we deal with the trace term

Z t Z V 2 = Duε (s,x),gε − Du(s,x),gε (x)dsdx ε s,x H s,x H ϕ (3.58) 0 Rd ε ε =: T1 − T2 , where

Z t Z T ε = Duε (s,x),gε (x)dsdx, 1 s,x H ϕ 0 Rd Z t Z T ε = Du(s,x),gε (x)dsdx. 2 s,x H ϕ 0 Rd

ε ε We will show that T1 and T2 converge to the same random variable as ε ↓ 0. ε We start with the term T2 . Note that

B ε 1 g ,g = 1 (r)1 x (z), 1 (r)1 (z) s,x s,x [0,s] [0,Bs−r] [s−ε, s+ε] [0,x] 2ε H x 1 = 1[0,s](r)Q Bs−r,x , 1[s−ε, s+ε](r) . 2ε H

45 1 γ x 2 −δ 1 1 Since Q Bs−·,x ∈C ([0,T]) for any 0 < δ < 2 , noticing that H > 2 − 4 and applying Lemma 3.20 we obtain

Z t Z T ε = EB u (Bx) W gB gB ,gε (x)dsdx lim 2 0 s exp s,x s,x s,x H ϕ ε→0 0 Rd Z t Z B x B 2H−1 = E u0 (Bs)exp W gs,x ϕ(x)[Q(x,x)Hs (3.59) 0 Rd Z s x 2H−2 + H (2H − 1) Q Bs−r,x − Q(x,x) r dr]dsdx. 0

ε On the other hand, for the term T1 , note that

Z 2ε ε,B ε 1 1 g ,g = 1 (r)1 x (z)dθ, 1 (r)1 (z) s,x s,x [θ−ε,θ+ε] [0,Bs−θ ] [s−ε, s+ε] [0,x] 0 2ε 2ε H Z 2ε 1 x 1 = 1[θ−ε,θ+ε](r)Q Bs−θ ,x dθ, 1[s−ε, s+ε](r) . 0 2ε 2ε H

Applying Lemma 3.21, we obtain

Z t Z T ε = EB u (Bx) W gε,B gε,B,gε (x)dsdx lim 1 0 s exp s,x s,x s,x H ϕ ε→0 0 Rd Z t Z B x B 2H−1 = E u0 (Bs)exp W gs,x ϕ(x)[Q(x,x)Hs (3.60) 0 Rd Z s x 2H−2 +H (2H − 1) Q Bs−r,x − Q(x,x) r dr]dsdx. 0

2 2 The convergence in L to zero of Vε follows from (3.60) and (3.59).

3.6 Skorohod type equation and Chaos expansion

d In this section, we consider the following heat equation on R

  ∂u 1 ∂W d  ∂t = 2 ∆u + u ∂t (t,x), t ≥ 0, x ∈ R , (3.61)  u(0,x) = u0(x).

46 The difference between the above equation and Equation (3.1) is that here we use the

Wick product . This equation is studied in Hu and Nualart [15] for the case H1 = ··· = 1 1 Hd = 2 , and in [17] for the case H1,...,Hd ∈ ( 2 ,1), 2H0 + H1 + ··· + Hd > d + 1. As in that paper, we can deﬁne the following notion of solution.

d Deﬁnition 3.14. An adapted random ﬁeld u = {u(t,x),t ≥ 0, x ∈ R } with Eu2 (t,x) < d ∞ for all (t,x) is a (mild) solution to Equation (3.61) if for any (t,x) ∈ [0,∞) × R , d the process pt−s (x − y)u(s,y)1[0,t] (s),s ≥ 0, y ∈ R is Skorohod integrable, and the following equation holds

Z t Z u(t,x) = pt f (x) + pt−s (x − y)u(s,y)δWs,y, (3.62) 0 Rd

R where p (x) denotes the heat kernel and p f (x) = d p (x − y) f (y)dy. t t R t

From [15], we know that the solution to Equation (3.61) exists with an explicit Wiener chaos expansion if and only if the Wiener chaos expansion converges. Note

B B x that g (r,z) := 1 (r)1 x (z) ∈ H. Formally, we can write g (r,z) = δ B − z t,x [0,t] [0,Bt−r] t,x t−r and we have

Z t Z t Z x B x W dr,Bs−r = W gt,x = δ Bt−r − z W (dr,z)dz. 0 0 Rd

Then in the same way as in Section 8 in [17] we can check that u(t,x) given by (3.63) below has the suitable Wiener chaos expansion, which has to be convergent because u(t,x) is square integrable. We state it as the following theorem.

1 1 Theorem 3.15. Suppose H > 2 − 4 γ and u0 is a bounded measurable function. Then the unique (mild) solution to Equation (3.61) is given by the process

B x B 1 B 2 u(t,x) = E u0 (B )exp(W g − g ) . (3.63) t t,x 2 t,x H

47 Remark 3.16. We can also obtain a Feynman-Kac formula for the coefﬁcients of the chaos expansion of the solution to Equation (3.1)

∞ 1 u(t,x) = ∑ In (hn (t,x)), n=0 n! with B x B B 1 B 2 hn (t,x) = E u0 (B )g (r1,z1)···g (rn,zn)exp g . t t,x t,x 2 t,x H

3.7 Appendix

H H In this section, we denote by B = {Bt ,t ∈ R} a mean zero Gaussian process with H H 1 2H 2H 2H covariance E(Bt Bs ) = 2 |t| + |s| − |t − s| . Denote by E the space of all step functions on [−T,T]. On E , we introduce the following scalar product h1[0,t],1[0,s]iH0 =

RH(t,s), where if t < 0 we assume that 1[0,t] = −1[t,0]. Let H0 be the closure of E with respect to the above scalar product.

For r > 0, ε > 0 and β > 0, let

1 h i f ε (r) := 2rβ − |r − 2ε|β − (r + 2ε)β . 4ε2

It is easy to see that

lim f ε (r) = β (β − 1)rβ−2. (3.64) ε↓0

Lemma 3.17. For any r > 0, ε > 0 and 0 < β < 2,

| f ε (r)| ≤ 64rβ−2. (3.65)

48 Proof. If 0 < r < 4ε, then |r − 2ε|β < (2ε)β , (r + 2ε)β < (6ε)β , and hence (noting that β < 2) | f ε (r)| ≤ 4β+1εβ−2 ≤ 64rβ−2.

On the other hand, if r ≥ 4ε, then

Z 1 rβ − |r − 2ε|β = 2εβ (r − 2λε)β−1 dλ, 0 Z 1 rβ − (r + 2ε)β = −2εβ (r + 2λε)β−1 dλ, 0 and hence

1 Z 1 h i f ε (r) = β (r − 2λε)β−1 − (r + 2λε)β−1 dλ 2ε 0 Z 1 Z 1 = 2β (β − 1) λ (r − 2λε + 4µλε)β−2 dµdλ. 0 0

Therefore, using β < 2 and r ≥ 4ε we obtain

r − 2ε β−2 | f ε (r)| ≤ 2β (r − 2ε)β−2 ≤ 4rβ−2 ≤ 16rβ−2. r

Lemma 3.18. For any s > 0, 0 < β < 1 and φ ∈ Cα ([0,T]) with α > 1 − β,

Z s Z s lim φ (r) f ε (r)dr = φ (0)βsβ−1 + β (β − 1) (φ (r) − φ (0))rβ−2dr. (3.66) ε→0 0 0

Moreover,

Z s ε β−1 α+β−1 φ (r) f (r)dr ≤ C (β,α) kφk∞ s + kφkα s . (3.67) 0

49 Proof. The lemma follows easily from (3.68) and (3.65) if we rewrite

Z s Z s Z s φ (r) f ε (r)dr = φ (0) f ε (r)dr + [φ (r) − φ (0)] f ε (r)dr. 0 0 0

Lemma 3.19. For any bounded function φ ∈ H0 and any s,t ≥ 0, we have

Z s h 2H−1 2H−1i 1[0,s]φ,1[0,t] = H φ (r) r + sign(t − r)|t − r| dr. (3.68) H0 0

If u < s < t, we have

Z s h 2H−1 2H−1i 1[0,s]φ,1[u,t] = H φ (r) (t − r) − sign(u − r)|u − r| dr. (3.69) H0 0

Proof. We only have to prove (3.68) since (3.69) follows easily. Without loss of general-

n ity, assume that φ = Σi=1ai1[ti−1,ti], where 0 = t0 ≤ t1 ≤ ... ≤ tn = s. (If t < s, we assume that t = ti for some 0 < i < n.) Then

n H H H 1[0,s]φ,1[0,t] = EΣi= ai Bt − Bt Bt H0 1 i i−1 1 = Σn a t2H −t2H + |t −t |2H − |t −t |2H i=1 i 2 i i−1 i−1 i Z s h i = H φ (r) r2H−1 + sign(t − r)|t − r|2H−1 dr. 0

Using Lemma 3.19 and similar arguments to those in the proof of Lemma 3.18, we can prove the following lemma.

50 Lemma 3.20. For any s > 0, for any φ ∈ Cα ([0,T]) with α > 1 − 2H,

Z s 1 2H−1 2H−2 lim 1[0,s]φ, 1[s−ε,s+ε] = φ (s)Hs + c0 (φ (s − r) − φ (s))r dr, ε→0 2ε 0 H0 where c0 = H(2H − 1). Moreover,

1 1 φ, 1 ≤ C (H,α)||φ|| s2H−1 + ||φ|| sα+2H−1. (3.70) [0,s] 2ε [s−ε,s+ε] ∞ α H0

Proof. Applying Lemma 3.19 and making a substitution, we get

1 1 φ, 1 [0,s] 2ε [s−ε,s+ε] H0 H Z s h i = φ (s − u) (u + ε)2H−1 − sign(u − ε)|u − ε|2H−1 du 2ε 0 Z s Z s = : Hφ (s) gε (u)du + H [φ (s − u) − φ (s)]gε (u)du, 0 0 where we let

1 h i gε (u) = (u + ε)2H−1 − sign(u − ε)|u − ε|2H−1 . 2ε

If 0 < u < 2ε, we have |gε (u)| ≤ 16r2H−2. On the other hand, if u > 2ε,

1 h 2H−1 2H−1i |gε (u)| = (u − ε) − (u + ε) 2ε 1 Z 1 = (1 − 2H) (u − λε)2H−2 dλ ≤ (1 − 2H)u2H−2. 2 −1

ε 2H−2 Then the lemma follows by noticing that limε→0 g (u) = (2H − 1)u .

51 Lemma 3.21. For any φ ∈ Cα ([0,T]) with α > 1 − 2H, for any s > 0,

1 Z s 1 lim 1[θ−ε,θ+ε]φ (θ)dθ, 1[s−ε,s+ε] (3.71) ε→0 2ε 0 2ε H0 Z s = φ (s)Hs2H−1 + H (2H − 1) (φ (s − r) − φ (s))r2H−2dr. 0

Moreover,

1 Z s 1 1[θ−ε,θ+ε]φ (θ)dθ, 1[s−ε,s+ε] (3.72) 2ε 0 2ε H0 2H−1 α+2H−1 ≤ C (H,α) ||φ||∞ Hs + ||φ||α s .

Proof. By Fubini’s theorem and making a substitution, we have

1 Z s 1 1[θ−ε,θ+ε]φ (θ)dθ, 1[s−ε,s+ε] 2ε 0 2ε H0 Z s 1 H H H H = 2 E φ (θ) Bθ+ε − Bθ−ε Bs+ε − Bs−ε dθ 4ε 0 Z s 1 h 2H 2H 2Hi = 2 φ (s − θ) 2r − |r − 2ε| − (r + 2ε) dr. 8ε 0

Then (3.72) and (3.73) follow from Lemma 3.18.

52 Chapter 4

Holder¨ continuity of the solution for a class of nonlinear

SPDE arising from one dimensional superprocesses

The Holder¨ continuity of the solution Xt(x) to a nonlinear stochastic partial differential equation (see (4.2) below) arising from one dimensional super process is studied in this chapter. It is proved that the Holder¨ exponent in time variable is as close as to 1/4, improving the result of 1/10 in [25]. The method is to use the Malliavin calculus. The Holder¨ continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Holder¨ continuity result is sharp since the corresponding linear heat equation has the same Holder¨ continuity.

4.1 Introduction

Consider a system of particles indexed by multi-indexes α in a random environment whose motions are described by

Z t Z α xα (t) = xα + B (t) + h(y − xα (u))W (du,dy), (4.1) 0 R

53 2 α where h ∈ L (R), (B (t);t ≥ 0)α are independent Brownian motions and W is a Brow- α nian sheet on R+ × R independent of B . For more detail about this model, we refer to Wang ([53], [54]) and Dawson, Li and Wang [6]. Under some speciﬁcations for the branching mechanism and in the limiting situation, Dawson, Vaillancourt and Wang [7] obtained that the density of the branching particles satisﬁes the following stochastic partial differential equation (SPDE):

Z t Z t Z Xt(x) = µ(x) + ∆Xu(x)dr − ∇x (h(y − x)Xu (x))W (du,dy) 0 0 R Z t p V (du,dx) + Xu (x) , (4.2) 0 dx where V is a Brownian sheet on R+ × R independent of W. The joint Holder¨ continuity of (t,x) 7−→ Xt(x) is left as an open problem in [7].

k n 2 (i) 2 o Let H2 (R) = u ∈ L (R);u ∈ L (R) for i = 1,2,...,k , the Sobolev space with 2 2 k (k) norm khkk,2 = ∑i=0 h . In a recent paper, Li, Wang, Xiong and Zhou [25] proved L2(R) 2 that Xt(x) is almost surely jointly Holder¨ continuous, under the condition that h ∈ H2 (R) 2 1 with khk1,2 < 2 and X0 = µ ∈ H2 (R) is bounded. More precisely, they showed that for ﬁxed t its Holder¨ exponent in x is in (0,1/2) and for ﬁxed x its Holder¨ exponent in t is in (0,1/10). Comparing to the Holder¨ continuity for the stochastic heat equation which has the Holder¨ continuity of 1/4 in time, it is conjectured that the Holder¨ continuity of

Xt(x) should also be 1/4. The aim of this chapter is to provide an afﬁrmative answer to the above conjecture. Here is the main result.

2 2 Theorem 4.1. Suppose that h ∈ H2 (R) and X0 = µ ∈ L (R) is bounded. Then the solution to Xt (x) is jointly Holder¨ continuous with the Holder¨ exponent in x in (0,1/2) and with the Holder¨ exponent in t in (0,1/4). That is, for any 0 ≤ s < t ≤ T, x,y ∈ R and

54 p ≥ 1, there exists a constant C depending only on p,T, khk and k k 2 such that 2,2 µ L (R)

2p −p p− 1 p − 1 E |Xt (y) − Xs (x)| ≤ C(1 +t )(|x − y| 2 + (t − s) 2 4 ). (4.3)

Note that the term t−p in the right hand side of (4.3) implies that the Holder¨ norm of

Xt(x) blows up as t → 0. This problem arises naturally since we only assume X0 = µ ∈

L2(R). When h = 0 the equation (4.2) is reduced to the famous Dawson-Watanabe equation (process). The study on the joint Holder¨ continuity for this equation has been studied by Konno and Shiga [21] and Reimers [49]. The starting point is to interpret the equation

(when h = 0) in mild form with the heat kernel associated with the Laplacian ∆ in (4.2). Then the properties of the heat kernel (Gaussian density) can be fully used to analyze the Holder¨ continuity. The straightforward extension of the mild solution concept and technique to general nonzero h case in (4.2) meets a substantial difﬁculty. To overcome this difﬁculty, Li et al [25] replace the heat kernel by a random heat kernel associated with

Z t Z t Z ∆Xu(x)dr − ∇x(h(y − x)Xu(x))W (du,dy). 0 0 R

The random heat kernel is given by the conditional transition function of a typical particle in the system with W given. To be more precise, consider the spatial motion of a typical particle in the system:

Z t Z ξt = ξ0 + Bt + h(y − ξu)W (du,dy), (4.4) 0 R where (Bt;t ≥ 0) is a Brownian motion.

55 For r ≤ t and x ∈ R, deﬁne the conditional (conditioned by W) transition probability by

r,x,W W Pt (·) ≡ P (ξt ∈ ·|ξr = x). (4.5)

W r,x,W Denote by p (r,x;t,y) the density of Pt (·). It is proved that Xt(y) has the following convolution representation:

Z Z t Z W W Xt(y) = µ(z)p (0,z;t,y)dz + p (r,z;t,y)Z (dr,dz) (4.6) R 0 R ≡ Xt,1(y) + Xt,2(y),

p where Z (dr,dz) = Xr (z)V (dr,dz). Then they introduce a fractional integration by parts technique to obtain the Holder¨ continuity estimates, using Krylov’s Lp theory (cf. Krylov [20]) for linear SPDE. In this chapter, we shall use the techniques from Malliavin calculus to obtain more precise estimates for the conditional transition function pW (r,x;t,y). This allows us to improve the Holder¨ continuity in the time variable for the solution Xt(x). The rest of the chapter is organized as follows: First we derive moment estimates for the conditional transition function in Section 4.2. Then we study the Holder¨ continuity in spatial and time variables of Xt(x) in Section 4.3 and Section 4.4 respectively. The proof of Theorem 4.1 is concluded in Section 4.4.

Along the paper, we shall use the following notations; k·kH denotes the norm on 2 2 p Hilbert space H = L ([0,T]), k·k (and k·kp) denotes the norm on L (R) (and on L (Ω)). The expectation on (Ω,F ,P) is denoted by E and the conditional expectation with respect to the process W is denoted by EB.

We denote by C a generic positive constant depending only on p, T, khk2,2 and k k 2 . µ L (R)

56 4.2 Moment estimates

In this section, we derive moment estimates for the derivatives of ξt and the conditional transition function pW (r,x;t,y). r,x Recall that ξt = ξt with initial value ξr = x is given by

t t ξt = x + Br + Ir (h) , 0 ≤ r < t ≤ T , (4.7)

where we introduced the notations

Z t Z t t Br ≡ Bt − Br, and Ir (h) ≡ h(y − ξu)W (du,dy). (4.8) r R

2 Since h ∈ H2 (R), by using the standard Picard iteration scheme, we can prove that such a solution ξt to the stochastic differential equation (4.7) exists, and by a regularization

2,2 argument of h we can prove that ξt ∈ D (here the Malliavin derivative is with respect to B). Taking the Malliavin derivative Dθ with respect to B, we have

Z t Z 0 Dθ ξt = 1[r,t] (θ) 1 − h (y − ξu)Dθ ξuW (du,dy) . (4.9) θ R

Note that Z t Z 0 Mθ,t := h (y − ξu)W (du,dy) θ R

0 2 is a martingale with quadratic variation hMiθ,t = kh k (t − θ) for t > θ. Thus

1 0 2 D ξt = 1 (θ)exp M ,t − h (t − θ) . (4.10) θ [r,t] θ 2

57 As a result, we have

1 0 2 Dη D ξt = 1 (θ)exp M ,t − h (t − θ) Dη M ,t = D ξt · Dη M ,t, (4.11) θ [r,t] θ 2 θ θ θ where D M = 1 ( )R t R h00 (y − )D W (du,dy). η θ,t [θ,t] η η R ξu η ξu 2 The next lemma gives estimates for the moments of Dξt and D ξt.

Lemma 4.2. For any 0 ≤ r < t ≤ T and p ≥ 1, we have

2 1 0 2 kkDξtkHk2p ≤ exp (2p − 1) h (t − r) (t − r) , (4.12)

2 00 0 2 3 D ξt ≤ Cp h exp (4p − 1) h (t − r) (t − r) 2 , (4.13) H⊗H 2p and for any γ > 0,

−2γ 2 0 2 −γ E(kDξtkH ) ≤ exp 2γ + γ h (t − r) (t − r) . (4.14)

Proof. Note that for any p ≥ 1 and r ≤ θ < t,

1 p 2 1 0 2 kD ξtk = E exp 2p M ,t − h (t − θ) θ 2p θ 2 0 2 = exp (2p − 1) h (t − θ) . (4.15)

Then (4.12) follows from Minkowski’s inequality and (4.15) since

1 Z t p p Z t 2 2 2 kkDξtkHk2p = E |Dθ ξt| dθ ≤ kDθ ξtk2p dθ. r r

58 Applying the Burkholder-Davis-Gundy inequality we have for any r ≤ θ ≤ η < t

1 Z t Z p p 2 00 2 Dη Mθ,t 2p ≤ Cp E h (y − ξu)Dη ξu dydu η R Z t 00 2 2 ≤ Cp h Dη ξu 2p du. (4.16) η

Combining (4.11), (4.15) and (4.16) yields for any r ≤ θ ≤ η < t

2 2 2 2 Dη Dθ ξt 2p = Dθ ξtDη Mθ,t 2p ≤ kDθ ξtk4p Dη Mθ,t 4p 00 2 0 2 ≤ Cp h exp 2(4p − 1) h (t − θ) (t − η). (4.17)

An application of Minkowski’s inequality implies that

2 Z t Z t 2 2 D ξt H⊗H ≤ Dη Dθ ξt 2p dθdη. 2p r r

This yields (4.13).

For the negative moments of kDξtkH , by Jensen’s inequality we have

Z t −γ Z t −2γ 2 −γ−1 −2γ E kDξtkH = E |Dθ ξt| dθ ≤ (t − r) E |Dθ ξt| dθ. r r

Then, (4.14) follows immediately.

The moment estimates of the Malliavin derivatives of the difference ξt − ξs can also be obtained in a similar way. The next lemma gives these estimates.

Lemma 4.3. For 0 ≤ s < t ≤ T and p ≥ 1, we have

1 2 kkD(ξt − ξs)kHk2p < C (t − s) , (4.18)

59 and

2 3 D (ξt − ξs) < C (t − s) 2 . (4.19) H⊗H 2p

Proof. Similar to (4.9), we have

Z t Z 0 Dθ ξt = Dθ ξs + 1[s,t](θ) − h (y − ξu)Dθ ξuW (du,dy) θ∨s R t 0 = Dθ ξs + 1[s,t](θ) − Iθ h Dθ ξ. , (4.20)

2 where henceforth for any process Y = (Yt,0 ≤ t ≤ T) and f ∈ L (R), we denote

Z t Z t Iθ ( fY·) = 1[s,t](θ) f (y − ξu)YuW (du,dy). θ R

Applying the Burkholder-Davis-Gundy inequality with (4.15), we obtain for s ≤ θ ≤ t

1 Z t Z p p t 0 2 0 2 Iθ h Dθ ξ. 2p ≤ E h (y − ξu)Dθ ξu dudy θ R 0 2 0 2 ≤ h exp (2p − 1) h (t − θ) (t − θ). (4.21)

Then (4.18) follows from (4.20) and (4.21) since

1 1 Z T p p 2p p t 0 2 E kDξt − DξskH = E 1[s,t](θ) + Iθ h Dθ ξ. dθ 0 Z T 1 t 0 2p p ≤ E 1[s,t](θ) + Iθ h Dθ ξ. dθ 0 Z t 1 t 0 2p p ≤ 2(t − s) + 2 E Iθ h Dθ ξ. dθ s 0 2 0 2 ≤ 2 1 + h exp (2p − 1) h (t − s) (t − s).

2 For moments of D (ξt − ξs), from (4.20) we have

2 t 0 t 00 t 0 2 Dη,θ (ξt − ξs) = −Dη Iθ h Dθ ξ. = Iη h Dθ ξ.Dη ξ. − Iη h Dη,θ ξ. . (4.22)

60 In a similar way as above we can get (4.19).

Next we derive some estimates for the density pW (r,x;t,y) of the conditional transition probability deﬁned in (4.5). Denote

Dξt ut ≡ 2 . (4.23) kDξtkH

The next two lemmas give estimates of the divergence of ut and ut − us, which are im- portant to derive the moment estimates of pW (r,x;t,y).

Lemma 4.4. For any p ≥ 1 and 0 ≤ r < t ≤ T, we have

1 − 2 kδ (ut)kp ≤ C (t − r) . (4.24)

Proof. Using the estimate (2.9) we obtain

1 1 p p B p p kδ (ut)kp = (E |δ (ut)| ) ≤ E E |δ (ut)| 1 h B p B p i p ≤ Cp E E ut H + E kDutkH⊗H ≤ Cp kkutkHkp + kDutkH⊗H p .

We have 2 2 D ξt D ξt,Dξt ⊗ Dξt H⊗H Dut = 2 − 2 4 , kDξtkH kDξtkH

D2 3k ξt kH⊗H and consequently kDutkH⊗H ≤ 2 . Hence, for any positive number α,β > 1 kDξt kH 1 1 1 such that a + β = p , applying (4.13) and (4.14) we obtain (4.24):

−1 2 −2 kδ (ut)kp ≤ Cp kDξtkH + 3 D ξt α kDξtkH p L (Ω,H⊗H) β 0 00 − 1 3 −1 ≤ C p, h , h ,T (t − r) 2 + (t − r) 2 (t − r) .

61 This proves the lemma.

Lemma 4.5. For p ≥ 1, and 0 ≤ r < s < t ≤ T,

1 1 1 2 − 2 − 2 kδ (ut − us)k2p ≤ C (t − s) (s − r) (t − r) . (4.25)

Proof. Using (4.20) we can write

Dξt Dξs ut − us = 2 − 2 = A1 + A2 + A3, kDξtkH kDξskH where

! ( ) t 0 1 1 1[s,t] θ Iθ (h Dθ ξ.) A1 = Dξs 2 − 2 ,A2 = 2 ,A3 = 2 . kDξskH kDξtkH kDξtkH kDξtkH

As a consequence, we have

3 kδ (ut − us)k2p ≤ ∑ kδAik2p . (4.26) i=1

For simplicity we introduce the following notation

2 i Vt ≡ kDξtkH , Nt ≡ D ξt H⊗H , Yi = D (ξt − ξs) H⊗i , i = 1,2.

Note that hDξt − Dξs,Dξt + Dξsi −2 −1 −1 kA1kH = 2 ≤ Y1 Vt +Vs Vt , kDξskH kDξtkH and

! D hD − D ,D + D i ξs ξt ξs ξt ξs kDA1kH⊗H = D kD k2 kD k2 ξs H ξt H H⊗H −2 −1 −1 −2 −1 −1 −2 ≤ Y1Ns Vs Vt +Vs Vt +Y2 Vs Vt +Vt

62 −1 −2 +Y1 (Nt + Ns)Vs Vt

−2 −1 −1 −2 −3 −1 −2 +2Y1 Ns Vs Vt +Vs Vt + Nt Vt +Vs Vt .

As a consequence, applying (2.9) and Holder’s¨ inequality we get

kδ (A1)k2p ≤ C kkA1kHk2p + kDA1kH⊗H 2p −2 −1 −1 ≤ C kY1k4p Vt 4p + Vt 8p Vs 8p −1 −2 −1 −2 +C kY1k8p kNsk8p Vt 8p Vs 8p + Vs 8p Vt 8p −1 −1 −2 +C kY2k4p Vt 8p Vs 8p + Vt 4p −2 −1 +C kY1k8p kNsk8p + kNtk8p Vt 8p Vs 8p −1 −2 −2 −1 +2C kY1k8p kNsk8p Vt 8p Vs 8p + Vt 8p Vs 8p −3 −2 −1 +2C kY1k8p kNtk8p Vt 4p + Vt 8p Vs 8p .

From Lemma 4.2 and Lemma 4.3 it follows that

1 1 1 2 − 2 − 2 kδ (A1)k2p ≤ C (t − s) (s − r) (t − r) . (4.27)

1[s,t](θ) −2 1 Note that kA2k = = kDξtk (t − s) 2 and H kDξ k2 H t H H

1 −3 2 2 kDA2kH⊗H ≤ 2kDξtkH D ξt H⊗H (t − s) .

Then, by (2.9), Holder’s¨ inequality and Lemma 4.2 we see that

kδ (A2)k2p ≤ C kkA2kHk2p + kDA2k2p 1 2 −2 2 −1 ≤ C (t − s) Vt 2p + D ξt 4p Vt 4p 1 −1 ≤ 2C (t − s) 2 (t − r) + 1 . (4.28)

63 For the term A3, we apply Minkowski’s inequality and the Burkholder-Davis-Gundy inequality and use (4.21). Thus for any p ≥ 1,

1 Z t p 2p t 0 t 0 2 Iθ h Dθ ξ. H = E Iθ (h Dθ ξ.) dθ 2p s 1 Z t 2 t 0 2 ≤ Cp Iθ h Dξ. 2p dθ s 0 0 2 1 ≤ Cp h exp (2p − 1) h (t − r) (t − s) 2 . (4.29)

From (4.22) it follows that

2 −2 kDA3kH⊗H ≤ D (ξt − ξs) H⊗H kDξtkH t 0 2 −3 +2 Iθ h Dξ. H D ξt H⊗H kDξtkH .

Combining this with (2.9), Holder’s¨ inequality, Lemma 4.3 and (4.29) we deduce

kδ(A3)k2p ≤ Cp kkA3kHk2p + kDA3k2p ≤ C It h0D V −2 + kY k V −2 p s θ ξ. H 2p t 2p 2 4p t 4p + C It h0D V −3 kN k 2 p s θ ξ. H 4p t 8p t 8p 1 −1 ≤ C (t − s) 2 (t − r) . (4.30)

Substituting (4.27), (4.28) and (4.30) into (4.26) yields (4.25).

Now we provide the moment estimates for the conditional transition probability density pW (r,x;t,y).

2 Lemma 4.6. Let c = 1 ∨ khk . For any 0 ≤ r < t ≤ T, y ∈ R and p ≥ 1,

1 2 ! W 2p 2p (x − y) E p (r,x;t,y) ≤ 2exp − kδ (ut)k . (4.31) 64pc(t − r) 4p

64 Proof. By Lemma 2.1 we can write

W B B p (r,x;t,y) = E 1 (u ) = E [1 t t (u )], (4.32) {ξt >y}δ t {Br+Ir(h)>y−x}δ t

t t where Br and Ir(h) are deﬁned in (4.8). Then, (2.14) implies

1 W 2p 2p E p (r,x;t,y) p 1 h B t t p B 2 i 2p ≤ E P Br + Ir (h) > |y − x| E |δ (ut)| 1 B t t 2p 4p ≤ kδ (ut)k4p E P Br + Ir (h) > |y − x| . (4.33)

Applying Chebyshev and Jensen’s inequalities, we have for p ≥ 1,

B t t 2p E P Br + Ir (h) > |y − x| ! 2p −2p(x − y)2 (Bt + It(h))2 B r r ≤ exp E E exp 32pc(t − r) 32pc(t − r) ! −(x − y)2 (Bt + It(h))2 ≤ exp E exp r r . (4.34) 16c(t − r) 16c(t − r)

Using the fact that for 0 ≤ ν < 1/8 and Gaussian random variables X,Y,

1 1 2 2 2 2 2 2 2 − 1 Eeν(X+Y) ≤ Ee2ν(X +Y ) ≤ Ee4νX Ee4νY = (1 − 8ν) 2 ,

t t and noticing that Br and Ir(h) are Gaussian, we have

2 − 1 (Bt + It(h)) 1 2 √ E exp r r ≤ 1 − ≤ 2. (4.35) 16c(t − r) 2c

Combining (4.33)–(4.35), we get (4.31).

65 4.3 Holder¨ continuity in spatial variable

In this section, we obtain the Holder¨ continuity of Xt(y) with respect to y. More precisely, we show that for t > 0 ﬁxed, Xt(y) is almost surely Holder¨ continuous in y with any exponent in (0,1/2). This result was proved in [25]. Here we provide a different proof

t t based on Malliavin calculus. We continue to use the notations Br, Ir(h) (deﬁned by (4.8)) and ut (deﬁned by (4.23)).

2 2 Proposition 4.7. Suppose that h ∈ H2 (R) and X0 = µ ∈ L (R) is bounded. Then, for any t ∈ (0,T], α ∈ (0,1) and p > 1, there exists a constant C depending only on p,T, khk and k k 2 such that 2,2 µ L (R)

2p −p α p E |Xt (y2) − Xt (y1)| ≤ C(1 +t )(y2 − y1) . (4.36)

Proof. We will use the convolution representation (4.6), where the two terms Xt,1 (y) and

Xt,2 (y) will be estimated separately. X y y y We start with t,2 ( ). Suppose 1 < 2 ∈ R. Note ﬁrst that 1{ξt >y1} − 1{ξt >y2} =

1{y1<ξt ≤y2} and

Z y2 EB PB {y y } pW r x t z dz 1{y1<ξt ≤y2} = 1 < ξt ≤ 2 = ( , ; , ) . y1

Therefore by (4.32) we have

2 2 pW r x t y pW r x t y EB u ( , ; , 1) − ( , ; , 2) = 1{y1<ξt

66 Hence,

1 W W 2(2p−1) 2p−1 E p (r,x;t,y1) − p (r,x;t,y2) Z y 2 2 W ≤ kδ (ut)k4(2p−1) p (r,x;t,z) 2(2p−1) dz. (4.37) y1

Lemma 4.4 and Lemma 4.6 yield

Z 1 W W 2(2p−1) 2p−1 E p (r,x;t,y1) − p (r,x;t,y2) dx R Z Z y 2 ! 3 2 −(z − x) ≤ C kδ (ut)k4(2p−1) exp dzdx R y1 32(2p − 1)c(t − r) −1 ≤ C (t − r) (y2 − y1). (4.38)

On the other hand, the left hand side of (4.38) can be estimated differently again by using Lemma 4.6:

Z 1 W W 2(2p−1) 2p−1 E p (r,x;t,y1) − p (r,x;t,y2) dx R Z 1 W 2(2p−1) 2p−1 ≤ 2 Σi=1,2 E p (r,x;t,yi) dx R 2 ! Z −(yi − x) 1 2 − 2 ≤ Cp Σi=1,2 kδ (ut)k4(2p−1) exp dx ≤ C (t − r) . (4.39) R 64pc(t − r)

Then (4.38) and (4.39) yield that for any α,β > 0 with α + β = 1

Z 1 W W 2(2p−1) 2p−1 −α− 1 β α E p (r,x;t,y1) − p (r,x;t,y2) dx ≤ C (t − r) 2 (y2 − y1) . R (4.40)

Since µ is bounded, it follows from [25, Lemma 4.1] that

Z t Z 2p W W 2 E p (r,x;t,y2) − p (r,x;s,y1) Z (drdx) 0 R

67 p Z t Z 2p−1! 2p−1 W W 2 ≤ C E p (r,x;t,y2) − p (r,x;s,y1) drdx , (4.41) 0 R for any p ≥ 1, 0 ≤ s ≤ t ≤ T and y1,y2 ∈ R. Then, applying Minkowski’s inequality we obtain for any 0 < α < 1,

1 2p p E Xt,2 (y2) − Xt,2 (y1) Z t Z 1 W W 2(2p−1) 2p−1 ≤ E p (r,x;t,y1) − p (r,x;t,y2) dxdr 0 R Z t −α− 1 β α α ≤ C (t − r) 2 (y2 − y1) dr ≤ C (y2 − y1) 0

− − 1 −(1+ )/2 since (t − r) α 2 β = (t − r) α is integrable for all 0 < α < 1.

Now we consider Xt,1(y) in (4.6). Applying Minkowski’s inequality and using (4.37) with 2p − 1 replaced by p we get

2p E Xt,1 (y2) − Xt,1 (y1) Z 1 2p W 2p 2p ≤ E p(0,x;t,y1) − p (0,x;t,y2) µ (x)dx R 2p (Z Z y 1/2 ) 2 W ≤ C p (0,x;t,z) 2p dz kδ (ut)k4p µ (x)dx R y1 p y 2 ! ! 2p 2p Z Z 2 (z − x) ≤ C kδ (ut)k kµk exp − dzdx 4p L2(R) R y1 64pct 2p −p p ≤ C kµk t (y2 − y1) . L2(R)

This completes the proof.

68 4.4 Holder¨ continuity in time variable

In this section we show that for any ﬁxed y ∈ R, Xt(y) is Holder¨ continuous in t with any exponent in (0,1/4).

2 2 Proposition 4.8. Suppose that h ∈ H2 (R) and X0 has a bounded density µ ∈ L (R). Then, for any p ≥ 1, 0 ≤ s < t ≤ T and y ∈ R,

2p −p p − 1 E |Xt (y) − Xs (y)| ≤ C(1 +t )(t − s) 2 4 ,

where the constant C depending only on p,T, khk and k k 2 . 2,2 µ L (R)

We need some preparations to prove the above result.

Suppose 0 < s < t. We start by estimating X·,2 (y) in (4.6) and we write

Z s Z W W Xt,2 (y) − Xs,2 (y) = p (r,x;t,y) − p (r,x;s,y) Z (drdx) 0 R Z t Z + pW (r,x;t,y)Z (drdx). (4.42) s R

We are going to estimate the two terms separately.

Lemma 4.9. For any 0 ≤ s < t ≤ T, y ∈ R and p ≥ 1, we have

Z s Z 2p p − 1 E pW (r,x;t,y) − pW (r,x;s,y)Z (drdx) ≤ C (t − s) 2 4 . (4.43) 0 R

Proof. From (4.32), we have for 0 < r < s < t ≤ T,

W W B p (r,x;t,y) − p (r,x;s,y) = E 1{ξt >y}δ (ut) − 1{ξs>y}δ (us) B = E 1{ξt >y} − 1{ξs>y} δ (ut) + 1{ξs>y}δ (ut − us) ,

69 Let I1 ≡ 1{ξt >y} − 1{ξs>y} δ (ut) and I2 ≡ 1{ξs>y}δ (ut − us). Then (4.41) implies

Z s Z 2p E pW (r,x;t,y) − pW (r,x;s,y)Z (drdx) 0 R p " Z s Z 2p−1# 2p−1 B 2 ≤ E E [I1 + I2] drdx 0 R p " Z s Z 2p−1# 2p−1 B 2 ≤ C ∑ E E Ii drdx . (4.44) i=1,2 0 R

First, we study the term I1. Note that

2 1{ξt >y} − 1{ξs>y} = 1{ξs≤y<ξt } + 1{ξt ≤y<ξs} =: A1 + A2.

Then we can write

1 " Z s Z 2p−1# 2p−1 B 2 E E I1 drdx 0 R 1 " Z s Z 2p−1# 2p−1 B 2 = E E [(A1 + A2)δ (ut)] drdx 0 R 1 " Z s Z 2p−1# 2p−1 B 2 ≤ 2 ∑ E E [Aiδ (ut)] drdx . (4.45) i=1,2 0 R

Applying Minkowski, Jensen and Holder’s¨ inequalities we deduce that for i = 1,2 and for any conjugate pair (p1,q1)

1 " Z s Z 2p−1# 2p−1 B 2 E E [Aiδ (ut)] drdx 0 R 1 Z s Z 2p−1! 2p−1 2 ≤ E Ai |δ (ut)| dx dr 0 R

70 1 1 Z s Z p1 Z q1 2q1 ≤ Aidx Ai |δ (ut)| dx dr 0 R R 2p−1 1 1 Z s Z p1 Z q1 2q1 ≤ Aidx Ai |δ (ut)| dx dr. (4.46) 0 R 2(2p−1) R 2(2p−1)

Notice that

t t s s {ξs ≤ y < ξt} = y − Br − Ir(h) < x ≤ y − Br − Ir (h) ,

s s t t {ξt ≤ y < ξs} = y − Br − Ir (h) < x ≤ y − Br − Ir(h) .

Then, for i = 1,2, we have Z t t Aidx = Bs + Is(h) . R

1 Hence for p1 = 1 − 2p ,

1 Z p 1 1 − 1 Aidx ≤ C (t − s) 2 4p . (4.47)

R 2(2p−1)

On the other hand, we have

s s t t {ξs ≤ y < ξt} = Br + Ir (h) ≤ y − x < Br + Ir(h)

t t s s ⊂ |x − y| ≤ Br + Ir(h) + |Br + Ir (h)| .

Similarly t t s s {ξt ≤ y < ξs} ⊂ |x − y| ≤ Br + Ir(h) + |Br + Ir (h)| .

Applying Chebyshev’s inequality and (4.35), we deduce that for i = 1,2,

B t t s s E (Ai) ≤ EP |x − y| ≤ Br + Ir(h) + |Br + Ir (h)|

71 ! ! −(x − y)2 |Bt + It(h)|2 |Bs + Is(h)|2 ≤ exp E exp r r + r r 32c(t − r) 16c(t − r) 16c(s − r) ! (x − y)2 ≤ 2exp − . (4.48) 32c(t − r)

Using Minkowski and Holder’s¨ inequalities, from (4.48) and Lemma 4.4 we obtain that for q1 = 2p ≤ 2(2p − 1),

1 1 ! q1 Z q1 Z A |δ (u )|2q1 dx ≤ A |δ (u )|2q1 dx i t i t 2(2p−1) R 2(2p−1) R q1 1 Z q1 q1 1 4(2p−1) 2q1 4p −1 ≤ (EAi) kδ (ut)k8(2p−1) dx ≤ C (t − r) . (4.49) R

Substituting (4.47) and (4.49) into (4.46) we obtain

1 " Z s Z 2p−1# 2p−1 B 2 E E [Aiδ (ut)] drdx 0 R Z s 1 − 1 1 −1 1 − 1 ≤ C (t − s) 2 4p (t − r) 4p dr ≤ C (t − s) 2 4p . (4.50) 0

Combining (4.45) and (4.50), we have

1 " 2p−1# 2p−1 Z s Z 1 1 B 2 2 − 4p E E I1 drdx ≤ C (t − s) . (4.51) 0 R

We turn into the term I2. From Lemma 2.2 we can deduce as in Lemma 4.6 that

1 2 ! 2(2p−1) 2p−1 −(x − y) E EBI ≤ 2exp kδ (u − u )k2 . 2 32(2p − 1)c(s − r) t s 4(2p−1)

72 Then applying Minkowski’s inequality and Lemma 4.5, we obtain

1 " Z s Z 2p−1# 2p−1 Z s Z 1 B 2 B 2(2p−1) 2p−1 E E I2 drdx ≤ E E I2 drdx 0 R 0 R Z s Z 2 ! (x − y) 2 ≤ 2 exp − kδ (ut − us)k4(2p−1) dxdr 0 R 32(2p − 1)(s − r) Z s 1 −1 −1 1 − 1 ≤ C (t − s) (s − r) 2 (t − r) dr ≤ C (t − s) 2 4p , (4.52) 0

−1 − 1 − − 1 + where in the last step we used that (t − r) ≤ (t − s) 2 ε (s − r) 2 ε for any ε > 0. Substituting (4.51) and (4.52) in (4.44) we obtain (4.43).

Lemma 4.10. For any 0 ≤ s < t ≤ T and any y ∈ R and p ≥ 1, we have

2p Z t Z p E pW (r,x;t,y)Z (drdx) ≤ C (t − s) 2 . (4.53) s R

Proof. Since µ is bounded, it follows from [25, Lemma 4.1] that

p Z t Z 2p Z t Z 2p−1! 2p−1 W 2 W 2 E p (r,x;t,y) Z (drdx) ≤ C E p (r,x;t,y) drdx , s R s R (4.54) for any p ≥ 1 and y ∈ R. Applying Minkowski’s inequality, Lemma 4.6 and Lemma 4.4 we obtain

1 " Z s Z 2p−1# 2p−1 W 2 E p (r,x;t,y) drdx 0 R Z t Z 1 W 2(2p−1) 2p−1 ≤ C E p (r,x;t,y) drdx s R Z t Z 2 ! (x − y) 2 ≤ C exp − kδ (ut)k4(2p−1) drdx s R 32c(t − r) Z t 1 −1 1 ≤ C (t − r) 2 dr ≤ C (t − s) 2 . s

73 Then (4.53) follows immediately.

In summary of the above two lemmas, we get

Proposition 4.11. For any p ≥ 1, 0 ≤ s < t ≤ T and y ∈ R, we have

2p p − 1 E Xt,2 (y) − Xs,2 (y) ≤ C (t − s) 2 4 .

Now we consider Xt,1 (y). Note that

Z 2p 2p W W E Xt,1 (y) − Xs,1 (y) = E p (0,z;t,y) − p (0,z;t,y) µ(z)dz R Z 2p B = E E 1{ξt >y}δ (ut) − 1{ξs>y}δ (us) µ(z)dz . R

Then, similar to the proof for X·,2 (y) we get estimates for X·,1 (y).

Proposition 4.12. For any p ≥ 1, 0 ≤ s < t ≤ T and any y ∈ R, we have

2p −p 1 p E Xt,1 (y) − Xs,1 (y) ≤ C 1 +t (t − s) 2 . (4.55)

Proof. Let I1 ≡ 1{ξt >y} − 1{ξs>y} δ (ut) and I2 ≡ 1{ξs>y}δ (ut − us). Then,

Z 2p 2p B E Xt,1 (y) − Xs,1 (y) = E µ(x)E [I1 + I2]dx . R

Noticing that 1{ξt >y} − 1{ξs>y} = 1{ξs≤y<ξt } + 1{ξt ≤y<ξs} =: A1 + A2, and applying Fubini’s theorem, Jensen, Holder¨ and Minkowski’s inequalities, we obtain

Z 2p Z 2p B B E µ(x)E |I1|dx ≤ ∑ E µ(x)E [Aiδ (ut)]dx R i=1,2 R Z p Z p 2 ≤ ∑ E[ |µ(x)δ (ut)| dx Aidx ] i=1,2 R R

74 Z p 1 2 2 t t 2p 2 ≤ ∑ |µ(x)| kδ (ut)k4p dx E Bs + Is(h) i=1,2 R 2 2p 1 p k k k k −p 2 ≤ C 1 + h µ L2 t (t − s) .

For the term I2, using Minkowski’s inequality, (4.31) and (4.25) with r = 0 we have

Z 2p Z 1 2p B B 2p 2p E µ(x)E |I2|dx = |µ(x)| E E 1{ξs>y}δ (ut − us) dx R R 2p Z 2 ! ! 2p (x − y) ≤ C kµk∞ exp − kδ (ut − us)k4p dx R 32cs 2p −p p ≤ C kµk∞ t (t − s) .

Then we can conclude (4.55).

Proof of Proposition 4.8. It follows from Proposition 4.11 and Proposition 4.12.

Proof of Theorem 4.1. It follows from Proposition 4.7 and Proposition 4.8.

75 Chapter 5

Convergence of densities of some functionals of Gaussian processes

The aim of this chapter is to establish the uniform convergence of the densities of a sequence of random variables, which are functionals of an underlying Gaussian process, to a normal density. Precise estimates for the uniform distance are derived by using the techniques of Malliavin calculus, combined with Stein’s method for normal approximation. We need to assume some non-degeneracy conditions. First, the study is focused on random variables in a ﬁxed Wiener chaos, and later, the results are extended to the uniform convergence of the derivatives of the densities and to the case of random vectors in some ﬁxed chaos, which are uniformly non-degenerate in the sense of Malliavin calculus. Explicit upper bounds for the uniform norm are obtained for random variables in the second Wiener chaos, and an application to the convergence of densities of the least square estimator for the drift parameter in Ornstein-Uhlenbeck processes is discussed.

5.1 Introduction

There has been a recent interest in studying normal approximations for sequences of multiple stochastic integrals. Consider a sequence of multiple stochastic integrals of

76 2 order q ≥ 2, Fn = Iq( fn), with variance σ > 0, with respect to an isonormal Gaussian process X = {X(h),h ∈ H} associated with a Hilbert space H. It was proved by Nualart and Peccati [41] and Nualart and Ortiz-Latorre [40] that Fn converges in distribution to the normal law N(0,σ 2) as n → ∞ if and only if one of the following three equivalent conditions holds:

4 4 (i) limn→∞ E[Fn ] = 3σ (convergence of the fourth moments).

(ii) For all 1 ≤ r ≤ q − 1, fn ⊗r fn converges to zero, where ⊗rdenotes the contraction of order r (see equation (2.5)).

2 2 2 (iii) kDFnkH (see deﬁnition in Section 2) converges to qσ in L (Ω) as n tends to inﬁnity.

A new methodology to study normal approximations and to derive quantitative results combining Stein’s method with Malliavin calculus was introduced by Nourdin and Peccati [34] (see also Nourdin and Peccati [35]). As an illustration of the power of this method, let us mention the following estimate for the total variation distance between the

2 2 2 law L (F) of F = Iq( f ) and distribution γ = N(0,σ ), where σ = E[F ]:

√ q q 2 2 2 q − 1 4 4 dTV (L (F),γ) ≤ Var kDFk ≤ √ E[F ] − 3σ . qσ 2 H σ 2 3q

This inequality can be used to show the above equivalence (i)-(iii). A recent result of Nourdin and Poly [38] says that the convergence in law for a sequence of multiple stochastic integrals of order q ≥ 2 is equivalent to the convergence in total variation if the limit is not constant. As a consequence, for a sequence Fn of nonzero multiple stochastic integrals of order q ≥ 2, the limit in law to is equivalent to the limit of the densities in

L1(R), provided the limit is not constant. A multivariate extension of this result has been derived in [33].

77 The aim of this paper is to study the uniform convergence of the densities of a sequence of random vectors Fn to the normal density using the techniques of Malliavin calculus, combined with Stein’s method for normal approximation. It is well-known that to guarantee that each Fn has a density we need to assume that the norm of the

Malliavin derivative of Fn has negative moments. Thus, a natural assumption to obtain uniform convergence of densities is to assume uniform boundedness of the negative moments of the corresponding Malliavin derivatives. Our ﬁrst result (Theorem 5.10) says that if F is a multiple stochastic integral of order q ≥ 2 such that E[F2] = σ 2 and

−6 M := E(kDFkH ) < ∞, we have

q 4 4 sup| fF (x) − φ(x)| ≤ C E[F ] − 3σ , (5.1) x∈R

2 where fF is the density of F, φ is the density of the normal law N(0,σ ) and the constant C depends on q, σ and M. We can also replace the expression in the right-hand side of

q 2 (5.1) by Var kDFkH . The main idea to prove this result is to express the density of F using Malliavin calculus:

−2 −2 fF (x) = E[1{F>x}qkDFkH F] − E[1{F>x}hDF,D(kDFkH )iH].

−2 Then, one can ﬁnd an estimate of the form (5.1) for the terms E[|hDF,D(kDFkH )iH|] −2 −2 and E[|qkDFkH − σ |]. On the other hand, taking into account that

−2 φ(x) = σ E[1{N>x}N], it sufﬁces to estimate the difference

E[1{F>x}F] − E[1{N>x}N],

78 which can be done by Stein’s method. The estimate (5.1) leads to the uniform convergence of the densities in the above equivalence of conditions (i) to (iii) if we assume that

−6 supn E(kDFnkH ) < ∞.

This methodology is extended in the paper in several directions. We consider the uniform approximation of the mth derivative of the density of F by the corresponding densities φ (m), in the case of random variables in a ﬁxed chaos of order q ≥ 2. In The- −β orem 5.13 we obtain an inequality similar to (5.1) assuming that E(kDFkH ) < ∞ for m some β > 6m + 6 b 2 c ∨ 1 . Again the proof is obtained by a combination of Malliavin calculus and the Stein’s method. Here we need to consider Stein’s equation for functions of the form of h(x) = 1{x>a} p(x), where p is a polynomial.

For a d dimensional random vector F = (F1,...,Fd) whose components are multiple stochastic integrals of orders q1,...,qd, qi ≥ 2, we assume non degeneracy condition −p E[detγF ] < ∞ for all p ≥ 1, where γF = (hDF,,DFi)1≤i, j≤d denotes the Malliavin matrix of F. Then, for any multi-index β = (β1,...,βk), 1 ≤ βi ≤ d, we obtain the estimate (see Theorem 5.17)

d ! 1 q 2 4 2 2 sup ∂β fF (x) − ∂β φ(x) ≤ C |V − I| + ∑ E[Fj ] − 3(E[Fj ]) , x∈Rd j=1 where V is the covariance matrix of F, φ is the standard d dimensional normal den-

∂ k sity, and ∂β = ∂x ···∂x . As a consequence, we derive the uniform convergence of the β1 βk densities and their derivatives for a sequence of vectors of multiple stochastic integrals, under the the assumption sup E[det −p] < ∞ for all p ≥ 1. A multivariate extension n γFn of Stein’s method is required for noncontinuous functions with polynomial growth (see Proposition 5.25). While univariate Stein’s equations with nonsmooth test functions have been extensively studied, relatively few results are available for the multivariate case, see [5,4, 28, 37, 48, 50], so this result has its own interest.

79 We also consider the case of random variables F such that E[F] = 0 and E[F2] = σ 2,

,s belonging to the Sobolev space D2 for some s > 4. In this case, under a nondegeneracy −1 −r assumption of the form E[|hDF,−DFL FiH| |] < ∞ for some r > 2, we derive an estimate for the uniform distance between the density of F and the density of the normal law N(0,σ 2).

The chapter is organized as follows. Section 5.2 brieﬂy introduces Stein’s method for normal approximations. Section 5.3 is devoted to density formulae with elementary estimates using Malliavin calculus. The density formulae themselves are well-known results, but we present explicit formulae with useful estimates, such as the Holder¨ continuity and boundedness estimates in theorems5 .3 and5 .5. The boundedness estimates enable us to prove the Lp convergence of the densities (see (5.43)). The Holder¨ continuity estimates can be used to provide a short proof for the convergence of densities based on a compactness argument, assuming convergence in law (see Theorem 5.31). Section 5.4 proves the convergence of densities of random variables in a ﬁxed Wiener chaos, and Section 5.5 discusses convergence of densities for random vectors. In Section 5.6, the convergence of densities for sequences of general centered square integrable random variables are studied.

The main difficulty in the application of the above results is to verify the existence of negative moments for the determinant of the Malliavin matrix. We provide explicit sufficient conditions for this condition for random variables in the second Wiener chaos in Section 5.7. As an application we derive the uniform convergence of the densities and their derivatives to the normal distribution, as time goes to infinity, for the least squares estimator of the parameter θ in the Ornstein-Uhlenbeck process: dXt = −θXtdt + γdBt, where B = {Bt,t ≥ 0} is a standard Brownian motion. Some technical results and proofs are included in Section 5.8.

80 Along this paper, we denote by C (maybe with subindices) a generic constant that might depend on quantities such as the order of multiple stochastic integrals q, the order of the derivatives m, the variance σ 2 or the negative moments of the Malliavin derivative.

p We denote by k·kp the norm in the space L (Ω).

5.2 Stein’s method of normal approximation

We shall now give a brief account of Stein’s method of univariate normal approximation and its connection with Malliavin calculus. For a more detailed exposition we refer to [5, 35, 52].

Let F be an arbitrary random variable and let N be a N(0,σ 2) distributed random variable, where σ 2 > 0. Consider the distance between the law of F and the law of N given by

dH (F,N) = sup |E[h(F) − h(N)]| (5.2) h∈H for a class of functions H such that E[h(F)] and E[h(N)] are well-deﬁned for h ∈ H . Notice ﬁrst the following fact (which is usually referred as Stein’s lemma): a random variable N is N(0,σ 2) distributed if and only if E[σ 2 f 0(N) − N f (N)] = 0 for all abso- lutely continuous functions f such that E[| f 0(N)|] < ∞. This suggests that the distance of

E[σ 2 f 0(F) − F f (F)] from zero may quantify the distance between the law of F and the law of N. To see this, for each function h such that E[|h(N)|] < ∞, Stein [52] introduced the Stein’s equation: x f 0(x) − f (x) = h(x) − E[h(N)] (5.3) σ 2

81 for all x ∈ R. For a random variable F such that E[|h(F)|] < ∞, any solution fh to Equation (5.3) veriﬁes

1 E[σ 2 f 0(F) − F f (F)] = E[h(F) − h(N)], (5.4) σ 2 h h and the distance deﬁned in (5.2) can be written as

1 2 0 dH (F,N) = 2 sup E[σ fh(F) − F fh(F)] . (5.5) σ h∈H

−x2/(2σ 2) The unique solution to (5.3) verifying limx→±∞ e f (x) = 0 is

Z x x2/(2σ 2) −y2/(2σ 2) fh(x) = e {h(y) − E[h(N)]}e dy. (5.6) −∞

From (5.5) and (5.6), one can get bounds for probability distances like the total variation distance, where we let H consist of all indicator functions of measurable sets, Kolmogorov distance, where we consider all the half-line indicator functions and Wasser- stein distance, where we take H to be the set of all Lipschitz-continuous functions with Lipschitz-constant equal to 1.

In the present chapter, we shall consider the case when h : R → R is given by h(x) =

1{x>z}Hk(x) for any integer k ≥ 1 and z ∈ R, where Hk(x) is the kth Hermite polynomial. More generally, we have the following lemma whose proof can be found in the Appendix.

Lemma 5.1. Suppose |h(x)| ≤ a|x|k + b for some integer k ≥ 0 and some nonnegative numbers a,b. Then, the solution fh to the Stein’s equation (5.3) given by (5.6) satisﬁes

k 0 k−i i fh(x) ≤ aCk ∑ σ |x| + 4b i=0 for all x ∈ R, where Ck is a constant depending only on k.

82 Nourdin and Peccati [34, 35] combined Stein’s method with Malliavin calculus to estimate the distance between the distributions of regular functionals of an isonormal

Gaussian process and the normal distribution N(0,σ 2). The basic ingredient is the fol-

, lowing integration by parts formula. For F ∈ D1 2 with E[F] = 0 and any function f ∈ C1 such that E[| f 0(F)|] < ∞, using (2.10) and (2.6) we have

E[F f (F)] = E[LL−1F f (F)] = E[−δDL−1F f (F)]

−1 0 −1 = E[ −DL F,D f (F) ] = E[ f (F) −DL F,DF H].

Then, it follows that

2 0 0 2 −1 E[σ f (F) − F f (F)] = E[ f (F)(σ − DF,−DL F H)]. (5.7)

Combining Equation (5.7) with (5.4) and Lemma 5.1 we obtain the following result.

k Lemma 5.2. Suppose h : R → R veriﬁes |h(x)| ≤ a|x| + b for some a,b ≥ 0 and some 2 1,2k integer k ≥ 0. Let N ∼ N(0,σ ) and let F ∈ D with kFk2k ≤ cσ for some c > 0. Then there exists a constant Ck,c depending only on k and c such that

−2 k 2 −1 |E[h(F) − h(N)]| ≤ σ [aCk,cσ + 4b] σ − DF,−DL F . H 2

k k−i i Proof. From (5.4), (5.7) and Lemma (5.1), it sufﬁces to notice that ∑i=0 σ |F| ≤ 2 k i k−i k ∑i=0 kFk2k σ ≤ Ck,cσ .

5.3 Density formulae

In this section, we present explicit formulae for the density of a random variable and its derivatives, using the techniques of Malliavin calculus.

83 5.3.1 Density formulae

We shall present two explicit formulae for the density of a random variable, with estimates of its uniform and Holder¨ norms.

2,s 2p −2r Theorem 5.3. Let F ∈ D such that E[|F| ] < ∞ and E[kDFkH ] < ∞ for p,r,s > 1 1 1 1 satisfying p + r + s = 1. Denote

2 −1 w = kDFkH , u = w DF.

1,p0 0 p Then u ∈ D with p = p−1 and F has a density given by

fF (x) = E 1{F>x}δ (u) . (5.8)

1 Furthermore, fF (x) is bounded and Holder¨ continuous of order p , that is

−1 −2 2 fF (x) ≤ Cp w r kFk2,s 1 ∧ (|x| kFk2p) , (5.9)

1+ 1 1+ 1 1 −1 p p p | fF (x) − fF (y)| ≤ Cp w r kFk2,s |x − y| (5.10) for any x,y ∈ R, where Cp is a constant depending only on p.

Proof. Note that −1 2 −2 2 Du = w D F − 2w D F ⊗1 DF ⊗ DF.

Applying Meyer’s inequality (2.9) and Holder’s¨ inequality we have

kδ (u)kp0 ≤ Cp kuk1,p0 ≤ Cp(kukp0 + kDukp0 )

−1 −1 2 ≤ Cp( w kDFkH 0 + 3 w D F ) p H⊗H p0 −1 2 ≤ 3Cp w r (kDFks + D F s). (5.11)

84 ,p0 Then u ∈ D1 and the density formula (5.8) holds (see, for instance, Nualart [39, Propo- sition 2.1.1]). From E[δ(u)] = 0 and Holder’s¨ inequality it follows that

1 1 p p −2p 2p E 1{F>x}δ (u) ≤ P(|F| > |x|) kδ (u)kp0 ≤ 1 ∧ (|x| kFk2p) kδ (u)kp0 . (5.12) Then (5.9) follows from (5.12) and (5.11).

Finally, for x < y ∈ R, noticing that 1{F>x} − 1{F>y} = 1{x

1 p | fF (x) − fF (y)| ≤ E[1{x

R y Applying (5.9) and (5.11) with the fact that E[1{x

With the exact proof of [39, Propositions 2.1.1], one can prove the following slightly more general result.

1,p Proposition 5.4. Let F ∈ D and h : Ω → H, and suppose that hDF,hiH 6= 0 a.s. and h ∈ 1,q(H) for some p,q > 1. Then the law of F has a density given by hDF,hiH D

h fF (x) = E 1{F>x}δ . (5.13) hDF,hiH

Our next goal is to take h to be −DL−1F in formula (5.13) and get a result similar to

−1 0 Theorem 5.3. First, to get a sufﬁcient condition for −DL F ∈ 1,p for some p0 > 1, DF,−DL−1F D h iH we need some technical estimates on DL−1F and D2L−1F. Estimates of this type have been obtained by Nourdin, Peccati and Reinert [36] (see also Nourdin and Peccati’s book [35, Lemma 5.3.8]), when proving an inﬁnite-dimensional Poincare´ inequality. More

85 ,p precisely, by using Mehler’s formula, they proved that for any p ≥ 1, if F ∈ D2 , then

−1 p p E[ DL F H] ≤ E[kDFkH]. (5.14)

2 −1 p −p 2 p E[ D L F op] ≤ 2 E[ D F op], (5.15)

2 where D F op denotes the operator norm of the Hilbert-Schmidt operator from H to 2 2 H : f 7→ f ⊗1 D F. Furthermore, the operator norm D F op satisﬁes the following “random contraction inequality”

2 4 2 2 2 2 4 D F op ≤ D F ⊗1 D F H⊗2 ≤ D F H⊗2 . (5.16)

The next proposition gives a density formula with estimates similar to Theorem 5.3. Let −1 −1 −1 w¯ = DF,−DL F H , u¯ = −w¯ DL F.

,s 2p −r Proposition 5.5. Let F ∈ D2 ,E[|F| ] < ∞ and suppose that E[|w¯| ] < ∞, where p > 1, 1 2 3 1,p0 0 p r > 2, s > 3 satisfy p + r + s = 1. Then u¯ ∈ D with p = p−1 and the law of F has a density given by fF (x) = E 1{F>x}δ (u¯) . (5.17)

1 Furthermore, fF (x) is bounded and Holder¨ continuous of order p , that is

−2 2 fF (x) ≤ K0 1 ∧ (|x| kFk2p) , (5.18)

1+ 1 1 p p | fF (x) − fF (y)| ≤ K0 |x − y| (5.19)

−1 −1 2 for any x,y ∈ R, where K0 = Cp w¯ r kFk2,s ( w¯ r kDFks + 1), and Cp depends only on p.

86 2 −1 2 −1 Proof. Note that Dw¯ = −D F ⊗1 DL F − DF ⊗1 D L F. Then, applying (5.14) and (5.15) we obtain

−s 2 kDw¯k s ≤ (1 + 2 ) D F kDFks . (5.20) 2 op s

Fromu ¯ = −w¯−1DL−1F we get Du¯ = −w¯−1D2L−1F + w−2Dw¯ ⊗ DL−1F. Then, using

1 1 1 (5.14)–(5.16) we have for t > 0 satisfying p0 = r + t ,

−1 −1 −1 ku¯kp0 ≤ w¯ DL F ≤ w¯ kDFkt , H p0 r and

−1 2 −1 −2 −1 kDu¯kp0 ≤ w¯ D L F + w¯ kDw¯kH DL F H⊗H p0 H p0 −1 2 −2 ≤ w¯ D F + w¯ kDw¯k s kDFk ,. r t r 2 s

2 2 Noticing that D F t ≤ D F s because t < s, and applying Meyer’s inequality (2.9) with (5.20) and (5.16) we obtain

kδ (u¯)kp0 ≤ Cp ku¯k1,p0 ≤ K0. (5.21)

,p0 Then u ∈ D1 and the density formula (5.17) holds. As in the proof of Theorem 5.3, (5.18) and (5.19) follow from (5.21) and

1 p −2 2 E 1{F>x}δ (u¯) ≤ P(|F| > |x|) kδ (u¯)kp0 ≤ 1 ∧ (|x| kFk2p) kδ (u¯)kp0 ,

1 p | fF (x) − fF (y)| ≤ E[1{x

87 5.3.2 Derivatives of the density

Next we present a formula for the derivatives of the density function, under additional conditions. A sequence of recursively deﬁned random variables given by G0 = 1 and

Gk+1 = δ(Gku) where u is an H-valued process, plays an essential role in the formula.

The following technical lemma gives an explicit formula for the sequence Gk, relating it to Hermite polynomials. To simplify the notation, for an H–valued random variable u, we denote

k D k−1 E δu = δ(u), DuG = hDG,uiH , DuG = D Du G ,u . (5.22) H

Recall Hk(x) denotes the kth Hermite polynomial. For λ > 0 and x ∈ R, we deﬁne the generalized kth Hermite polynomial as

k x Hk (λ,x) = λ 2 Hk(√ ). (5.23) λ

0 From the property Hk(x) = kHk−1(x) it follows by induction that the kth Hermite poly- k−2i nomials has the form Hk(x) = ∑0≤i≤bk/2c ck,ix , where we denote by bk/2c the largest integer less than or equal to k/2. Then (5.23) implies

k−2i i Hk(λ,x) = ∑ ck,ix λ . (5.24) 0≤i≤bk/2c

m,p Lemma 5.6. Fix an integer m ≥ 1 and a number p > m. Suppose u ∈ D (H). We deﬁne m recursively a sequence {Gk}k=0 by G0 = 1 and Gk+1 = δ(Gku). Then, these variables m−k, p are well-deﬁned and for k = 1,2,...,m, Gk ∈ D k and

Gk = Hk(Duδu,δu) + Tk, (5.25)

88 where we denote by Tk the higher order derivative terms which can be deﬁned recursively as follows: T1 = T2 = 0 and for k ≥ 2,

2 Tk+1 = δuTk − DuTk − ∂λ Hk(Duδu,δu)Duδu. (5.26)

The following remark is proved in the Appendix.

Remark 5.7. From (5.26) we can deduce that for k ≥ 3

i ik−1 i0 i1 2 2 k−1 Tk = ∑ ai0,i1,...,ik−1 δu (Duδu) Duδu ··· Du δu , (5.27) (i0,...,ik−1)∈Jk

where the coefﬁcients ai0,i1,...,ik−1 are real numbers and Jk is the set of multi-indices k (i0,i1,...,ik−1) ∈ N satisfying the following three conditions

k−1 k−1 k − 1 (a) i0 + ∑ ji j ≤ k − 1; (b) i2 + ··· + ik−1 ≥ 1; (c) ∑ i j ≤ b c. j=1 j=1 2

j From (b) we see that every term in Tk contains at least one factor of the form Duδu with some j ≥ 2. We shall show this type of factors will converge to zero. For this reason we call these terms high order terms.

Proof of Lemma 5.6. First, we prove by induction on k that the above sequence Gk is m−k, p well-deﬁned and Gk ∈ D k . Suppose ﬁrst that k = 1. Then, Meyer’s inequality implies m−1,p m−k, p that G1 = δu ∈ D . Assume now that for k ≤ m − 1, Gk ∈ D k . Then it follows from Meyer’s and Holder’s¨ inequalities (see [39, Proposition 1.5.6]) that

0 kGk+1k p ≤ Cm,p kGkuk p ≤ C kGkk p kuk < ∞. m−k−1, k+1 m−k, k+1 m,p m−k, k m−k,p

Let us now show, by induction, the decomposition (5.25). When k = 1 (5.25) is true because G1 = δu and T1 = 0. Assume now (5.25) holds for k ≤ m − 1. Noticing that

89 0 ∂xHk(λ,x) = kHk−1(λ,x) (since Hk(x) = kHk−1(x)), we get

2 DuHk(Duδu,δu) = kHk−1(Duδu,δu)Duδu + ∂λ Hk(Duδu,δu)Duδu.

Hence, applying the operator Du to both sides of (5.25),

DuGk = kHk−1(Duδu,δu)Duδu + Tek+1, where

2 Tek+1 = DuTk + ∂λ Hk(Duδu,δu)Duδu. (5.28)

From the deﬁnition of Gk+1 and using (2.7) we obtain

Gk+1 = δ(uGk) = Gkδu − DuGk

= δuHk(Duδu,δu) + δuTk − kHk−1(Duδu,δu)Duδu − Tek+1.

Note that Hk+1(x) = xHk(x)−kHk−1(x) implies xHk(λ,x)−kλHk−1(λ,x) = Hk+1(λ,x). Hence,

Gk+1 = Hk+1(Duδu,δu) + δuTk − Tek+1.

The term Tk+1 = δuTk −Tek+1 has the form given in (5.26). This completes the proof.

Now we are ready to present some formulae for the derivatives of the density function under certain sufﬁcient conditions on the random variable F. For a random variable F in

, D1 2 and for any β ≥ 1 we are going to use the notation

1 −β β Mβ (F) = E[kDFkH ] . (5.29)

90 m+ , Proposition 5.8. Fix an integer m ≥ 1. Let F be a random variable in D 2 ∞ such m 2 DF that Mβ (F) < ∞ for some β > 3m + 3(b 2 c ∨ 1). Denote w = kDFkH and u = w . m+1,p m+1 Then, u ∈ D (H) for some p > 1, and the random variables {Gk}k=0 introduced in Lemma 5.6 are well-deﬁned. Under these assumptions, F has a density f of class Cm with derivatives given by

(k) k fF (x) = (−1) E[1{F>x}Gk+1] (5.30) for k = 1,...,m.

m+1 Proof. It is enough to show that {Gk}k=0 are well-deﬁned, since it follows from [39, Exercise 2.1.4] that the kth derivative of the density of F is given by (5.30). To do this

1 we will show that Gk deﬁned in (5.25) are in L (Ω) for all k = 1,...,m + 1. From (5.25) we can write

E[|Gk|] ≤ E[|Hk(Duδu,δu)|] + E[|Tk|].

Recall the explicit expression of Hk(λ,x) in (5.24). Since β > 3(m + 1), we can choose β β r0 < 3 ,r1 < 6 such that

k − 2i i 3(k − 2i) 6i 3k 1 ≥ + > + = , r0 r1 β β β for any 0 ≤ i ≤ bk/2c and 1 ≤ k ≤ m+1. Then, applying Holder’s¨ inequality with (5.24), (5.147) and (5.148) we have

E[|H (D δ ,δ )|] ≤ C kδ kk−2i kD δ ki < ∞. k u u u k ∑ u r0 u u r1 0≤i≤bk/2c

91 To prove that E [|Tk|] < ∞, applying Holder’s¨ inequality to the expression (5.27) and choosing r j > 0 for 0 ≤ j ≤ k − 1 such that

i k−1 i 3i k−1 (3 j + 3)i 1 ≥ 0 + ∑ j > 0 + ∑ j , r0 j=1 r j β j=1 β we obtain that, (assuming k ≥ 3, otherwise Tk = 0)

k−1 i0 j i j E [|Tk|] ≤ C kδuk D δu . ∑ r0 ∏ u r j (i0,...,ik)∈Jk j=1

β Due to (5.147) and (5.148), this expression is ﬁnite, provided r j < 3 j+3 for 0 ≤ j ≤ k−1.

We can choose (r j,0 ≤ j ≤ k−1) satisfying the above conditions because β > 3(k−1)+ k−1 3b 2 c for all 1 ≤ k ≤ m+1, and from properties (a) and (c) of Jk in Remark 5.7 we have

3i k−1 (3 j + 3)i 3(k − 1) + 3b k−1 c 0 + ∑ j ≤ 2 . β j=1 β β

This completes the proof.

Example 5.9. Consider a random variable in the ﬁrst Wiener chaos N = I1(h), where ∈ 2 H with khkH = σ. Then N has the normal distribution N ∼ N(0,σ ) with density denoted by (x). Clearly kDNk = , u = h , = N and D = h . Then G = H ( 1 , N ) φ H σ σ 2 δu σ 2 uδu σ 2 k k σ 2 σ 2 and from (5.30) we obtain the formula

1 N φ (k)(x) = (−1)kE 1 H , , (5.31) {N>x} k+1 σ 2 σ 2 which can also be obtained by analytic arguments.

92 5.4 Random variables in the qth Wiener chaos

In this section we establish our main results on uniform estimates and uniform convergence of densities and their derivatives. We shall deal ﬁrst with the convergence of densities and later we consider their derivatives.

5.4.1 Uniform estimates of densities

q Let F = Iq( f ) for some f ∈ H and q ≥ 2. To simplify the notation, along this section we denote

2 −1 w = kDFkH , u = w DF.

Note that LF = −qF and using (2.7) and (2.10) we can write

−1 −1 δu = δ(u) = qFw − Dw ,DF H . (5.32)

q Theorem 5.10. Let F = Iq( f ), q ≥ 2, for some f ∈ H be a random variable in the

2 2 qth Wiener chaos with E[F ] = σ . Assume that M6(F) < ∞, where M6(F) is deﬁned in 2 (5.29). Let φ(x) be the density of N ∼ N(0,σ ). Then F has a density fF (x) given by (5.8). Furthermore,

q 4 4 sup| fF (x) − φ(x)| ≤ C E[F ] − 3σ , (5.33) x∈R

−1 2 3 −3 where the constant C has the form C = Cq σ M6(F) + M6(F) + σ and Cq depends only on q.

k l We begin with a lemma giving an estimate for the contraction D F ⊗1 D F with k + l ≥ 3.

93 2 Lemma 5.11. Let F = Iq( f ) be a random variable in the qth Wiener chaos with E[F ] =

2 σ . Then for any integers k ≥ l ≥ 1 satisfying k + l ≥ 3, there exists a constant Ck,l,q depending only on k,l,q such that

k l 2 2 D F ⊗1 D F ≤ Ck,l,q qσ − kDFkH . (5.34) 2 2

k Proof. Note that D F = q(q − 1)···(q − k + 1)Iq−k( f ). Applying (2.4), we get

k l 2 2 2 D F ⊗1 D F = q (q − 1) ······(q − l + 1) (q − l)···(q − k + 1) q−k q − kq − l × ∑ r! I2q−k−l−2r( f ⊗e r+1 f ). r=0 r r

Taking into account the orthogonality of multiple integrals of different orders, we obtain

2 4 k l (q!) E[ D F ⊗1 D F ] = H⊗(k+l−2) (q − l)!2 (q − k)!2 q−k 2 2 2 q − k q − l 2 × ∑ r! (2q − k − l − 2r)! f ⊗e r+1 f H⊗2q−2−2r . (5.35) r=0 r r

Applying (5.35) with k = l = 1, we obtain

4 2 E[kDFkH] = E[|DF ⊗1 DF| ] (5.36) q−1 4 4 2 q − 1 2 = q ∑ r! (2q − 2 − 2r)! f ⊗e r+1 f H⊗2q−2−2r r=0 r q−2 4 4 2 q − 1 2 2 2 4 = q ∑ r! (2q − 2 − 2r)! f ⊗e r+1 f H⊗2q−2−2r + q q! k f kH⊗q . r=0 r

2 2 2 Taking into account that σ = E[F ] = q!k f kH⊗q , we obtain that for any k + l ≥ 3, there exists a constant Ck,l,q such that

2 k l 2 4 2 4 E[ D F ⊗1 D F ] ≤ Ck,l,qE[kDFkH − q σ ]. H⊗(k+l−2)

94 2 2 2 Meanwhile, it follows from E[kDFkH] = qk f kH⊗q = qσ that

4 2 4 4 2 2 2 4 2 2 2 E[kDFkH − q σ ] = E[kDFkH − 2qσ kDFkH + q σ ] = E[(kDFkH − qσ ) ]. (5.37)

Combining (5.35), (5.36) and (5.37) we have

2 k l 2 2 2 2 E[ D F ⊗1 D F ] ≤ Ck,l,qE[(kDFkH − qσ ) ], H⊗(k+l−2) which completes the proof.

Proof of Theorem 5.10. It follows from Theorem 5.3 that F admits a density fF (x) = E 1 (u). By (5.31) with k = 1 we can write (x) = 1 E[1 N]. Then, using {F>x}δ φ σ 2 {N>x} (5.32), for all x ∈ R we obtain

−2 fF (x) − φ(x) = E 1{F>x}δ (u) − σ E[1{N>x}N] q = E[1 (F( − σ −2) − Dw−1,DF )] + σ −2E F1 − N1 {F>x} w H {F>x} {N>x}

= A1 + A2. (5.38)

For the ﬁrst term A1,Holder’s¨ inequality implies

q |A | = E[1 (F( − σ −2) − Dw−1,DF )] 1 {F>x} w H 3 −2 −1 2 − 2 2 ≤ σ E Fw (w − qσ ) + 2E[w D F ⊗1 DF H] 3 −2 −1 2 − 2 2 ≤ σ w kFk3 w − qσ + 2 w D F ⊗1 DF . 3 3 2 H 2

Note that (2.12) implies

2 2 w − qσ 3 ≤ C w − qσ 2

95 and kFk3 ≤ C kFk2 = Cσ. Combining these estimates with (5.34) we obtain

3 −1 −1 −1 2 2 |A1| ≤ C(σ w 3 + w 3 ) w − qσ 2 . (5.39)

For the second term A2, applying Lemma 5.2 to the function h(z) = z1{z>x}, which satisﬁes |h(z)| ≤ |z|, we have

−2 |A2| = σ E F1{F>x} − N1{N>x}

−3 2 −1 −3 2 ≤ Cσ σ − DF,−DL F ≤ Cσ qσ − w . (5.40) H 2 2

Combining (5.38) with (5.39)–(5.40) we obtain

3 −1 −1 −1 2 −3 2 sup| fF (x) − φ(x)| ≤ C(σ w 3 + w 3 + σ ) w − qσ 2 . x∈R

Then (5.33) follows from (2.13). This completes the proof.

Using the estimates shown in Theorem 5.10 we can deduce the following uniform convergence and convergence in Lp of densities for a sequence of random variables in a ﬁxed qth Wiener chaos.

Corollary 5.12. Let {F } be a sequence of random variables in the qth Wiener chaos n n∈N 2 2 2 2 2 with q ≥ 2. Set σn = E[Fn ] and assume that limn→∞ σn = σ , 0 < δ ≤ σn ≤ K for all n, 4 4 limn→∞ E[Fn ] = 3σ and

1/6 −6 M := sup E[kDFnkH ] < ∞. (5.41) n

96 2 Let φ(x) be the density of the law N(0,σ ). Then, each Fn admits a density fFn ∈ C(R) and there exists a constant C depending only on q,σ,δ and M such that

1 4 4 2 sup| fFn (x) − φ(x)| ≤ C E[Fn ] − 3σn + |σn − σ| . (5.42) x∈R

1 Furthermore, for any p ≥ 1 and α ∈ ( 2 , p),

p−α 1 p 4 4 2 k f − φk p ≤ C E[F ] − 3σ + |σ − σ| , (5.43) Fn L (R) n n n where C is a constant depending on q,σ,M, p,α and K.

2 Proof. Let φn(x) be the density of N(0,σn ). Then Theorem 5.10 implies that

1 4 4 2 sup| fFn (x) − φn(x)| ≤ C E[Fn ] − 3σn . x∈R

2 On the other hand, if Nn ∼ N(0,σn ), it is easy to see that

sup|φn(x) − φ(x)| ≤ C |σn − σ|. x∈R

Then (5.42) follows from triangle inequality. To show (5.43), ﬁrst notice that (5.9) implies −2 fFn (x) ≤ C(1 ∧ |x| ).

1 α Therefore, if α > 2 the function ( fFn (x) + φ(x)) is integrable. Then, (5.43) follows from (5.42) and the inequality

p p−α α | fFn (x) − φ(x)| ≤ | fFn (x) − φ(x)| ( fFn (x) + φ(x)) .

97 5.4.2 Uniform estimation of derivatives of densities

In this subsection, we establish the uniform convergence for derivatives of densities of random variables to a normal distribution. We begin with the following theorem which estimates the uniform distance between the derivatives of the densities of a random variable F in the qth Wiener chaos and the normal law N(0,E[F2]).

Theorem 5.13. Let m ≥ 1 be an integer. Let F be a random variable in the qth Wiener

2 2 m chaos, q ≥ 2, with E[F ] = σ and Mβ := Mβ (F) < ∞ for some β > 6m + 6(b 2 c ∨ 1) 2 (Recall the deﬁnition of Mβ (F) in (5.29)). Let φ(x) be the density of N ∼ N(0,σ ). m Then F has a density fF (x) ∈ C (R) with derivatives given by (5.30). Moreover, for any k = 1,...,m q (k) (k) −k−3 4 2 sup fF (x) − φ (x) ≤ σ C E[F ] − 3σ , x∈R where the constant C depends on q, β, m, σ and Mβ with polynomial growth in σ and

Mβ .

To prove Theorem 5.13, we need some technical results. Recall the notation we introduced in (5.22), where we denote δu = δ(u), Duδu = hDδu,uiH.

Lemma 5.14. Let F be a random variable in the qth Wiener chaos with E[F2] = σ 2. Let

2 −1 w = kDFkH and u = w DF.

2β (i) If Mβ (F) < ∞ for some β > 6, then for any 1 < r ≤ β+6

−2 −1 3 2 δu − σ F r ≤ Cσ (Mβ ∨ 1) qσ − w 2 . (5.44)

2β (ii) If Mβ (F) < ∞ for some β > 12, then for any 1 < r < β+12

−2 −2 6 2 Duδu − σ r ≤ Cσ (Mβ ∨ 1) qσ − w 2 , (5.45)

98 where the constant C depends on σ.

−1 −1 Proof. Recall that δu = qFw − DDF w . Using Holder’s¨ inequality and (5.139) we can write

−2 −2 −1 2 −1 δu − σ F r ≤ σ Fw (qσ − w) r + DDF w r −2 −1 3 2 ≤ C σ kFw ks + (Mβ ∨ 1) qσ − w 2 ,

1 1 1 provided r = s + 2 . By the hypercontractivity property (2.11) kFkγ ≤ Cq,γ kFk2 for any 1 1 1 γ ≥ 2. Thus, by Holder’s¨ inequality, if s = γ + p

−1 −1 2 kFw ks ≤ kFkγ kw kp ≤ Cq,γ σM2p.

Choosing p such that 2p < β we get (5.44).

We can compute Duδu as

−1 −1 −1 −1 2 −1 −1 2 −1 Duδu = qw + qFw DDF w − w DDF w − w D F,DF ⊗ Dw H .

Applying Holder’s¨ inequality we obtain

−2 −1 −2 2 −1 2 −1 Duδu − σ r ≤ w σ qσ − w + qFDDF w − DDF w r −2 −1 2 −1 −1 2 −1 ≤ σ kw k 2r qσ − w +Cσ kw kp DDF w + DDF w , 2−r 2 s s

1 1 1 2β if r > p + s . Then, using (5.139) and (5.140) with k = 2 and assuming that s < β+8 and that 2p < β we obtain (5.45).

m−1 Proof of Theorem 5.13. Proposition 5.8 implies that fF (x) ∈C (R) and for k = 0,1,...,m− 1,

(k) k fF (x) = (−1) E[1{F>x}Gk+1],

99 where G0 = 1 and Gk+1 = δ(Gku) = Gkδ(u) − hDGk,uiH. From (5.31),

(k) k −2 −2 φ (x) = (−1) E[1{N>x}Hk+1(σ ,σ N)].

Then, the identity Gk+1 = Hk+1(Duδu,δu) + Tk+1 (see formula (5.25)), suggests the following triangle inequality

(k) (k) −2 −2 fF (x) − φ (x) = E[1{F>x}Gk+1 − 1{N>x}Hk+1(σ ,σ N)] −2 −2 ≤ E[1{F>x}Gk+1 − 1{F>x}Hk+1(σ ,σ F)]

−2 −2 −2 −2 + E[1{F>x}Hk+1(σ ,σ F) − 1{N>x}Hk+1(σ ,σ N)]

= A1 + A2 .

We ﬁrst estimate the term A2. Note that kFk2k+2 ≤ Cq,k kFk2 = Cq,kσ by the hyper- −2 −2 contractivity property (2.11). Applying Lemma 5.2 with h(z) = 1{z>x}Hk+1(σ ,σ z), k+1 −k−1 which satisﬁes |h(z)| ≤ Ck(|z| + σ ), we obtain

A2 = |E[h(F) − h(N)]|

−2 k −k−1 2 −1 ≤ Cq,kσ σ + 4σ σ − DF,−DL F H 2 −k−3 2 ≤ Cq,k,σ σ qσ − w 2 , (5.46)

−1 w where in the second inequality we used the fact that DF,−DL F H = q .

For the term A1, Lemma 5.6 implies

−2 −2 A1 ≤ E[ Hk+1(Duδu,δu) − Hk+1(σ ,σ F) ] + E[|Tk+1|]. (5.47)

100 To proceed with the ﬁrst term above, applying (5.24) we have

−2 −2 Hk+1(Duδu,δu) − Hk+1(σ ,σ F) (5.48) b k+1 c 2 k+1−2i i −2 k+1−2i −2i ≤ ∑ |ck,i| δu (Duδu) − σ F σ i=0 b k+1 c 2 h i k+1−2i −2 k+1−2i i −2 k+1−2i i −2i ≤ Ck ∑ δu − σ F |Duδu| + σ F (Duδu) − σ . i=0

k k k−1− j j Using the fact that x − y ≤ Ck |x − y|∑0≤ j≤k−1 |x| |y| and applying Holder’s¨ inequality and the hypercontractivity property (2.11) we obtain

1 i k−2i− j 1 i k−2i provided 1 ≥ r + s + p , which is implied by 1 ≥ r + s + p . In order to apply the 1 3 1 1 6 1 3 estimates (5.44), (5.148) (with k = 1) and (5.147) we need r > β + 2 , s > β and p > β , respectively. These are possible because β > 6k + 6. Then we obtain an estimate of the form

h k+1−2i i E k+1−2i − −2F |D |i ≤ C −k(M3k+3 ∨ 1)kq 2 − wk . (5.50) δu σ uδu q,k,σ σ β σ 2

Similarly,

h i −2 k+1−2i i −2i E σ F (Duδu) − σ " # −2 k+1−2i −2 j −2(i−1− j) ≤ Cq,k,σ E σ F Duδu − σ ∑ |Duδu| σ 0≤ j≤i−1

101 −(k−1) −2 j ≤ Cq,k,σ σ Duδu − σ r ∑ kDuδuks , (5.51) 0≤ j≤i−1

1 j provided 1 > r + s . In order to apply the estimates (5.45) and (5.148) (with k = 1) we 1 6 1 1 6 need r > β + 2 and s > β , respectively. This implies

1 j 6 + 6 j 1 + > + . r s β 2

1 3k+3 Notice that 6 + 6 j ≤ 6i ≤ 3k + 3. So, we need 1 > 2 + β . The above r,s and p exist because β > 6k + 6. Thus, we obtain an estimate of the form

h k+1−2i i E −2F (D )i − −2i ≤ C −(k−1)(M3k+3 ∨ 1)kq 2 − wk . (5.52) σ uδu σ q,k,σ,β σ β σ 2

Combining (5.50) and (5.52) we have

−2 −2 −k 3k+3 2 E Hk+1(Duδu,δu) − Hk+1(σ ,σ F) ≤ Cq,k,σ,β σ (Mβ ∨ 1)kqσ − wk2. (5.53) Applying Holder’s¨ inequality to the expression (5.27) we obtain, (assuming k ≥ 2 , otherwise Tk+1 = 0)

k i0 j i j E [|Tk+ |] ≤ C kδuk D δu , 1 q,k,σ,β ∑ r0 ∏ u r j (i0,...,ik)∈Jk+1 j=1

i where 1 = i0 + ∑k j . From property (b) in Remark 5.7 there is at least one factor of r0 j=1 r j j the form Duδu with j ≥ 2. We apply the estimate (5.149) to one of these factors, s j and the estimate (5.148) to all the remaining factors. We also use the estimate (5.147) to control kδ k . Notice that u r0

i k i 3i k i (3 j + 3) 1 1 = 0 + ∑ j > 0 + ∑ j + , r0 j=1 r j β j=1 β 2

102 and, on the other hand, using properties (a) and (c) in Remark 5.7

3i k i (3 j + 3) 1 3k + 3b k c 1 0 + ∑ j + ≤ 2 + . β j=1 β 2 β 2

k We can choose the r j’s satisfying the above properties because β > 6k + 6b 2 c, and we obtain k 3k+3b 2 c 2 E |Tk+1| ≤ Cq,k,σ,β (Mβ ∨ 1) qσ − w 2 . (5.54)

Combining (5.53) and (5.54) we complete the proof.

Corollary 5.15. Fix an integer m ≥ 1. Let {F } be a sequence of random variables n n∈N 2 2 in the qth Wiener chaos with q ≥ 2 and E[Fn ] = σn . Assume limn→∞ σn = σ, 0 < δ ≤ 2 4 4 σn ≤ K for all n, limn→∞ E[Fn ] = 3σ and

1 −β β M := sup E[kDFnkH ] < ∞ (5.55) n

m 2 for some β > 6(m) + 6(b 2 c ∨ 1). Let φ(x) be the density of N(0,σ ). Then, each Fn m admits a probability density function fFn ∈ C (R) with derivatives given by (5.30) and for any k = 1,...,m,

q (k) (k) 4 4 sup f (x) − φ (x) ≤ C E[F ] − 3σ + |σn − σ| , Fn n n x∈R where the constant C depends only on q,m,β,M,σ,δ and K.

2 Proof. Let φn(x) be the density of N(0,σn ). Then Theorem 5.13 implies that

q (k) (k) 4 4 sup f (x) − φn (x) ≤ C E[F ] − 3σ . Fn q,m,β,M,σ n n x∈R

103 On the other hand, by the mean value theorem we can write

(k) (k) (k) 1 (k+2) φn (x) − φ (x) ≤ |σn − σ| sup ∂γ φγ (x) = |σn − σ| sup γ φγ (x) , σ 2 σ γ∈[ 2 ,2σ] γ∈[ 2 ,2σ]

2 where φγ (x) is the density of the law N(0,γ ). Then, using the expression

(k+2) −2 −2 φγ (x) = E[1N>xHk+3(γ ,γ Z)],

2 where Z ∼ N(0,γ ) and the explicit form of Hk+3(λ,x), we obtain

(k+2) sup γ φγ (x) ≤ Ck,σ . σ γ∈[ 2 ,2σ]

Therefore,

(k) (k) sup φn (x) − φ (x) ≤ Ck,σ |σn − σ|. x∈R This completes the proof.

5.5 Random vectors in Wiener chaos

5.5.1 Main result

In this section, we study the multidimensional counterpart of Theorem5 .15. We begin with a density formula for a smooth random vector.

∞ A random vector F = (F1,...,Fd) in D is called nondegenerate if its Malliavin −1 p matrix γF = ( DFi,DFj H)1≤i, j≤d is invertible a.s. and (detγF ) ∈ ∩p≥1L (Ω). For any multi-index k β = (β1,β2,...,βk) ∈ {1,2,...,d}

104 ∂ k of length k ≥ 1, the symbol ∂β stands for the partial derivative ∂x ...∂x . For β of length β1 βk d 0 we make the convention that ∂β f = f . We denote by S (R ) the Schwartz space of rapidly decreasing smooth functions, that is, the space of all inﬁnitely differentiable

d m f → d |x| f (x) < functions : R R such that supx∈R ∂β ∞ for any nonnegative integer m and for all multi-index β. The following lemma (see Nualart [39, Proposition 2.1.5]) gives an explicit formula for the density of F.

Lemma 5.16. Let F = (F1,...,Fd) be a nondegenerate random vector. Then, F has a d density fF ∈ S (R ), and fF and its partial derivative ∂β fF , for any multi-index β =

(β1,β2,...,βk) of length k ≥ 0, are given by

fF (x) = E[1{F>x}H(1,2,...,d)(F)], (5.56)

k ∂β fF (x) = (−1) E[1{F>x}H(1,2,...,d,β1,β2,...,βk)(F)], (5.57)

d where 1{F>x} = ∏i=1 1{Fi>xi} and the elements Hβ (F) are recursively deﬁned by

  H (F) = 1, if k = 0;  β  (5.58)   β j  H (F) = d H (F) −1 1 DF , if k ≥ .  (β1,β2,...,βk) ∑ j=1 δ (β1,β2,...,βk−1) γF j 1

Fix d natural numbers 1 ≤ q1 ≤ ··· ≤ qd. We will consider a random vector of

qi multiple stochastic integrals: F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)), where fi ∈ H . Denote

V = E[FiFj] 1≤i, j≤d , Q = diag(q1,...,qd)(diagonal matrix of elements q1,...,qd). (5.59)

105 Along this section, we denote by N = (N1,...,Nd) a standard normal vector given by

Ni = I1(hi), where hi ∈ H are orthonormal. We denote by I the d-dimensional identity matrix, and by | · | the Hilbert-Schmidt norm of a matrix. The following is the main theorem of this section.

Theorem 5.17. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)) be nondegenerate and let φ be the density of N. Then for any multi-index β of length k ≥ 0, the density fF of F satisﬁes

! q 4 2 2 sup ∂β fF (x) − ∂β φ(x) ≤ C |V − I| + ∑ E[Fj ] − 3(E[Fj ]) , (5.60) x∈Rd 1≤ j≤d

−1 where the constant C depends on d,V,Q,k and (detγF ) . (k+4)2k+3

k Proof. Note that ∂β φ(x) = (−1) E[1{N>x}H(1,2,...,d,β1,β2,...,βk)(N)]. Then, in order to estimate the difference between ∂β fFn and ∂β φ, it sufﬁces to estimate

E[1{F>x}Hβ (F)] − E[1{N>x}Hβ (N)] for all multi-index β of length k for all k ≥ d. Fix a multi-index β of length k for some k ≥ d. For the above standard normal random vector N, we have γN = I and δ(DNi) = Ni. We can deduce from the expression (5.58) d that Hβ (N) = gβ (N), where gβ (x) is a polynomial on R (see Remark 5.19). Then,

E[1{F>x}Hβ (F)] − E[1{N>x}Hβ (N)] ≤ E[1{F>x}gβ (F)] − E[1{N>x}gβ (N)] + E Hβ (F) − gβ (F)

= A1 + A2. (5.61)

106

The term A1 = E[1{F>x}gβ (F) − 1{N>x}gβ (N)] will be studied in Subsection 5.5.3 by using the multivariate Stein’s method. Proposition 5.25 will imply that A1 is bounded by the right-hand side of (5.60). Consider the term A2 = E Hβ (F) − gβ (F) . We introduce an auxiliary term Kβ (F), −1 −1 which is deﬁned similar to Hβ (F) with γF replaced by (VQ) . That is, for any multi- index β = (β1,β2,...,βk) of length k ≥ 0, we deﬁne

  K (F) = 1 if k = 0;  β  (5.62)   K (F) = K (F) (VQ)−1 DF k ≥ .  β δ (β1,β2,...,βk−1) if 1 βk

We have

A2 ≤ E Hβ (F) − Kβ (F) + E[ Kβ (F) − gβ (F) ] =: A3 + A4 . (5.63)

Lemma 5.26 below shows that the term A3 = E Hβ (F) − Kβ (F) is bounded by the right-hand of (5.60).

It remains to estimate A4. For this we need the following lemma which provides an explicit expression for the term Kβ (F). Before stating this lemma we need to introduce some notation. For any multi-index β = (β1,β2,...,βk), k ≥ 1, denote by βbi1...im the multi-index obtained from β after taking away the elements βi1 ,βi2 ,...,βim . For example,

βb14 = (β2,β3 ,β5 ,...,βk). For any d-dimensional vector G we denote by Gβ the product G G ···G G = Sm ≤ m ≤ b k c β1 β2 βk and set β 1 if the length of β is 0. Denote by k ;0 2 the

107 following sets

  S−1 = S0 =  k k ∅    2m  {(i1,i2),...,(i2m−1,i2m)} ∈ {1,2,...,k} :  (5.64)  Sm =  k   i2l−1 < i2l for 1 ≤ l ≤ mand il 6= i j if l 6= j 

m For each element {(i1,i2),...,(i2m−1,i2m)} ∈ Sk , we emphasize that the m pairs of in- m dices are unordered. In other words, for m ≥ 1, the set Sk can be viewed as the set of all partitions of {1,2,...,k} into m pairs and k − 2m singletons.

−1 −1 −1 Denote M = V γFV Q for V and Q given in (5.59) and denote Mi j the (i, j)-th entry of M. Denote by D the Malliavin derivative in the direction of V −1Q−1DF = βi βi −1 −1 V Q DFβi , that is, D −1 −1 E Dβ G = DG, V Q DF (5.65) i βi H

, for any random variable G ∈ D1 2.

Lemma 5.18. Let F be a nondegenerate random vector. For a multi-index β = (β1,...,βk) of length k ≥ 0,Kβ (F) deﬁned by (5.62) can be computed as follows:

Kβ (F) = Gβ (F) + Tβ (F), (5.66) where

bk/2c m −1 G (F) = (−1) V F Mβ β ···Mβ β , (5.67) β ∑ ∑ βbi ···i i1 i2 i2m−1 2m m m 1 2m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk and Tβ (F) are deﬁned recursively by

−1 Tβ (F) = V F T (F) − Dβ T (F) (5.68) βk βbk k βbk

108 −1 − V F D Mβ β ···Mβ β , ∑ βb βk i1 i2 i2m−1 2m m ki1···i2m {(i1,i2),...,(i2m−1,i2m)}∈Sk−1 for k ≥ 2 and T1(F) = T2(F) = 0.

Proof. For simplicity, we write Kβ , Gβ and Tβ for Kβ (F), Gβ (F) and Tβ (F), respectively. By using the fact that δ (VQ)−1 DF = V −1F we obtain from (5.62) βi βi that −1 Kβ = V F K − Dβ K . (5.69) βk βbk k βbk

If if k = 1, namely, β = (β1), then

K = V −1F = G . β β1 β

If k = 2, namely, β = (β1,β2), then

K V −1F M G β = β − β1β2 = β .

Hence, the identity (5.66) is true for k = 1,2. Assume now (5.66) is true for all multi- index of length less than or equal to k. Let β = (β1,...,βk+1). Then, (5.69) implies

−1 Kβ = V F G + T − Dβ G + T . (5.70) βk+1 βbk+1 βbk+1 k+1 βbk+1 βbk+1

Noticing that

D V −1F = V −1F M , βk+1 β ∑ β β jβk+1 b(k+1)i1···i2m b(k+1) ji1···i2m j∈{1,...,k}\{i1,...,i2m} we have

Dβ G = Bβ (5.71) k+1 βbk+1

109 bk/2c m −1 + (−1) V F D Mβ β ···Mβ β , ∑ ∑ βb βk+1 i1 i2 i2m−1 2m m m (k+1)i1···i2m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk where we let

bk/2c m −1 B = (−1) V F Mβ β Mβ β ···Mβ β . β ∑ ∑ β j k+1 i1 i2 i2m−1 i2m m bj(k+1)i1···i2m m=0 {(i1,i2),...,(i2m−1,i2m)}∈Sk , j∈{1,...,k}\{i1,...,i2m}

Substituting the expression (5.71) for Dβ G into (5.70) and using (5.68) we obtain k+1 βbk+1

−1 Kβ = V F G − Bβ + Tβ . βk+1 βbk+1

To arrive at (5.66) it remains to verify

−1 Gβ = V F G − Bβ . (5.72) βk+1 βbk+1

Introduce the following notation

m m−1 Ck+1 = {(i1,i2),...,(i2m−3,i2m−2),( j,k + 1)} : {(i1,i2),...,(i2m−3,i2m−2)} ∈ Sk (5.73)

k m for 1 ≤ m ≤ b 2 c. Then, Sk+1 can be decomposed as follows

m m m Sk+1 = Sk ∪Ck+1. (5.74)

We consider ﬁrst the case when k is even. In this case, noticing that for any element k b 2 c in {(i1,i2),...,(i2m−1,i2m)} ∈ Sk , {1,...,k}\{i1,...,i2m} = ∅. For any collection of indices i1,...,i2m ⊂ {1,2,...,k}, we set

−1 Φ = V F Mβ β ···Mβ β . i1...i2m i i i m− i m βbi1···i2m 1 2 2 1 2

110 Then, we have

k b 2 c−1 −B = (−1)m+1 β ∑ ∑ Φ j(k+1)i1...i2m m m=0 {(i1,i2),...,(i2m−1,i2m)}∈Sk , j∈{1,...,k}\{i1,...,i2m} k b 2 c = (−1)n (5.75) ∑ ∑ Φ j(k+1)i1...i2m−2 m=1 m−1 {(i1,i2),...,(i2m−3,i2m−2)}∈Sk , j∈{1,...,k}\{i1,...,i2n−2} k+1 b 2 c = (−1)m , ∑ ∑ Φ j(k+1)i1...i2m−2 m=1 {(i1,i2),...,(i2m−3,i2m−2),( j,k+1)} m ∈Ck+1

k k+1 where in the last equality we used (5.73) and the fact that b 2 c = b 2 c since k is even. Taking into account that V −1F V −1F = V −1F , we obtain from β β β k+1 b(k+1)i1···i2m bi1···i2m (5.67) that

bk/2c −1 m −1 V F G = (−1) V F Mβi βi ···Mβi βi . βk+1 βbk+1 ∑ ∑ βbi ···i 1 2 2m−1 2m m m 1 2m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk (5.76)

k k+1 Now combining (5.75) and (5.76) with (5.74) and using again b 2 c = b 2 c we obtain

b(k+1)/2c −1 m V F G − Bβ = (−1) Φi1...i2m βk+1 βbk+1 ∑ ∑ m m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk b(k+1)/2c m + ∑ (−1) ∑ Φi1...i2m m= m 1 {(i1,i2),...,(i2m−3,i2m−2),( j,k+1)}∈Ck+1 b(k+1)/2c m = ∑ (−1) ∑ Φi1...i2m m= m 0 {(i1,i2),...,(i2m−1,i2m)}∈Sk+1 = Gβ as desired. This veriﬁes (5.72) for the case k is even. The case when k is odd can be veriﬁed similarly. Thus, we have proved (5.66) by induction.

111 Remark 5.19. For the random vector N ∼ N(0,I), we have γN = VQ = I, so Hβ (N) =

Kβ (N). Then, it follows from Lemma 5.18 that Hβ (N) = Kβ (N) = gβ (N) with the func- d tion gβ (x) : R → R given by

bk/2c m g (x) = (−1) x δβ β ···δβ β , (5.77) β ∑ ∑ βbi ···i i1 i2 i2m−1 2m m m 1 2m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk where we used δi j to denote the Kronecker symbol (without confusion with the divergence operator). Notice that d

gβ (x) = ∏Hki (xi), i=1 where Hki is the kith Hermite polynomial and for each i = 1,...,d, ki is the number of components of β equal to i.

Let us return to the proof of Theorem 5.17 of estimating the term A4. From (5.66) we can write

A4 = E Kβ (F) − gβ (F) ≤ E |Gβ (F) − gβ (F)| + E |Tβ (F)| . (5.78)

Observe from the expression (5.68) that Tβ (F) is the sum of terms of the following form

r −1 V F D D ···D ( Mβ j β ) (5.79) βi βi ···βi βk1 βk2 βkt ∏ i li 1 2 s i for some {i1,...,is,k1,...,kt, j1,l1,... jr,lr} ⊂ {1,2,...,k} and t ≥ 1. Applying Lemma 5.20 with (2.11) and (2.12) we obtain

1 2 2 2 E Tβ (F) ≤ C ∑ kDFlkH − qlE[Fl ] . (5.80) 1≤l≤d 2

112 In order to compare gβ (F) with Gβ (F), from (5.77) we can write gβ (F) as

bk/2c m g (F) = (−1) F δ ···δβ β . β ∑ ∑ βbi ···i βi1 βi2 i2m−1 2m m m 1 2m =0 {(i1,i2),...,(i2m−1,i2m)}∈Sk

Then, it follows from hypercontractivity property (2.11) that

−1 E Gβ (F) − gβ (F) ≤ C V − I + kM − Ik2 , where the constant C depends on k,V and Q. From V −1 − I = V −1 (I −V) we have

−1 −1 −1 −1 V − I ≤C |V − I|, where C depends on V. We also have M−I =V (γF −VQ)V Q + V −1 − I. Then, Lemma 5.20 implies that

−1 kM − Ik2 ≤ C kγF −VQk2 + |V − I| !

2 2 ≤ C ∑ kDFlkH − qlEFl + |V − I| , 1≤l≤d 2 where the constant C depends on k,V and Q. Therefore

! 1 2 2 2 E Gβ (F) − gβ (F) ≤ C |V − I| + ∑ kDFlkH − qlEFl . (5.81) 1≤l≤d 2

Combining it with (5.80) we obtain from (5.78) that

! 1 2 2 2 A4 ≤ C |V − I| + ∑ kDFlkH − qlEFl , 1≤l≤d 2 where the constant C depends on d,V,Q. This completes the estimation of the term

A4.

113 5.5.2 Sobolev norms of the inverse of the Malliavin matrix

−1 In this subsection we estimate the Sobolev norms of γF , the inverse of the Malliavin matrix γF for a random variable F of multiple stochastic integrals. We begin with the following estimate on the variance and Sobolev norms of (γF )i j = DFi,DFj H, 1 ≤ i, j ≤ d, following the approach of [31, 35, 37].

p q Lemma 5.20. Let F = Ip( f ) and G = Iq(g) with f ∈ H and g ∈ H for p,q ≥ 1. Then for all k ≥ 0 there exists a constant Cp,q,k such that

√ k D hDF,DGiH − pqE[FG] (5.82) 2 1 1 2 2 2 2 2 2 2 2 ≤ Cp,q,k(kFk2 + kGk2)( kDFkH − pE F + kDGkH − pE G ). 2 2

Proof. Without lost of generality, we assume p ≤ q. Applying (2.4) with the fact that

DIp( f ) = pIp−1( f ) we have

hDF,DGiH = pq Ip−1( f ),Iq−1(g) H (5.83) p−1 p − 1q − 1 = pq ∑ r! Ip+q−2−2r f ⊗]r+1g r=0 r r p p − 1 q − 1 = pq ∑ (r − 1)! Ip+q−2r f ⊗frg . r=1 r − 1 r − 1

Note that E[FG] = 0 if p < q and E[FG] = h f ,giH⊗p = f ⊗fpg if p = q. Then

p √ p − 1 q − 1 hDF,DGiH − pqE(FG) = pq ∑(1 − δqr)(r − 1)! Ip+q−2r f ⊗frg , r=1 r − 1 r − 1 where δqr is again the Kronecker symbol. It follows that

√ 2 E hDF,DGiH − pqE[FG] (5.84)

114 p 2 2 2 2 2 p − 1 q − 1 2 = p q ∑(1 − δqr)(r − 1)! (p + q − 2r)! f ⊗frg H⊗(p+q−2r) . r=1 r − 1 r − 1

Note that if r < p ≤ q, then (see also [35, (6.2.7)])

2 2 f ⊗frg H⊗(p+q−2r) ≤ k f ⊗r gkH⊗(p+q−2r) = f ⊗p−r f ,g ⊗q−r g H⊗2r 1 2 2 ≤ ( f ⊗p−r f ⊗2r + g ⊗q−r g ⊗2r ), (5.85) 2 H H and if r = p < q,

2 2 2 f ⊗fpg H⊗(q−p) ≤ f ⊗p g H⊗(q−p) ≤ k f kH⊗p g ⊗q−p g H⊗2p . (5.86)

From (5.36) and (5.37) it follows that

2 p−1 p − 12 kDFk2 − pE F2 = p4 (r − ) 2 ( p − r) k f ⊗ f k2 H ∑ 1 ! 2 2 ! r H⊗(2p−2r) . (5.87) 2 r=1 r − 1

Combining (5.84)–(5.87) we obtain

√ 2 E hDF,DGiH − pqE [FG] 2 2 2 2 2 2 ≤ Cp,q( kDFkH − pE F + kFk2 kDGkH − pE G ). 2 2

2 2 2 Then (5.82) with k = 0 follows from kDFkH − pE F ≤ Cp kFk2, which is implied 2 by (2.12). From (5.83) we deduce

p+q−k p∧ 2 k p − 1 q − 1 p + q − 2r D hDF,DGiH = pq ∑ (r − 1)! Ip+q−k−2r f ⊗frg . r=1 r − 1 r − 1 p + q − k − 2r

115 Then it follows from (5.85)–(5.87) that

2 k E D hDF,DGiH H⊗k p∧[(p+q−k)/2] 2 2 2 2 2 2 p − 1 q − 1 (p + q − 2r)! 2 = p q ∑ (r − 1)! f ⊗frg H⊗(p+q−2r) r=1 r − 1 r − 1 (p + q − k − 2r)! 2 2 2 2 2 2 ≤ Cp,q( kDFkH − pE F + kFk2 kDGkH − pE G ). 2 2

This completes the proof.

−1 The following lemma gives estimates on the Sobolev norms of the entries of γF .

Lemma 5.21. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)) be nondegenerate and let γF = DFi,DFj . Set V = E[FiFj] . Then for any real number p > 1, H 1≤i, j≤d 1≤i, j≤d

−1 −1 γF ≤ C (detγF ) , (5.88) p 2p

where the constant C depends on q1,...,qd,d, p and V. Moreover, for any integer k ≥ 1 and any real number p > 1

k+1 d −1 ≤ C (det )−1 kDF k2 − q E F2 , (5.89) γF k,p γF ∑ i H i i (k+2)2p i=1 2 where the constant C depends on q1,...,qd,d, p,k and V.

∗ Proof. Let γF be the adjugate matrix of γF . Note that Holder¨ inequality and (2.12) imply

DFi,DFj ≤ kDFik2p DFj ≤ CV,p H p 2p for all 1 ≤ i, j ≤ d, p ≥ 1. Applying Holder’s inequality we obtain that the p norm

∗ of γF is also bounded by a constant. A further application of Holder’s inequality to

116 −1 −1 ∗ γF = (detγF ) γF yields

−1 −1 ∗ −1 γF ≤ (detγF ) k γF k2p ≤ CV,p (detγF ) , (5.90) p 2p 2p which implies (5.88).

−1 ∞ Since F is nondegenerate, then (see [39, Lemma 2.1.6]) γF i j belongs to D for all i, j and d −1 −1 −1 D γF i j = − ∑ γF im γF n j D(γF )mn . (5.91) m,n=1 Then, applying Holder’s¨ inequality we obtain

−1 −1 2 D γF p ≤ γF 3p kDγF k3p . 2 d −1 2 2 ≤ CV,p (detγF ) ∑ kDFikH − qiE Fi , 6p i=1 2 where in the second inequality we used (5.88) and

d 2 2 kDγF k3p ≤ CV,p kDγF k2 ≤ CV,p ∑ kDFikH − qiE Fi i=1 2 for all p ≥ 1, which follows from (2.12) and (5.82). This implies (5.89) with k = 1. For higher order derivatives, (5.89) follows from repeating the use of (5.91), (2.12) and (5.82).

−1 −1 −1 The following lemma estimates the difference γF −V Q .

Lemma 5.22. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)) be a nondegenerate random q vector with 1 ≤ q1 ≤ ··· ≤ qd and fi ∈ H i . Let γF be the Malliavin matrix of F. Recall the notation of V and Q in (5.59). Then, for every integer k ≥ 1 and any real number

117 p > 1 we have

k+1 1 −1 −1 −1 −1 2 2 2 γF −V Q ≤ C (detγF ) kDFlkH − qlE Fl , (5.92) k,p (k+ ) p ∑ 2 2 1≤l≤d 2 where the constant C depends on d,V,Q, p and k.

Proof. In view of Lemma 5.21, we only need to consider the case when k = 0 because V and Q are deterministic matrices. Note that

−1 −1 −1 −1 −1 −1 γF −V Q = γF (VQ − γF )V Q .

Then, applying Holder’s inequality we have

−1 −1 −1 −1 γF −V Q p ≤ CV,Q γF 2p kVQ − γF k2p .

Note that (2.12) and (5.82) with k = 0 imply

d 1 2 2 2 kVQ − γF k2p ≤ CV,Q,p kVQ − γF k2 ≤ CV,Q,p ∑ kDFikH − qiE Fi . i=1 2

Then, applying (5.90) we obtain

d 1 −1 −V −1Q−1 ≤ C kDF k2 − q E F2 2 (5.93) γF p d,V,Q,p ∑ i H i i i=1 2 as desired.

118 5.5.3 Technical estimates

In this subsection, we study the terms A1 = |E[h(F)] − E[h(N)]| in Equation (5.61) and A3 = E Hβ (F) − Kβ (F) in (5.63). For A1, we shall use the multivariate Stein’s method to give an estimate for a large class of non-smooth test functions h.

d Lemma 5.23. Let h : R → R be an almost everywhere continuous function such that m |h(x)| ≤ c(|x| + 1) for some m,c > 0. Let F = (F1,...,Fd) be nondegenerate with

E[Fi] = 0,1 ≤ i ≤ d and denote N ∼ N(0,I). Then there exists a constant Cm,c depending on m and c such that

d |E[h(F)] − E[h(N)]| ≤ C (kFkm + ) A −1 DF , m,c 2m 1 ∑ δ i j γF jk k (5.94) i, j,k=1 2

−1 where γF is the inverse of the Malliavin matrix of F and

−1 Ai j = δi j − DFj,−DL Fi H . (5.95)

Proof. For ε > 0, let

Z hε (x) = (1{|·|< 1 }h) ∗ ρε (x) = 1|y|< 1 h(y)ρε (x − y)dy. ε Rd ε where ρ (x) = 1 ρ( x ), with ρ(x) = C1 exp( 1 ) and the constant C is such that ε εd ε {|x|<1} |x|2−1 R d (x)dx = 1. Then h is Lipschitz continuous. Hence, the solution f to the following R ρ ε ε Stein’s equation:

f (x) − hx, f (x)i d = h (x) − E[h (N)] (5.96) ∆ ε ∇ ε R ε ε exists and its derivative has the following expression [35, Page 82]

∂ Z 1 1 √ √ ∂i fε (x) = E[hε ( tx + 1 −tN)]dt (5.97) ∂xi 0 2t

119 Z 1 √ √ 1 = E[hε ( tx + 1 −tN)Ni] √ √ dt. 0 2 t 1 −t

It follows directly from the polynomial growth of h that

m |hε (x)| ≤ C1 |x| +C2 (5.98)

for all ε < 1, where C1,C2 > 0 are two constants depending on c and m. Then, from (5.96) we can write m |∂i fε (x)| ≤ C1 |x| +C2, with two possibly different constants C1, and C2. Hence,

m k∂i fε (F)k2 ≤ C1 kFk2m +C2. (5.99)

Meanwhile, note that for 1 ≤ i ≤ d,

−1 −1 E[Fi∂i fε (F)] = E[LL Fi∂i fε (F)] = E[ −DL Fi,D∂i fε (F) ] d −1 = ∑ E[ −DL Fi,∂i j fε (F)DFj ]. j=1

Then, replacing x by F and taking expectation in Equation (5.96) yields

d 2 |E[hε (F)] − E[hε (N)]| = ∑ E ∂i j fε (F)Ai j . (5.100) i, j=1

Notice that

* d + d 2 2 hDFi,D∂i fε (F)iH = DFi, ∑ ∂i j fε (F)DFj = ∑ ∂i j fε (F) DFi,DFj H j=1 H j=1

120 for all 1 ≤ i,k ≤ d, which implies

d −1 ∂i j fε (F) = ∑ γF jk hDFk,D∂i fε (F)iH , k=1 and hence

d d " * d + # 2 −1 ∑ E ∂i j fε (F)Ai j = ∑ E Ai j ∑ γF jk DFk,D∂i fε (F) i, j=1 i, j=1 k=1 H d " d !# −1 = ∑ E ∂i fε (F)δ Ai j ∑ γF jk DFk . i, j=1 k=1

Substituting this expression in (5.100) and using (5.99) we obtain

d h −1 i |E[hε (F)] − E[hε (N)]| = ∑ E ∂i fε (F)δ Ai j γF jk DFk i, j,k=1 d ≤ k f (F)k A −1 DF ∑ ∂i ε 2 δ i j γF jk k i, j,k=1 2 d ≤ (C kFkm +C ) A −1 DF . 1 2m 2 ∑ δ i j γF jk k i, j,k=1 2

Then, we can conclude the proof by observing that

lim |E[hε (F)] − E[hε (N)]| = |E[h(F)] − E[h(N)]|, ε→0

which follows from (5.98) and the fact that hε → h almost everywhere. −1 The next lemma gives an estimate for δ Ai j γF DFk when F is a vector of jk 2 multiple stochastic integrals.

qi Lemma 5.24. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)), where fi ∈ H , be nondegenerate and denote N ∼ N(0,I). Recall the notation of V and Q in (5.59) and Ai j in

121 (5.95). Then, for all 1 ≤ i, j,k ≤ d we have

3 −1 −1 δ Ai j γF DFk ≤ C (detγF ) (5.101) jk 2 12 ! d 1 2 2 2 × |V − I| + ∑ kDFikH − qiE Fi , i=1 2 where the constant C depends on d,V,Q.

Proof. Applying Meyer’s inequality (2.9) we have

−1 −1 −1 δ Ai j γF DFk ≤ Ai j γF DFk + D Ai j γF DFk . jk 2 jk 2 jk 2

Applying Holder’s inequality and (2.12) we have

−1 −1 −1 Ai j γF DFk ≤ Ai j γF kDFkk4 ≤ Cd,V,Q Ai j γF . jk 2 2 jk 4 2 jk 4

Similarly, Holder’s inequality and (2.12) imply

−1 −1 D Ai j γF DFk ≤ Cd,V,Q[ DAi j γF jk 2 2 jk 4

−1 −1 + Ai j D γF + Ai j γF ]. 2 jk 4 2 jk 4

Combining the above inequalities we obtain

−1 −1 δ Ai j γF DFk ≤ Cd,V,Q Ai j γF . jk 2 1,2 jk 1,4

Note that

−1 1 Ai j = δi j − DFj,−DL Fi H = δi j −Vi j +Vi j − DFj,−DFi H . qi

122 Then, it follows from Lemma 5.20 that

! d 1 A ≤ C |V − I| + kDF k2 − q E F2 2 . i j 1,2 d,V,Q ∑ i H i i i=1 2

Then, the lemma follows by taking into account of (5.89) with k = 1.

As a consequence of the above two lemmas, we have the following result.

d Proposition 5.25. Let h : R → R be an almost everywhere continuous function such m that |h(x)| ≤ c(|x| + 1) for some m,c > 0. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)), q where fi ∈ H i , be nondegenerate and denote N ∼ N(0,I). Recall the notation of V and Q in (5.59). Then

3 −1 |E[h(F)] − E[h(N)]| ≤ C (detγF ) (5.102) 12 ! d 1 2 2 2 × |V − I| + ∑ kDFikH − qiE Fi , i=1 2 where the constant C depends on d,V,Q,m,c.

In the following, we estimate the term A3 = E Hβ (F) − Kβ (F) in (5.63), where

Hβ (F) and Kβ (F) are deﬁned in (5.58) and (5.62), respectively.

Lemma 5.26. Let F = (F1,...,Fd) = (Iq1 ( f1),...,Iqd ( fd)) be nondegenerate. Let β =

(β1,...,βk) be a multi-index of length k ≥ 1. Let Hβ (F) and Kβ (F) be deﬁned by (5.58) and (5.62), respectively. Then there exists a constant C depending on d,V,Q,k such that

k(k+2) −1 E Hβ (F) − Kβ (F) ≤ C (detγF ) (5.103) (k+4)2k+3 d 1 2 2 2 × ∑ kDFikH − qiE Fi . i=1 2

123 Proof. To simplify notation, we write Hβ and Kβ for Hβ (F) and Kβ (F), respectively. From (5.58) and (5.62) we see that

H − K = δ H γ−1DF − K (VQ)−1 DF , β β βb F βk βb k k βk

where βbk = (β1,...,βk−1). For any s ≥ 0, p > 1, using Meyer’s inequality (2.9) we obtain

Hβ − Kβ s,p

−1 −1 ≤ Cs,p H γ DF − K (VQ) DF βbk F βk βbk βk s+1,p

−1 ≤ Cs,p H − K (VQ) DF βbk βbk βk s+1,p

−1 −1 +Cs,p H γ − (VQ) DF . βbk F βk s+1,p

Then, Holder’s¨ inequality yields

Hβ − Kβ s,p

−1 ≤ H − K (VQ) DF βbk βbk s+1,2p βk s+1,2p

−1 −1 + H γ − (VQ) DF . βbk F s+1,2p βk s+1,2p

−1 Note that (2.12) implies (VQ) DF ≤ Cd,V,Q,s,p. Also note that (2.12), βk s+1,2p Holder’s¨ inequality and (5.92) indicate

s+2 −1 −1 −1 γF − (VQ) DF ≤ Cd,V,Q,s,p∆ (detγF ) . βk s+1,2p (s+3)8p where we denote 1 2 2 2 ∆ := ∑ kDFlkH − qlE Fl 1≤l≤d 2

124 to simplify notation. Thus we obtain

Hβ − Kβ ≤ Cd,V,Q,s,p H − K (5.104) s,p βbk βbk s+1,2p s+2 −1 +Cd,V,Q,s,p∆ H (detγF ) . βbk s+1,2p (s+3)8p

Similarly, from Meyer’s inequality (2.9), Holder’s¨ inequality and (2.12) we obtain by iteration

−1 Hβ ≤ Cs,p H γF DF s,p βbk βk s+1,p s+2 −1 ≤ Cd,V,Q,s,p H (detγF ) βbk s+1,2p (s+3)8p ··· k(s+k) −1 ≤ Cd,V,Q,s,p,k (detγF ) . (5.105) (s+k+1)2k+2 p

Applying (5.105) into (5.104) and by iteration we can obtain

k(2s+k+2) −1 Hβ − Kβ ≤ Cd,V,Q,s,p,k (detγF ) ∆. s,p (2s+k+4)2k+2 p

Now (5.103) follows by taking s = 0, p = 2 in the above inequality.

5.6 Uniform estimates for densities of general random

variables

In this section, we study the uniform convergence of densities for general random variables. We ﬁrst characterize the convergence of densities with quantitative bounds for a sequence of centered random variables, using the density formula (5.17). In the second part of this section, a short proof of the uniform convergence of densities (without quan-

125 titative bounds) is given, using a compactness argument based on the assumption that the sequence converges in law.

5.6.1 Convergence of densities with quantitative bounds

In this subsection, we estimate the rate of uniform convergence for densities of general random variables. The idea is to use the density formula (5.17). We use the following notations throughout this section.

−1 −1 −1 w¯ = DF,−DL F H , u¯ = −w¯ DL F.

The following technical lemma is useful.

,s Lemma 5.27. Let F ∈ D2 with s ≥ 4 such that E [F] = 0 and E[F2] = σ 2. Let m be the s largest even integer less than or equal to 2 . Then there is a positive constant Cm such that for any t ≤ m,

2 2 w¯ − σ t ≤ w¯ − σ m ≤ Cm kDw¯km ≤ Cm kDw¯ks/2 . (5.106)

Proof. It sufﬁces to show the above second inequality. From the integration by parts formula in Malliavin calculus it follows

2 2 h −1 i σ = E[F ] = E DF,−DL F H = E [w¯] .

1, s Note that from (5.16) and (5.20) we havew ¯ ∈ D 2 . Then the lemma follows from the following inﬁnite-dimensional Poincare´ inequality [35, Lemma 5.3.8]:

m m/2 m E[(G − E [G]) ] ≤ (m − 1) E kDGkH ,

126 ,m for any even integer m and G ∈ D1 .

The next theorem gives a bound for the uniform distance between the density of a random variable F and the normal density.

,s Theorem 5.28. Let F ∈ D2 with s ≥ 8 such that E [F] = 0,E[F2] = σ 2. Suppose r −r −1 2 4 M := E |w¯| < ∞, where w¯ = DF,−DL F H and r > 2. Assume r + s = 1. Then

F admits a density fF (x) and there is a constant Cr,s,σ,M depending on r,s,σ and M such that

2 2 sup| fF (x) − φ(x)| ≤ Cr,s,σ,M kFk1,s D F , (5.107) op ,s x∈R 0 2 2 where φ(x) is the density of N ∼ N(0,σ ) and D F op indicates the operator norm of D2F introduced in (5.16).

Proof. It follows from Proposition 5.5 that F admits a density fF (x) = E 1{F>x}δ (u¯) . Then

−2 sup| fF (x) − φ(x)| = sup E 1{F>x}δ (u¯) − σ E[1{N>x}N] . (5.108) x∈R x∈R

Note that, from (2.7)

−1 −1 −1 −1 −1 δ(u¯) = δ(−DL Fw¯ ) = Fw¯ + Dw¯ ,DL F H .

Then

2 E σ 1{F>x}δ (u¯) − E[1{N>x}N] h i ≤ E Fw−1( 2 − w) + 2E Dw−1,DL−1F ¯ σ ¯ σ ¯ H + E F1{F>x} − N1{N>x} . (5.109)

127 1 3 −1 s s Note that for t = r + s , we have 2 −t ≥ 2, so there exists an even integer m ∈ [t, 2 ]. 1 1 1 Also, we have r + s + t = 1. Then, we can apply Holder’s¨ inequality and (5.106) to obtain

−1 2 −1 2 E Fw¯ (w¯ − σ ) ≤ kFks w¯ r w¯ − σ t −1 ≤ Cr,s kFks w¯ r kDw¯ks/2 . (5.110)

Meanwhile, applying Holder’s¨ inequality and (5.14) we have

h −2 −1 i −1 2 −1 E w¯ Dw¯,−DL F ≤ w¯ kDw¯k s DL F s H r 2 2 −1 2 ≤ w¯ kDw¯k s kDFk . (5.111) r 2 s

Also, applying Lemma 5.2 for h(y) = y1{y>x} and (5.106) we have

2 |E [F1F>x − N1N>x]| ≤ Cσ σ − w¯ 2 ≤ Cσ kDw¯ks/2 . (5.112)

Applying the estimates (5.110)-(5.112) to (5.109) we have

2 E σ 1F>xδ (u¯) − E[1N>xN] ≤ Cr,s,σ,M kFk1,s kDw¯ks/2 . (5.113)

Combining (5.108), (5.113) and (5.20) one gets

2 2 sup| fF (x) − φ(x)| ≤ Cr,s,σ,M kFk1,s D F . op s x∈R

This completes the proof.

Corollary 5.29. Let {F } ⊂ 2,s with s ≥ 8 such that E [F ] = 0 and lim E[F2] = n n∈N D n n→∞ n 2 2 2 4 σ . Assume E[Fn ] ≥ δ > 0 for all n. For r > 2 such that r + s = 1, assume

128 (i)M 1 = supn kFnk1,s < ∞.

−r (ii)M = E DF ,−DL−1F < . 2 supn n n H ∞

2 s (iii)E D Fn op → 0 as n → ∞.

Then each Fn admits a density fFn (x) and ,

2 2 2 sup| fFn (x) − φ(x)| ≤ C D Fn + E[Fn ] − σ , (5.114) op s x∈R where the constant C depends on σ,M1,M2 and δ. Moreover, if M3 = supn kFnk2s < ∞, 1 then for any k ≥ 1 and α ∈ ( 2 ,k),

k−α 2 2 2 k k fFn − φkLk( ) ≤ C D Fn + E[Fn ] − σ , R op s

where the constant C depends on σ,M1,M2,M3,α and δ.

Remark 5.30. By the “random contraction inequality” (5.16), a sufﬁcient condition for

2 2 s/2 2 s (iii) is E D Fn ⊗1 D Fn H⊗2 → 0 or E D Fn H⊗2 → 0.

Proof of Corollary 5.29. It follows from Theorem 5.28 and Proposition 5.5 with an argument similar to Corollary 5.12.

5.6.2 Compactness argument

In general, convergence in law does not imply convergence of the corresponding densities even if they exist. The following theorem speciﬁes some additional conditions which ensure that convergence in law will imply convergence of densities.

129 Theorem 5.31. Let {F } be a sequence of random variables in 2,s satisfying any n n∈N D one of the following two conditions:

−2 supkFnk2,s + supkFnk2p + sup kDFnkH < ∞ (5.115) n n n r

1 1 1 for some p,r,s > 1 satisfying p + r + s = 1, or

−1 kF k + DF ,−DL−1F < sup n 2,s sup n n H ∞ (5.116) n n r

2 4 for some r,s > 1 satisfying r + s = 1. 2 Suppose in addition that Fn → N ∼ N(0,σ ) in law. Then each Fn admits a density fFn ∈ C(R) given by either (5.8) or (5.17), and

sup| fFn (x) − φ(x)| → 0 x∈R as n → ∞, where φ is the density of N.

Proof. We assume (5.115). The other condition can be treated identically. From Theo- rem 5.3 it follows that the density formula (5.8) holds for each n and for all x,y ∈ R

−2 | fFn (x)| ≤ C(1 ∧ x ),

1 p | fFn (x) − fFn (y)| ≤ C |x − y| .

Hence the sequence { fFn } ⊂ C(R) is uniformly bounded and equicontinuous. Then applying Azela-Ascoli` theorem, we obtain a subsequence { f } which converges uni- Fnk formly to a continuous function f on such that 0 ≤ f (x) ≤ C(1 ∧ x−2). Then f → f R Fnk 1 in L ( ) as k → ∞ with k f k 1 = lim fF = 1. This implies that f is a density R L (R) k nk 1 L (R)

130 function. Then f must be φ because Fn converges to N in law. Since the limit is unique for any subsequence, we get the uniform convergence of fFn to φ.

Corollary 5.32. Let {F } be a sequence of centered random variables in 2,4 with n n∈N D ∞ the following Wiener chaos expansions: Fn = ∑q=1 JqFn. Suppose that

∞ 2 (i) limQ→∞ limsupn→∞ ∑q=Q+1 E[ JqFn ] = 0.

2 2 (ii) for every q ≥ 1, limn→∞ E[ JqFn ] = σq .

∞ 2 2 (iii) ∑q=1 σq = σ .

2 2 (iv) for all q ≥ 1, D JqFn ,D(JqFn) H −→ qσq , in L (Ω) as n → ∞.

−8 (v) supn kFnk2,4 + supn E[kDFnkH ] < ∞.

Then each Fn admits a density fFn (x) and

sup| fFn (x) − φ(x)| → 0 x∈R as n → ∞, where φ is the density of N(0,σ 2).

Proof. It has been proved by Nualart and Ortiz-Latorre in [40, Theorem 8] that under

2 conditions (i)–(iv), Fn converges to N ∼ N(0,σ ) in law. The condition (v) implies (5.115) with s = 4, p = 2,r = 4. Then we can conclude from Theorem 5.31.

5.7 Applications

The main difﬁculty in applying Theorem 5.10 or Theorem 5.17 is the veriﬁcation of the −p −p non-degeneracy condition of the Malliavin matrix: supn E[kDFnkH ] < ∞ or supn E[|detγFn | ] < ∞, respectively. In this section we consider the particular case of random variables in the

131 −p second Wiener chaos and we ﬁnd sufﬁcient conditions for supn E[kDFnkH ] < ∞. As an application we consider the problem of estimating the drift parameter in an Ornstein- Uhlenbeck process.

A general approach to verify E[G−p] < ∞ for some positive random variable and for some p ≥ 1 is to obtain a small ball probability estimate of the form

α P(G ≤ ε) ≤ Cε for some α > p and for all ε ∈ (0,ε0), (5.117)

where ε0 > 0 and C > 0 is a constant that may depend on ε0 and α. We refer to the paper by Li and Shao [23] for a survey on this topic. However, ﬁnding upper bounds of this type is a challenging topic, and the application of small ball probabilities to Malliavin calculus is still an unexplored domain.

5.7.1 Random variables in the second Wiener chaos

A random variable F in the second Wiener chaos can always be written as F = I2( f ) where f ∈ H 2. Without loss of generality we can assume that

∞ f = ∑ λiei ⊗ ei, (5.118) i=1 where {λi,i ≥ 1} verifying |λ1| ≥ |λ2| ≥ ··· ≥ |λn| ≥ ... are the eigenvalues of the

Hilbert-Schmidt operator corresponding to f and {ei,i ≥ 1} are the corresponding eigen-

∞ 2 vectors forming an orthonormal basis of H. Then, we have F = I2( f ) = ∑i=1 λi(I1(ei) − 1), ∞ DF = 2 ∑ λiI1(ei)ei (5.119) i=1 and ∞ 2 2 2 kDFkH = 4 ∑ λi I1(ei) . (5.120) i=1

132 The following lemma provides necessary and sufﬁcient conditions for a random variable of the form (5.120) to have negative moments.

1 ∞ 2 2 2 Lemma 5.33. Let G = ∑i=1 λi Xi , where {λi}i≥1 satisﬁes |λi| ≥ |λi+1| for all i ≥ 1 −2α and {Xi}i≥1 are i.i.d standard normal. Fix an α > 1. Then, E[G ] < ∞ if and only if there exists an integer N > 2α such that |λN| > 0 and in this case there exists a constant

Cα depending only on α such that

−2α −α −2α E[G ] ≤ Cα N |λN| . (5.121)

− 1 R ∞ − y − −tX2 1 Proof. Notice λ α = e λ yα 1dy and E[e i ] = √ for all t > 0. If there Γ(α) 0 1+2t exists N > 2α such that |λN| > 0, then

− " N ! α # 1 Z ∞ N 2 2 −2α 2 2 −y∑ λi Xi α−1 E[G ] ≤ E ∑ λi Xi = E e i=1 y dy i=1 Γ(α) 0 Z ∞ N 1 α−1 2 − 1 = y ∏(1 + 2λi y) 2 dy. (5.122) Γ(α) 0 i=1

2 2 Since λi is non increasing in i and N > 2α, using the change of variables 1 + 2λNy = z we have

Z ∞ N Z ∞ α−1 2 − 1 α−1 2 − N y ∏(1 + 2λi y) 2 dy ≤ y (1 + 2λNy) 2 dy 0 i=1 0 Z ∞ Z ∞ α−1 2−α α−1 − N 2−α z − 1 α−1− N = 2λN (z − 1) z 2 dz = 2λN z 2 dz 1 1 z Z 1 N 2−α α−1 N −α−1 2−α Γ(α)Γ( 2 − α) = 2λN (1 − x) x 2 dx = 2λN , 0 Γ(N/2) which implies (5.121).

On the other hand, if |λi| = 0 for all i > 2α, let N ≤ 2α be the largest nonnegative integer such that |λN| > 0. Then, the inequality in (5.122) becomes an equality. Using

133 2 again that λi i≥1 is a decreasing sequence we have

Z ∞ N Z 1 Z ∞ α−1 2 − 1 2 − N α−1 α−1− N y ∏(1 + 2λi y) 2 dy ≥ (1 + 2λ1 ) 2 ( y dy + y 2 dy) = ∞, 0 i=1 0 1 and we conclude that E[G−2α ] = ∞. This completes the proof.

The following theorem describes the distance between the densities of F = I2( f ) and N(0,E[F2]).

2 Theorem 5.34. Let F = I2( f ) with f ∈ H given in (5.118). Assume that there exists m N > 6m + 6 b 2 c ∨ 1 , for some integer m ≥ 0, such that λN 6= 0. Then F admits an m times continuously differentiable density fF . Furthermore, if φ(x) denotes the density of N(0,E[F2]), then for k = 0,1,...,m,

! 1 ∞ 2 1 (k) (k) 4 4 2 2 2 sup fF (x) − φ (x) ≤ C ∑ λi ≤ C E[F ] − 3 E[F ] , x∈R i=1 where the constant C depends on N and λN.

Proof. Taking into account of (5.120), we have

2 ∞ ∞ 2 2 2 4 Var kDFkH = E 4 ∑ λi I1(ei) − 1 = 32 ∑ λi . (5.123) i=1 i=1

From (5.120) and Lemma 5.33 it follows that

−β −β/2 −β E[kDFkH ] ≤ Cβ/2N |λN| , (5.124) for all β < N. Then, the theorem follows from Theorem 5.13, taking into account (5.123).

134 Now we are ready to prove convergence of densities of random variables in the sec-

2 ond Wiener chaos. Consider a sequence Fn = I2( fn) with fn ∈ H , which can be written as ∞ fn = ∑ λn,ien,i ⊗ en,i, (5.125) i=1 where {λn,i,i ≥ 1} veriﬁes |λn,i| ≥ λn,i+1 for all i ≥ 1 and {en,i,i ≥ 1} are the corresponding eigenvectors.

Theorem 5.35. Let F = I ( f ) with f ∈ H 2 given by (5.125). Assume that {λ } n 2 n n n,i n,i∈N satisﬁes

2 ∞ 2 (i) σ := 2limn→∞ ∑i=1 λn,i > 0;

∞ 4 (ii) limn→∞ ∑i=1 λn,i = 0; √ (iii) infn sup m |λn,i| i > 0 for some integer m ≥ 0. i>6m+6(b 2 c∨1)

m Then, each Fn admits a density function fFn ∈C (R). Furthermore, for k = 0,1,...,m and if denotes the density of the law N(0, 2), the derivatives of f (k) converge uni- φ σ Fn formly to the the derivatives of φ with a rate given by

 ! 1 1  ∞ 2 ∞ 2 sup f (k)(x) − φ (k)(x) ≤ C λ 4 + 2 λ 2 − σ 2 , Fn  ∑ n,i ∑ n,i  x∈R i=1 i=1 where C is a constant depending only on m and the inﬁmum appearing in condition (iii).

2 2 Proof of Theorem 5.35. Note that E[(I1(en,i) − 1)(I1(en, j) − 1)] = 2δi j. Thus,

∞ 2 2 1 2 ∑ λn,i = k fnkH 2 = E[Fn ]. i=1 2

Then, the result follows from (5.123), (5.124) and Corollary 5.15.

135 Condition (iii) in Theorem5 .35 means that there exist a positive constant δ > 0 such m p that for each n we can ﬁnd an index i(n) > 6m + 6 b 2 c ∨ 1 with |λn,i(n)| i(n) ≥ δ.

5.7.2 Parameter estimation in Ornstein-Uhlenbeck processes

Consider the following Ornstein-Uhlenbeck process

Z t Xt = −θ Xsds + γBt, 0

where θ > 0 is an unknown parameter, γ > 0 is known and B = {Bt,0 ≤ t < ∞} is a standard Brownian motion. Assume that the process X = {Xt,0 ≤ t ≤ T} can be observed continuously in the time interval [0,T]. Then the least squares estimator (or the maximum R T XtdXt likelihood estimator) of θ is given by θ = 0 . It is known (see for example, [26], bT R T 2 0 Xt dt [22]) that, as T tends to inﬁnity, θbT converges to θ almost surely and

√ TF T(θ − θ) = − T →L N(0,2θ), (5.126) bT R T 2 0 Xt dt where Z T Z T FT = I2( fT ) = fT (t,s)dBtdBs, (5.127) 0 0 with 2 γ −θ|t−s| fT (t,s) = √ e . (5.128) 2 T

Recently, Hu and Nualart [16] extended this result to the case where B is a fractional

1 3 Brownian motion with Hurst parameter H ∈ [ 2 , 4 ), which includes the standard Brownian 1 R T 2 1 2 −1 motion case. Since T 0 Xt dt → 2 γ θ almost surely as T tends to inﬁnity, the main effort in proving (5.126) is to show the convergence in law of FT to the normal law

136 γ4 N(0, 2θ ). We shall prove that the density of FT converges as T tends to inﬁnity to the γ4 density of the normal distribution N(0, 2θ ).

2 Theorem 5.36. Let FT be given by (5.128) and let φ be the density of the law N(0,σ ),

2 γ4 where σ = 2θ . Then for each T > 0,FT has a smooth probability density fFT and for any k ≥ 0,

(k) (k) − 1 sup f (x) − φ (x) ≤ CT 2 , FT x∈R where the constant C depends on k, γ and θ.

Before proving the theorem, let us ﬁrst analyze the asypmototic behavior of the eigen-

2 values of fT . The Hilbert space corresponding to Brownian motion B is H = L ([0,T]).

2 2 Let QT : L ([0,T]) → L ([0,T]) be the Hilbert-Schmidt operator associated to fT , that is,

Z T (QT ϕ)(t) = fT (t,s)ϕ(s)ds (5.129) 0

2 ∞ 2 for ϕ ∈ L [0,T]. The operator QT has eigenvalues λT,1 > λT,2 > ··· ≥ 0 and ∑i=1 λT,i < ∞. The following lemma provides upper and lower bounds for these eigenvalues.

Lemma 5.37. Fix T > 0. Let fT be given by (5.128) and QT be given by (5.129). The eigenvalues λT,i of QT (except maybe one) satisfy the following estimates

γ2θ γ2θ < λT,i < . (5.130) √ π 2 √ π 2 2 iπ+ 2 2 iπ− 2 T θ + T T θ + T

Proof. Consider the eigenvalue problem QT ϕ = λϕ, that is,

Z T 2 Z t Z T γ −θ(t−s) −θ(s−t) fT (t,s)ϕ(s)ds = √ e ϕ(s)ds + e ϕ(s)ds = λϕ(t). 0 2 T 0 t (5.131)

137 Then, φ is differentiable and

γ2θ Z t Z T √ − e−θ(t−s)ϕ(s)ds + e−θ(s−t)ϕ(s)ds = λϕ0(t). (5.132) 2 T 0 t

Differentiating again we have

γ2θ Z t Z T √ −2ϕ(t) + θ e−θ(t−s)ϕ(s)ds + θ e−θ(s−t)ϕ(s)ds = λϕ00(t). 2 T 0 t

Comparing this expression with (5.131), we obtain

γ2θ (θ 2 − √ )ϕ(t) = ϕ00(t). (5.133) Tλ

Also, from (5.131) and (5.132) it follows that

ϕ(0) = θϕ0(0), ϕ(T) = −θϕ0(T). (5.134)

Equations (5.133) and (5.134) form a Sturm-Liouville system. Its general solution is of the form

ϕ(t) = C1 sin µt +C2 cos µt, where C1 and C2 are constants, and µ > 0 is an eigenvalue of the Sturm-Liouville system.

By eliminating the constants C1 and C2 from (5.133) and (5.134) we obtain

γ2θ − µ2 = θ 2 − √ . (5.135) Tλ

138 Then, the desired estimates on the eigenvalues of QT ϕ = λϕwill follow form estimates on µ. Note that the Neumann condition (5.134) yields

(µ2θ 2 − 1)sin µT = 2µθ cos µT.

If we write x = µθ > 0 (since µ,θ > 0), the above equation becomes

x x (x2 − 1)sin T = 2xcos T. θ θ

1 x The solution x = 1 corresponds to the eigenvalue µ = θ . If x 6= 1, then cos θ T 6= 0 and

x 2x tan T = . (5.136) θ x2 − 1

xi π For any i ∈ Z+, there is exactly one solution xi to (5.136) such that θ T ∈ (iπ − 2 ,iπ +

π xi 2 ). Corresponding to each xi is an eigenvalue µi = θ of the Sturm-Liouville system, iπ − π iπ + π satisfying 2 < µ < 2 . The corresponding eigenvalue λ of Q obtained from T i T i T Equation (5.135) satisﬁes the estimate (5.130).

Proof of Theorem 5.36. For each T, let us compute the second moment of FT .

Z T Z T 2 2 2 E FT = k fT kH⊗2 = fT (t,s) dsdt 0 0 γ4 Z T Z t γ4 γ4 = e−2θ(t−s)dsdt = − (1 − e−2θT ). 4T 0 0 2θ 8θT

2 Also, noticing that FT = I2( fT ) = δ ( fT ) and

3 2 DsDtFT = 3FT fT (t,s) + 6FT I1( f (·,t)) ⊗ I1( f (·,s)),

139 and using the duality between δ and D, we can write

4 h 2 3 i 2 E FT = E fT ,D FT H⊗2 = 3E FT h fT , fT iH⊗2 +6E FT h fT (t,s),I1( fT (·,t)) ⊗ I1( fT (·,s))iH⊗2

22 = 3 E FT + 6A, where

A = E FT h fT (t,s),I1( fT (·,t)) ⊗ I1( fT (·,s))iH⊗2

= fT (u,v),h fT (t,s), fT (u,t) ⊗ fT (v,s)iH⊗2 H⊗2 8 Z T Z T Z T Z T γ −θ(|u−v|+|t−s|+|u−t|+|v−s|) = 2 e dudvdtds. 16T 0 0 0 0

Because the integrand is symmetric, we have

8 Z T Z u Z v Z s γ −2θ(u−t) −1 A = 2 4! du dv ds dt e ≤ CT . 16T 0 0 0 0

Then, in order to complete the proof by applying Corollary 5.15, we only need to verify that condition (iii) of Theorem 5.35 holds for any integer m ≥ 1, which implies the uniform boundedness of the negative moments

h −β i sup E kDFT kH < ∞ T>0 for any β > 0. Fix β > 0, and for each T, let i(T) = bβ + 1c + bTc. Then, the lower bound in (5.130) yields

p p i(T)γ2/θ i(T)λ ≥ T,i(T) 2 √ (i+1/2)π T 1 + Tθ

140 p i(T)γ2/θ γ2/θ ≥ ≥ > 0, √ 2 2 max −1 g(r) T 1 + i(T) 4 π (β+2) ≤r≤1 T θ 2

−1 i(T) where in the last inequality we made the substitution r = T and set

√ π2 g(r) := r(1 + r−24 ). θ 2

This implies condition (iii) and the proof of the theorem is complete.

5.8 Appendix

In this section, we present the omitted proofs and some technical results.

R ∞ −y2/(2σ 2) Proof of Lemma 5.1. Since −∞{h(y) − E[h(N)]}e dy = 0, we have

Z x 2 2 Z ∞ 2 2 {h(y) − E[h(N)]}e−y /(2σ )dy = − {h(y) − E[h(N)]}e−y /(2σ )dy. −∞ x

Hence

Z x Z ∞ −y2/(2 2) k −y2/(2 2) {h(y) − E[h(N)]}e σ dy ≤ [ay + b + E |h(N)|]e σ dy. −∞ |x|

By using the representation (5.6) of fh and Stein’s equation (5.3) we have

Z x 0 |x| x2/(2σ 2) −y2/(2σ 2) fh(x) ≤ |h(x) − E[h(N)]| + 2 e {h(y) − E[h(N)]}e dy σ −∞ 1 2 2 Z ∞ 2 2 ≤ a|x|k + b + E |h(N)| + ex /(2σ ) y[ayk + b + E |h(N)|]e−y /(2σ )dy σ 2 |x| 1 a = a|x|k + (b + E |h(N)|) 1 + s (x) + s (x), (5.137) σ 2 1 σ 2 k+1

x2/(2σ 2) R ∞ k −y2/(2σ 2) where we let sk(x) = e |x| y e dy for any integer k ≥ 0.

141 k k Note that E |h(N)| ≤ aE |N| + b ≤ Ckaσ + b and

2 2 2 Z ∞ y x /(2σ ) − 2 2 s1(x) = e ye 2σ dy = σ x for all x ∈ R. Using integration by parts, we see by induction that for any integer k ≥ 1,

Z ∞ x2/(2σ 2) k+1 −y2/(2σ 2) sk+1(x) = e y e dy |x| Z ∞ 2 x2/(2σ 2) k −y2/(2σ 2) 2 k = σ e y d(−e ) = σ [|x| + k sk−1(x)]. |x|

Then if k ≥ 1 is even, we have

k 2 k 2 k−2 k−2 2 k−i i sk+1(x) ≤ Ckσ [|x| + σ |x| + ··· + σ s1(x)] ≤ Ckσ ∑ σ |x| . i=0

If k ≥ 1 is odd, we have

k 2 k 2 k−2 k−1 2 k−i i sk+1(x) ≤ Ckσ [|x| + σ |x| + ··· + σ (|x| + s0(x))] ≤ Ckσ ∑ σ |x| , i=0

q π where we used the fact that s0(x) ≤ s0(0) = 2 σ for all x ∈ R (indeed, when x ≥ 0 we y2 0 x x2/(2 2) R ∞ −y2/(2 2) x2/(2 2) R ∞ y − have s (x) = e σ e σ dy−1 ≤ e σ e 2σ2 dy−1 = 0; similarly 0 σ 2 x x σ 2 0 when x < 0, s0(x) ≥ 0). Putting the above estimates into (5.137) we complete the proof.

Proof of Remark 5.7. We shall prove these properties by induction. From T1 = T2 = 0, 2 2 (5.24) and (5.26) we know that T3 = Duδu, with J3 = {(0,0,1)}; and T4 = δuDuδu+ 3 Duδu, with J4 = {(1,0,1,0),(0,0,0,1)}. Now suppose the statement is true for all Tl with l ≤ k − 1 for k ≥ 5. We want to prove the multi-indices of Tk satisfy (a)–(c). This will

142 2 be done by studying the three operations, δuTk−1, DuTk−1 and ∂λ Hk−1(Duδu,δu)Duδu, in expression (5.26).

2 For the term ∂λ Hk−1(Duδu,δu)Duδu, we observe from (5.24) that

2 2 k−1−2i i−1 ∂λ Hk−1(Duδu,δu)Duδu = Duδu ∑ ick−1,iδu (Duδu) , 1≤i≤b(k−1)/2c

k k−1 whose terms have multi-indices (k − 1 − 2i,i − 1,1,0,...,0) ∈ N for 1 ≤ i ≤ b 2 c. Then, it is straightforward to check that these multi-indices satisfy (a), (b) and (c).

The term δuTk−1 shifts the multi-index (i0,i1,...,ik−2) ∈ Jk−1 to multi-index (i0 + k 1,i1,...,ik−2,0) ∈ N , which obviously satisﬁes (a), (b) and (c), due to the induction hypothesis.

The third term DuTk−1 shifts the multi-index (i0,i1,...,ik−2) ∈ Jk−1 to either α = k (i0 − 1,i1 + 1,...,ik−2,0) ∈ N if i0 ≥ 1, or to

  (i0,i1,...,i j0 − 1,i j0+1 + 1,...,ik−2,0), for 1 ≤ j0 ≤ k − 3; β =  (i0,i1,...,i j0 − 1,1), for j0 = k − 2,

if i j0 ≥ 1. It is easy to check that β satisfies properties (a), (b) and (c) and α satisfies properties (b) and (c). We are left to verify that α satisfies property (c). That is, we want to show that k−2 k − 1 1 + ∑ i j ≤ b c. (5.138) j=1 2

If k is odd, say k = 2m + 1 for some m ≥ 2, (5.138) is true because (i0,i1,...,ik−2) ∈ k−2 k−2 Jk−1, which implies by induction hypothesis that ∑ j=1 i j ≤ b 2 c = m − 1. If k is even, say k = 2m + 2, (5.138) is true because the following claim asserts that if i0 ≥ 1, then k−2 k−2 ∑ j=1 i j < b 2 c = m. 2m Claim: For (i0,i1,...,i2m) ∈ J2m+1 with m ≥ 1, if ∑ j=1 i j = m then i0 = 0.

143 2m Indeed, suppose (i0,i1,...,i2m) ∈ J2m+1, ∑ j=1 i j = m and i0 ≥ 1. We are going to 2m show that leads to a contradiction. First notice that i1 ≥ 1, otherwise i1 = 0 and ∑ j=2 i j = m, which is not possible because

2m i0 + 2m ≤ i0 + ∑ ji j ≤ 2m. j=1

Also, we must have i2m = 0, because otherwise property (a) implies i2m = 1 and i0 = i1 = ··· = i2m−1 = 0. Now we trace back to its parent multi-indices in J2m by reversing 2 the three operations. Of the three operations, we can exclude ∂λ H2m(Duδu,δu)Duδu and 2 δuT2m, because ∂λ H2m(Duδu,δu)Duδu generates (2m − 2 j, j − 1,1,0,...,0) with 1 ≤ j ≤ m, where j must be m; and δuT2m traces it back to (i0 − 1,i1,...,i2m−1) ∈ J2m, where 2m−1 i1 + ··· + i2m−1 = m > b 2 c. Therefore, its parent multi-index in J2m must come from the operation DuT2m and hence must be (i0 +1,i1 −1,...,i2m−1) ∈ J2m. Note that for this multi-index, i1 − 1 + ··· + i2m−1 = m − 1. Repeating the above process we will end up at

(i0 + i1,0,i2 ...,i2m−i1 ) ∈ J2m+1−i1 with i2 + ··· + i2m−i1 = m − i1, which contradicts the property (b) of J2m+1−i1 because

2m−i1 i0 + 2m − i1 ≤ i0 + i1 + ∑ ji j ≤ 2m − i1. j=2

−1 −1 k −1 D k−1 −1 E Recall that for any k ≥ 2 we denote DDF w = Dw ,DF and DDF w = D(DDF w ),DF . H H p k −1 The following lemma estimates the L (Ω) norms of DDF w .

2 2 Lemma 5.38. Let F = Iq( f ) with q ≥ 2 satisfying E[F ] = σ . For any β ≥ 1 we deﬁne 1/β −β 2 and Mβ = E kDFkH . Set w = kDFkH.

144 2β (i) If Mβ < ∞ for some β ≥ 6, then for any 1 ≤ r ≤ β+6

−1 3 2 DDF w r ≤ CMβ qσ − w 2 . (5.139)

2β (ii) If k ≥ 2 and Mβ < ∞ for some β ≥ 2k + 4, then for any 1 < r < β+2k+4

k −1 2k−2 k+2 2 DDF w ≤ C σ ∨ 1 M ∨ 1 qσ − w . (5.140) r β 2

β (iii) If k ≥ 1and Mβ < ∞ for any β > k + 2, then for any 1 < r < k+2

k −1 2k k+2 DDF w ≤ C σ ∨ 1 M ∨ 1 . (5.141) r β

−1 −1 −2 2 Proof. Note that DDF w = Dw ,DF H = −2w D F ⊗1 DF,DF . Then

3 −1 − 2 2 DDF w ≤ 2w D F ⊗1 DF H .

1 1 1 Applying Holder’s¨ inequality with r = p + 2 , yields

1 3p p −1 − 2 2 DDF w r ≤ 2 E(w ) D F ⊗1 DF 2 , which implies (5.139) by choosing p ≤ β/3 and taking into account (5.34). Notice that

1 3 1 β+6 we need 1 ≥ r ≥ β + 2 = 2β . Consider now the case k ≥ 2. From the pattern indicated by the ﬁrst three terms,

−1 −1 DDF w = Dw ,DF H , 2 −1 2 −1 ⊗2 −1 2 DDF w = D w ,(DF) H⊗2 + Dw ⊗ DF,D F H⊗2 , 3 −1 3 −1 ⊗3 2 −1 2 DDF w = D w ,(DF) H⊗3 + 3 D w ⊗ DF,D F ⊗ DF H⊗3

145 −1 2 2 −1 ⊗2 3 + Dw ⊗ D F,D F ⊗ DF H⊗3 + Dw ⊗ (DF) ,D F H⊗3 , we can prove by induction that

  k k i Dk w−1 ≤ C Diw−1 kDFki D jF j . DF ∑ H⊗i H  ∑ ∏ H⊗ j  i=1 k j=1 ∑ j=1 i j=k−i

j By (2.12), for any p > 1, D F p ≤ C kFk2 = Cσ. Applying Holder’s¨ inequality and assuming that s > r, we have,

k Dk w−1 ≤ C Diw−1 kDFki k−i. DF ∑ H⊗i H σ (5.142) r i=1 s

i i −1 We are going to see that kDFkH will contribute to compensate the singularity of D w H⊗i . First by induction one can prove that for 1 ≤ i ≤ m, Diw−1 has the following expression

i l i −1 l −(l+1) O α j β j D w = ∑(−1) ∑ w D F ⊗1 D F , (5.143) l=1 (α,β)∈Ii,l j=1

2l l where Ii,l = {(α,β) ∈ N : α j + β j ≥ 3,∑ j=1(α j + β j) = i + 2l}. In fact, for i = 1,

−1 −2 2 Dw = −2w D F ⊗1 DF,

which is of the above form because I1,1 = {(1,2),(2,1)}. Suppose that (5.143) holds for some i ≤ m − 1. Then,

i l i+1 −1 l+1 −(l+2) 2 O α j β j D w = ∑(−1) 2(l + 1) ∑ w (D F ⊗1 DF) D F ⊗1 D F l=1 (α,β)∈Ii,l j=1 i l l −(l+1) α j+1 β j α j β j+1 + ∑(−1) ∑ w ∑ (D F ⊗1 D F + D F ⊗1 D F) l=1 (α,β)∈Ii,l h=1

146 l O α j β j × D F ⊗1 D F , j=1, j6=h which is equal to

i+1 l l −(l+1) O α j β j ∑(−1) ∑ w D F ⊗1 D F . l=1 (α,β)∈Ii+1,l j=1

From (5.143) for any i = 1,...,k we can write

i l i i i −1 −(l+1)+ 2 α j β j D w ⊗i kDFkH ≤ w D F ⊗1 D F ⊗ + − , (5.144) H ∑ ∑ ∏ H α j β j 2 l=1 (α,β)∈Ii,l j=1

l l l where Ii,l = {(α,β) ∈ N ×N : α j +β j ≥ 3,∑ j=1(α j +β j) = i+2l}. Note that by (2.12),

α j β j 2 2 D F ⊗1 D F ≤ C kFk2 = Cσ p

for all p ≥ 1 and all α j,β j. This inequality will be applied to all but one of the contraction

l α j β j terms in the product ∏ j=1 D F ⊗1 D F ⊗ + −2 . We decompose the sum in (5.144) H α j β j i into two parts. If the index l satisﬁes l ≤ 2 − 1, then the exponent of w is nonnegative, 2 i and the p-norm of w can be estimated by a constant times σ , while for 2 − 1 < l this 1 1 1 exponent is negative. Then, using Holder’s¨ inequality and assuming that s = p + 2 , we obtain

i −1 i D w ⊗i kDFkH H s   i i−2 −(l+1)+ 2 2(l−1) α1 β1 ≤ C 1{i≥2}σ + ∑ w σ  D F ⊗1 D F (5.145). i p 2 2 −1

147 i k i Note that for l ≤ i ≤ k, l + 1 − 2 ≤ 2 + 1. Therefore, for 2 − 1 < l ≤ i

i −(l+1)+ 2 2l+2−i 2l+2−i k+2 w = M i ≤ M(k+2)p ≤ M(k+2)p ∨ 1. p 2(l+1− 2 )p

Therefore, using (5.34) we obtain

i −1 i 2i−2 k+2 2 D w ⊗i kDFkH ≤ C (σ ∨ 1)(M ∨ 1) qσ − w . (5.146) H s (k+2)p 2

Combining (5.146) and (5.142) and choosing p such that (k + 2)p ≤ β we get (5.140). Note that we need 1 k + 2 1 β + 2k + 4 1 > > + = , r β 2 2β

2β which holds if 1 < r < β+2k+4 . The proof of part (iii) is similar and omitted.

k The next lemma gives estimates on Duδu for k ≥ 0.

2 2 Lemma 5.39. Let F = Iq( f ) with q ≥ 2 satisfying E[F ] = σ . For any β ≥ 1 we deﬁne 1/β −β 2 Mβ = E kDFkH and denote w = kDFkH.

β (i) If Mβ < ∞ for some β > 3, then for any 1 < s < 3 ,

2 3 kδuks ≤ C(σ ∨ 1)(Mβ ∨ 1). (5.147)

β (ii) If k ≥ 1 and Mβ < ∞ for some β > 3k + 3, then for any 1 < s < 3k+3 ,

k 3k+3 kDuδuks ≤ Cσ (Mβ ∨ 1). (5.148)

2β (iii) If k ≥ 2 and Mβ < ∞ for some β > 6k + 6, then for any 1 < s < β+6k+6 ,

k 3k+3 2 kDuδuks ≤ Cσ (Mβ ∨ 1)kqσ − wk2. (5.149)

148 −1 −1 Proof. Recall that δu = qFw − DDF w . Then for any r > s,

−1 −1 kδuks ≤ C σkw kr + kDDF w ks .

−1 2 Then, kw kr = M2r and the result follows by applying Lemma 5.38 (iii) with k = 1 and β by choosing r < 3 . k To show (ii) and (iii) we need to ﬁnd a useful expression for Duδu. Consider the −1 k operator Du = w DDF . We claim that for any k ≥ 1 the iterated operator Du can be expressed as k "k−l # k −l i j −1 i0 Du = ∑ w ∑ bi ∏ DDF w DDF , (5.150) l=1 i∈Il,k j=1 where bi > 0 are real numbers and

k−l Il,k = {i = (i0,i1,...,il) : i0 ≥ 1,i j ≥ 0 ∀ j = 1,...,l, ∑ i j = k}. j=0

In fact, this is clearly true for k = 1. Assume (5.150) holds for a given k. Then

k+1 −1 k Du = w DDF D u k "k−l # −l −1 i j −1 i0 = ∑ lw DDF w ∑ bi ∏ DDF w DDF l=1 i∈Il,k j=1 k "k−l k−l # −l−1 ih+1 −1 i j −1 i0 + ∑ w ∑ bi ∑ DDF w ∏ DDF w DDF l=1 i∈Il,k h=1 j=1, j6=h k "k−l # −l−1 i j −1 i0+1 + ∑ w ∑ bi ∏ DDF w DDF . l=1 i∈Il,k j=1

Shifting the indexes, this can be written as

k "k−l # k+1 −l −1 i j −1 i0 Du = ∑ lw DDF w ∑ bi ∏ DDF w DDF l=1 i∈Il,k j=1

149 k+1 "k+1−l k+1−l # −l ih+1 −1 i j −1 i0 + ∑ w ∑ bi ∑ DDF w ∏ DDF w DDF l=2 i∈Il−1,k h=1 j=1, j6=h k+1 "k+1−l # −l i j −1 i0+1 + ∑ w ∑ bi ∏ DDF w DDF . l=2 i∈Il−1,k j=1

It easy to check that this coincides with

k+1 "k+1−l # −l i j −1 i0 ∑ w ∑ bi ∏ DDF w DDF . l=1 i∈Il,k+1 j=1

−1 −1 Also, note that δu = qFw + DDF w and

−1 2 −1 DDF δu = q + qFDDF w + DDF w .

By induction we can show that for any i0 ≥ 1

i0−1 i0 i0−1− j j −1 i0 −1 i0+1 −1 DDF δu = qδ1i0 + q ∑ ci, jDDF wDDF w + qFDDF w + DDF w , (5.151) j=1

where δ1i0 is the Kronecker symbol. Combining (5.150) and (5.151) we obtain

k "k−l # " k −l i j −1 Duδu = ∑ w ∑ bi ∏ DDF w × qδ1i0 l=1 i∈Il,k j=1

i0−1 # i0−1− j j −1 i0 −1 i0+1 −1 +q ∑ ci,0 jDDF wDDF w + qFDDF w + DDF w . j=1

k Next we shall apply Holder’s¨ inequality to estimate Duδu s. Notice that for l = k, i0 = k ≥ 2. Therefore,

k−1 k−l i Dkδ ≤ C kw−lk kD j w−1k δ + max kDh w−1k u u σ ∑ ∑ p ∏ DF r j 1i0 DF r0 s 1≤h≤i0+1 l=1 i∈Il,k j=1 −k h −1 +Cσ kw kp max kDDF w kρ = B1 + B2, 1≤h≤k+1 0

150 assuming that for l = 1,...,k−1, 1 > 1 +∑k−l 1 and 1 > 1 + 1 , and where C denotes s p j=0 r j s p ρ0 σ a function of σ of the form C(1 + σ M).

Let us consider ﬁrst the term B1. Note that if i0 = 1 there is at least one factor of the

r j −1 k−l form kDDF w kr j in the above product, because ∑ j=1 i j = k−1 ≥ 1. Then, we will apply the inequality (5.140) to one of these factors and the inequality (5.141) to the remaining i + ones. The estimate (5.141) requires 1 > j 2 for j = 1,...,k − l and 1 > i0+3 . On r j β r0 β the other hand, the estimate (5.140) requires 1 > i j+2 + 1 for j = 1,...,k − l and 1 > r j β 2 r0 i0+3 1 k−l β + 2 . Then, choosing p such that 2pl < β, and taking into account that ∑ j=0 i j = k we obtain the inequalities

1 1 k−l i + 2 i + 3 1 3k + 3 1 > + ∑ j + 0 + > + . s p j=1 β β 2 β 2

2β Hence, if s < β+6k+6 we can write

k−1 k−l 2l i j+2 i0+3 2 −1 B1 ≤ Cσ ∑ Mβ ∏(Mβ ∨ 1)(Mβ ∨ 1)kqσ − w k2 l=1 j=1 3k+3 2 −1 ≤ Cσ (Mβ ∨ 1)kqσ − w k2.

For the term B2 we use the estimate (5.140) assuming 2pk < β and

1 1 k + 3 1 3k + 3 1 > + + > + . s p β 2 β 2

This leads to the same estimate and the proof of (5.149) is complete. To show the estimate (5.148) we proceed as before but using the inequality (5.141) for all the factors. In this

1 case the summand 2 does not appear and we obtain (5.148).

151 Chapter 6

Non-degeneracy of some Sobolev pseudo-norms of fBms

As we have seen in the previous chapters, non-degeneracy of a random variable plays an fundamental role when applying the density formula in Malliavin calculus. In this chapter, we shall apply an upper bound estimate in small deviation theory to prove the non-degeneracy of some fucntionals of fractional Brownian motion (fBm).

6.1 Introduction

H H Let B = Bt : t ≥ 0 be a fractional Brownian motion (fBm) on (Ω,F ,P). That is, H Bt : t ≥ 0 is a centered Gaussian process of Hurst parameter H ∈ (0,1) with covariance 1 R (t,s) = E(BHBH) = (|t|2H + |s|2H − |t − s|2H). (6.1) H t s 2

Consider the random variable F given by a functional of BH:

Z 1 Z 1 H H 2p Bt − Bt0 0 F = 0 q dtdt , (6.2) 0 0 |t −t | where p,q ≥ 0 satisfy (2p − 2)H > q − 1.

152 1 H In the case of H = 2 , B is a Brownian motion, and the random variable F is the Sobolev norm on the Wiener space considered by Airault and Malliavin in [1]. This norm plays a central role in the construction of surface measures on the Wiener space. Fang [9] showed that F is non-degenerate (see its deﬁnition below). Then it follows from

1 the well-known density formula in Malliavin calculus that the law of F 2 has a smooth density. In this note we shall show that for all H ∈ (0,1), F is non-degenerate. In order to state our result precisely, we need some notations from Malliavin calculus

(for which we refer to Nualart [39, Section 1.2]). Denote by E the set of all step functions on [0,1]. Let H be the Hilbert space deﬁned as closure of E with respect to the scalar product

h1[0,t],1[0,s]iH = RH(t,s), for s,t ∈ [0,1].

H Then the mapping 1[0,t] 7→ Bt extends to a linear isometry between H and the Gaussian space spanned by BH. We denote this isometry by BH. Then, for any h,g ∈ H, BH( f ) and

H H H B (g) are two centered Gaussian random variables with E[B (h)B (g)] = hh,giH. We , deﬁne the space D1 2 as the closure of the set of smooth and cylindrical random variable of the form

H H G = f (B (h1),...,B (hn))

∞ n with hi ∈ H, f ∈ Cp (R ) ( f and all its partial derivatives has polynomial growth) under the norm q 2 2 kGk1,2 = E[G ] + E[kDGkH], where the DF is the Malliavin derivative of F deﬁned as

n ∂ f H H DG = ∑ (B (h1),...,B (hn))hi. i=1 ∂xi

153 1,2 We say that a random vector V =(V1,...,Vd) whose components are in D is non- −1 degenerate if its Malliavin matrix γV = DVi,DVj H is invertible a.s. and (detγV) ∈ Lk(X), for all k ≥ 1 (see for instance [39, Deﬁnition 2.1.1]). Our main result is the following theorem.

Theorem 6.1. For all H ∈ (0,1), the functional F of a fBm BH given in (6.2) is nondegenerate. That is,

−1 k kDFkH ∈ L (X), for all k ≥ 1. (6.3)

We shall follow the same scheme introduced in [9] to prove Theorem 6.1. That is, it sufﬁces to prove that for any integer n, there exists a constant Cn such that

n P(kDFkH ≤ ε) ≤ Cnε (6.4) for all ε small. This kind of inequality is called upper bound estimate in small deviation theory (also called small ball probability theory, for which we refer to [23] and the reference there in), which is still a challenging topic. To prove (6.4), we will need an upper bound estimate of the small deviation for the path variance of the fBm, which is introduced in the following section. We comment that Li and Shao [24, Theorem 4] proved that

1 1 H H 2p Z Z Bt − Bs C P( q dtds ≤ ε) ≤ exp{− } (6.5) 0 0 |t − s| εβ for p > 0, 0 ≤ q < 1 + 2pH, q 6= 1 and β = 1/(pH − max{0,q − 1}. But (6.5) gives the small ball probability of F, not of kDFkH.

154 6.2 An estimate on the path variance of fBm

H H Lemma 6.2 (Estimate of the path variance of the fBm). Let B = Bt : t ≥ 0 be a fBm. H For a > b ≥ 0, consider the path variance V[a,b](B ) deﬁned by

Z b Z b H H 2 dt H dt 2 V[a,b](B ) = Bt − ( Bt ) . a b − a a b − a

− 2H+1 1 π 2H 2H Then for cH = H (2H + 1)sin 2H+1 (Γ(2H + 1)sin(πH)) ,

1 H 2 lim ε H logP(V[a,b](B ) ≤ ε ) = −(b − a)cH. (6.6) ε→0

Actually, we will only need

1 H 2 limsupε H logP(V[a,b](B ) ≤ ε ) < ∞. (6.7) ε→0

1 In the case of H = 2 , this estimate of the path variance for Brownian motion was introduced by Malliavin [27, Lemma 3.3.2], by using the following Payley–Wiener expansion of Brownian motion:

√ ∞ 1 Bt = tG + 2 ∑ (Xk cos2πkt +Yk sin2πkt), a.s. for all t ∈ [0,1], (6.8) k=1 2πk where G, Xk,Yk, k ∈ N, are i.i.d. standard Gaussian random variables. Then the estimate (6.7) follows by observing that V (B) = 1 ∞ 1 (X2 +Y 2), a sum of 2(1) random [0,1] 2π2 ∑k=1 2πk k k χ variables. The above expansion of Brownian motion can be obtained by integrating n √ √ o an expansion of white noise on the orthonormal basis 1, 2cos2πkt, 2sin2πkt of L2[0,1]. Payley–Wiener expansion of fBm has been proved recently by Dzhaparidze and

155 van Zanten [8]:

∞ H 1 Bt = tX + ∑ [Xk(cos2ωkt − 1) +Yk sin2ωkt], (6.9) k=1 ωk where 0 < ω1 < ω2 < ... are the real zeros of J−H (the Bessel function of the ﬁrst kind of order −H), and X, Xk,Yk, k ∈ N, are independent centered Gaussian random variables with variance

2 2 2 2 2 EX = σH,EXk = EYk = σk ,

3 2 Γ( 2 −H) 2 2 2 ωk 2H with σH = 1 and σk = σH(2 − 2H)Γ (1 − H) 2 J−H(ωk). Because 2HΓ(H+ 2 )Γ(3−2H) H the path variance V[0,1](B ) becomes difﬁcult to evaluate, it is clear that the techniques of [27, Lemma 3.3.2] does no longer work. Fortunately, the recent developments of small deviation theory makes (6.6) ready to be reached.

Proof. In [30, Theorem 3.1 and Remark 3.1] Nazarov and Nikitin proved that for any square integrable random variable G and any nonnegative function ψ ∈ L1[0,1],

2H+1 Z 1 Z 1 2H 1 H 2 2 1 lim ε H logP( (Bt − G) ψ(t)dt ≤ ε ) = −cH ψ(t) 2H+1 dt . (6.10) ε→0 0 0

Noticing that by the self-similarity property of fBm,

Z b 2 Z 1 2 H H H dt H H du V[a,b](B ) = Bt − B = b Bbu − B a b − a a/b b − a

2 2H+1 R 1 H −H H du has the same distribution as b a/b Bu − b B b−a . Then, Lemma 6.2 follows −H H from (6.10) by taking G = b B and ψ(t) = 1[a/b,1](t).

We comment that Bronski [3] proved (6.10) for G = 0 (which can be dropped, since a random variable G doesn’t contribute to the asymptotics of the Karhunen–Loeve eigen-

156 values) and ψ ≡ 1 by estimating the asymptotics of the Karhunen–Loeve eigenvalues of fBm.

6.3 Proof of the main theorem

In this section we prove (6.3) by estimating P(kDFkH ≤ ε) for ε small. For simplicity, we denote

I = t,t0 ∈ [0,1]2,t0 ≤ t ,

~t = t,t0, d~t = dtdt0.

Lemma 6.3. Let Q(~t,~s) = h1[t0,t],1[s0,s]iH. Then the operator Q deﬁned by

Z Q f (~t) = Q(~t,~s) f (~s)d~s, f ∈ L2(I) I is a symmetric positive compact operator on L2(I).

Proof. Compactness follows from that Q(~t,~s) ∈ L2(I × I). The function Q(~t,~s) is symmetric, so is the operator Q. Finally, Q is positive because for any f ∈ L2(I),

2 Z Z Z hQ f , f iL2(I) = Q(~t,~s) f (~s)d~s f (~t)d~t = 1[t0,t] f (~t)d~t, . I I I H

Then, it follows that Q has a sequence of decreasing eigenvalues { } , i.e. ≥ λn n∈N λ1 ··· ≥ > 0, and → 0. The corresponding normalized eigen–functions { } λn λn ϕn n∈N 2 form an orthonormal basis of L (I). Each of them are continuous because φn(~t) =

157 −1 R λn I Q(~t,~s)φn(~s)d~s and Q(~t,~s) is continuous. We can write

Q(~t,~s) = ∑ λnϕn(~t)ϕn (~s). (6.11) n≥1

From the deﬁnition of Malliavin derivative we have

Z BH − BH 2p−1 t t0 H H ~0 DrF = 2p 0 q sign(Bt − Bt0 )1[t0,t](r)dt [0,1]2 |t −t | Z BH − BH 2p−1 t t0 H H ~0 = 4p 0 q sign(Bt − Bt0 )1[t0,t](r)dt . I |t −t |

2p−1 BH −BH 0 | t t0 | H H Then denoting ∆(t,t ) = |t−t0|q sign(Bt − Bt0 ) we have

Z 2 2 2 0 0 kDFkH = 16p 1[t0,t](·)∆(t,t )dtdt (6.12) I H Z 2 0 0 = 16p h1[t0,t],1[s0,s]iH∆(t,t )∆(s,s )d~td~s. I×I

Applying (6.11) with (6.12) we have

2 2 2 kDFkH = 16p ∑ λiVi , (6.13) i≥1 where we denote

Z H H 2p−1 0 Bt − Bt0 H H 0 Vi = ϕi(t,t ) 0 q sign(Bt − Bt0 )dtdt . (6.14) I |t −t |

n−1 n For each β = (β1,...,βn) ∈ S (the unit sphere in R ), let Ψβ = β1ϕ1 +···+βnϕn. We denote Z BH − BH 2p−2 2 ~ t t0 ~ Gβ = Ψβ (t) 0 q dt. (6.15) I |t −t |

158 n−1 Lemma 6.4. There exists a constant Cp,H > 0 such that for all β ∈ S and ε > 0,

n 1 o − 2H(p−1) P Gβ ≤ ε ≤ exp −Cp,Hε . (6.16)

n−1 Proof. For any β ∈ S , Ψβ 6≡ 0 since ϕi,...,ϕn are linearly independent. Since Ψβ is ~ 0 continuous on I, there exists tβ = (tβ ,tβ ) ∈ I, δβ and ρβ such that

~ ~ 0 0 Ψβ (t) ≥ ρβ > 0, for all t ∈ Aβ := [tβ − δβ ,tβ + δβ ] × [tβ − δβ ,tβ + δβ ] ⊂ I.

Meanwhile, C = 2maxi∈{1,...,n} sup~t∈I ϕ(~t) < ∞ and

2 ~ 2 ~ 0 Ψβ (t) − Ψβ 0 (t) ≤ C β − β .

0 0 Then for any~t ∈ Aβ and any β,β satisfying kβ − β k ≤ ρβ /2C, one has

2 ~ 2 ~ 2 ~ 2 ~ Ψβ (t) ≥ Ψβ 0 (t) − Ψβ (t) − Ψβ 0 (t) ≥ ρβ 0 /2. (6.17)

ρ n−1 n−1 m i βi ~ ~ Note that S has a ﬁnite cover S ⊂ ∪i=1B(β , 2C ). Denote ρi = ρβ i , δi = δβ i , ti = tβ i n−1 i and Ai = Aβ i . Then it follows from (6.17) that for any β ∈ S , there exists a β such that 2 ~ ~ Ψβ (t) ≥ ρi/2, for all t ∈ Ai.

Then noticing that |t −t0| ≤ 1 and applying Jensen’s inequality we obtain

H H 2p−2 ρ Z B − B 0 ρ Z 2p−2 i t t ~ i H H ~ Gβ ≥ 0 q dt ≥ Bt − Bt0 dt 2 Ai |t −t | 2 Ai p−1 ρ Z 2 i H H ~ ≥ p−2 Bt − Bt0 dt . (6.18) 2(2δi) Ai

159 1 R b Note that for f ∈ C[a,b] with average f = b−a a f (ξ)dξ, we have

1 Z b 1 Z b ( f (ξ) − f )2dξ ≤ ( f (ξ) − c)2dξ b − a a b − a a for any number c. Then

0 Z Z ti+δi Z ti +δi Z ti+δi 2 H H2 H H2 0 H H B − B 0 d~t = B − B 0 dtdt ≥ 2δi B − B dt t t 0 t t t Ai ti−δi ti −δi ti−δi (6.19) where BH = R ti+δi BHdt. Combining (6.18)–(6.19) and applying Lemma 6.2 we obtain ti−δi t

Z ti+δi 2 1 1 H H − p−1 p−1 P Gβ ≤ ε ≤ P( Bt − B dt ≤ (ρiδi) ε ) ti−δi 1 − 1 ≤ exp{−cHδi (ρiδi) 2H(p−1) ε 2H(p−1) }.

1 for any i = 1,...,m. Let Cp,H = cH infi δi (ρiδi) 2H(p−1) , one gets (6.16).

Remark: At the ﬁrst glance, it seems that (6.16) can be obtained by applying (6.5) to the ﬁrst inequality in (6.18). But (6.5) can only be applied to square interval on the diagonal like [a,b]×[a,b] (after applying the scaling and self-similarity property of fBm),

0 0 and here the interval Ai = [ti − δi,ti + δi] × [ti − δi,ti + δi] is off diagonal.

Lemma 6.5. For any integer n, the random vector V = (V1,...,Vn) deﬁned in (6.14) is non-degenerate.

Proof. Denote by M = DVi,DVj H the Malliavin covariance of V. We want to show −1 k n that (detM) ∈ L , for any k ≥ 1. Note that detM ≥ γ1 , where γ1 > 0 is the smallest −1 nk eigenvalue of the positive deﬁnite matrix M. Then it sufﬁces to show that γ1 ∈ L , for

160 any k ≥ 1, for which is enough to estimate P(γ1 ≤ ε) for ε small. We have

! 2 n γ1 = inf (Mβ,β) = inf D βiVi . (6.20) kβk=1 kβk=1 ∑ i=1 H

n−1 n For any β = (β1,...,βn) ∈ S , let Ψβ (~t) = ∑i=1 βiϕi(~t),

! 2p−2 n Z BH − BH ~ t t0 ~ Dr ∑ βiVi = (2p − 1) Ψβ (t) 0 q 1[t0,t](r)dt. i=1 I |t −t |

Applying (6.11) as we computed (6.13),

! 2 2p−2 !2 n Z 1 Z BH − BH 2 ~ t t0 ~ D ∑ βiVi = (2p − 1) dr Ψβ (t) 0 q 1[t0,t](r)dt 0 I |t −t | i=1 H 2p−2 !2 Z BH − BH 2 ~ ~ t t0 ~ = (2p − 1) ∑ λi ϕi(t)Ψβ (t) 0 q dt i≥1 I |t −t | n 2 2 ≥ (2p − 1) ∑ λiqi , i=1

2p−2 (BH −BH ) R ~ ~ t t0 ~ n where qi = I ϕi(t)Ψβ (t) |t−t0|q dt. Recall (6.15), we have Gβ = ∑i=1 βiqi. Since

λ1 ≥ ··· ≥ λn > 0, we have

n n n 2 2 2 2 λn 2 ∑ λiqi ≥ λn ∑ qi ≥ λn ∑ βi qi ≥ Gβ , i=1 i=1 i=1 n

n 2 1 n 2 where in the third inequality we used the fact that ∑i=1 ai ≥ n (∑i=1 ai) . Therefore

! 2 n λ D β V ≥ (2p − 1)2 n G2 . (6.21) ∑ i i n β i=1 H

161 Combining (6.20) and (6.21) we have

2 λn 2 γ1 = inf (Mβ,β) ≥ (2p − 1) inf Gβ . (6.22) kβk=1 n kβk=1

1 For any ε > 0 and 0 < α < 2H(p−1) , let

Wβ = Gβ ≥ ε ,

n 2 −α o Wn = kDVikH ≤ expε ,i = 1,...,n .

0 n−1 On Wn, for any β,β ∈ S we have

0 0 0 1 (Mβ,β) − Mβ ,β ≤ Cn β − β exp , εα

0 where Cn is a constant independent of β,β and ε.

m i 2 n−1 i n−1 Note that we can ﬁnd a ﬁnite cover ∪i=1B(β ,exp(− εα )) of S with β ∈ S and

2n m ≤ C exp . εα

n−1 i Then on Wn, for any β ∈ S , there exists a β such that

i i 1 2 (Mβ,β) ≥ Mβ ,β −Cn exp exp(− ). εα εα

2 λn On Wβ i ∩Wn, applying (6.22) with An = (2p − 1) n and taking ε small enough,

2 1 An 2 (Mβ,β) ≥ Anε −Cn exp(− ) ≥ ε . εα 2

162 m Hence, on ∩i=1Wβ i ∩Wn,

An 2 γ1 = inf (Mβ,β) ≥ ε > 0. (6.23) kβk=1 2

On the other hand, applying Lemma 6.4, we have

m √ C P(∪m W c ) ≤ P(∪m W c ) ≤ m 2exp(− p,H ) i=1 β i ∑ i=1 β i 1/2H(p−1) i=1 ε 2n C C ≤ C exp exp(− p,α ) ≤ C exp(− ). (6.24) εα ε1/2H(p−1) ε1/2H(p−1)

Also, by Chebyshev’s inequality,

1 P(W c) ≤ C exp(− ). (6.25) n εα

Then it follows from (6.23)–(6.25) that for ε small,

A 1 P(γ < n ε2) ≤ C exp(− ). 1 2 εα

This completes the proof.

Proof of Theorem 6.1. Note that

n 2 2 2 2 2 kDFkH = 16p ∑ λiVi ≥ 16p λn ∑ Vi , (6.26) i≥1 i=1

2 n 2 for any integer n. Then, denoting |V| = ∑i=1 Vi we have

ε P(kDFkH < ε) ≤ P |V| < √ . 4p λn

163 Since V = (V1,...,Vn) is nondegenerate, then it has a smooth density fVn (x). Then we have ε n P |V| < √ ≤ Cn,pε , 4p λn √ 2πn/2 −n where Cn,p = n 4p λn max|x|≤1 fVn (x). Now the theorem follows. nΓ( 2 )

164 Bibliography

[1] H. Airault and P. Malliavin. Interation´ geom´ etrique´ sur l’espace de Wiener. (French) [Geometric integration on the Wiener space] Bull. Sci. Math. (2) 112 (1988), no. 1, 3–52. Cited on7, 153

[2] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman-Kac formula and intermittence. J. Statist. Phys. 78 (1995), no. 5-6, 1377–1401. Cited on 17

[3] J. C. Bronski. Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theoret. Probab. 16 (2003), no. 1, 87–100. Cited on 156

[4] S. Chatterjee and E. Meckes. Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008), 257–283. Cited on6, 79

[5] Louis H.Y. Chen, L. Goldstein, and Q.-M. Shao. Normal approximation by Stein’s method. Probability and its Applications (New York). Springer, Heidelberg, 2011. Cited on 79, 81

[6] D. A. Dawson, Z. Li, and H. Wang. Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab. 6 (2001), no. 25, 1–33. Cited on 54

[7] D. A. Dawson, J. Vaillancourt, and H. Wang. Stochastic partial differential equations for a class of interacting measure-valued diffusions. Ann. Inst. H. Poincare´ Probab. Statist. 36 (2000), no. 2, 167–180. Cited on4, 54

[8] K. Dzhaparidze and H. van Zanten. Krein’s spectral theory and the Paley-Wiener expansion for fractional Brownian motion. Ann. Probab. 33 (2005), no. 2, 620–644. Cited on 156

[9] S. Fang. Non-deg´ en´ erescence´ des pseudo-normes de Sobolev sur l’espace de Wiener. (French) [Nondegeneracy of Sobolev pseudonorms on Wiener space] Bull. Sci. Math. 115 (1991), no. 2, 223–234. Cited on7, 153, 154

[10] J. Hoffmann-Jørgensen, L. A. Shepp, and R. M. Dudley. On the lower tail of Gaus- sian seminorms. Ann. Probab. 7 (1979), no. 2, 319–342. Cited on

165 [11] Y. Hu, M. Jolis, and S. Tindel. On Stratonovich and Skorohod stochastic calculus for Gaussian processes. preprint, 2011. Cited on 18

[12] Y. Hu, F. Lu, and D. Nualart. Holder¨ continuity of the solution for a class of nonlinear SPDEs arising from one-dimensional superprocesses. Probab. Theory Relat. Fields, to appear. Cited on4

[13] Y. Hu, F. Lu, and D. Nualart. Feynman-Kac formula for a stochastic heat equation driven by fractional noise. Ann. Probab., 40:1041–1068, 2012. Cited on2

[14] Y. Hu and D. Nualart. Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab., 33:948–983, 2005. Cited on5

[15] Y. Hu and D. Nualart. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields, 143:285–328, 2009. Cited on 17, 18, 39, 47

[16] Y. Hu and D. Nualart. Parameter estimation for fractional Ornstein-Uhlenbeck processes. Statist. Probab. Lett., 80:1030–1038, 2010. Cited on 136

[17] Y. Hu, D. Nualart, and J. Song. Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab., 39:291–326, 2011. Cited on3, 16, 17, 18, 39, 47

[18] M. Kac, On distributions of certain Wiener functionals. Trans. Amer. Math. Soc. 65:1–13, 1949. Cited on3

[19] I. Kruk and F. Russo. Malliavin-Skorohod calculus and Paley-Wiener integral for covariance singular processes arxiv. preprint, 2010. Cited on 18

[20] N. V. Krylov. An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives. Math. Surveys Monogr., 64, 185-242,. Amer. Math. Soc., Providence, RI, 1999. Cited on4, 56

[21] N. Konno and T. Shiga. Stochastic partial differential equations for some measure- valued diffusions. Probab. Theory Related Fields, 79:201–225, 1988. Cited on 55

[22] Kutoyants, Yu. A. Statistical inference for ergodic diffusion processes. Springer Series in Statistics. Springer-Verlag, 2004. Cited on 136

[23] W. V. Li and Q. Shao. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, Handbook of Statist.,19, North-Holland, Amsterdam:533–597, 2001. Cited on 132, 154

[24] W.V. Li and Q. Shao. Small ball estimates for Gaussian processes under Sobolev type norms. J. Theoret. Probab., 12:699–720, 1999. Cited on 154

166 [25] Z. Li, H. Wang, J. Xiong, and X. Zhou. Joint continuity for the solutions to a class of nonlinear SPDE. Probab. Theory Related Fields, 153. Cited on4, 53, 54, 55, 66, 67, 73

[26] Liptser, R. S.; Shiryaev, A. N. Statistics of random processes. II. Applications. Second edition in Applications of Mathematics (New York), 6. Springer-Verlag, 2001. Cited on 136

[27] P. Malliavin. Ck-hypoellipticity with degeneracy. II. stochastic analysis. Proc. Inter- nat. Conf., Northwestern Univ., Evanston, Ill., 1978, pages 199–214 and 327–340, Academic Press, New York–London, 1978, 1978. Cited on 155, 156

[28] E. Meckes. An inﬁnitesimal version of stein’s method of exchangeable pairs. Ph.D. thesis, Stanford University, 2006. Cited on 79

[29] O. Mocioalca and F. Viens. Skorohod integration and stochastic calculus beyond the fractional Brownian scale. J. Funct. Anal., 222:385–434, 2005. Cited on 16

[30] A. I. Nazarov and Ya. Yu. Nikitin. Logarithmic l2-small ball asymptotics for some fractional Gaussian processes. . Theory Probab. Appl., 49:645–658, 2005. Cited on 156

[31] S. Noreddine and I. Nourdin. On the Gaussian approximation of vector-valued multiple integrals. J. Multivariate Anal., 102:1008–1017, 2011. Cited on 114

[32] I. Nourdin. Yet another proof of the Nualart-Peccati criterion. Electron. Comm. Probab., 16:467–481, 2011. Cited on5

[33] I. Nourdin, D. Nualart, and G. Poly. Absolute continuity and convergence of densities for random vectors on wiener chaos. preprint. Cited on 77

[34] I. Nourdin and G. Peccati. Stein’s method on wiener chaos. Probab. Theory Related Fields, 145:75–118, 2009. Cited on5, 77, 83

[35] I. Nourdin and G. Peccati. Normal approximations with Malliavin calculus. From Stein’s method to universality. Cambridge University Press., Cambridge tracts in mathematics. edition, 2012. Cited on5,8, 13, 77, 81, 83, 85, 114, 115, 119, 126

[36] I. Nourdin, G. Peccati, and G. Reinert. Second order Poincare´ inequalities and CLTs on Wiener space. J. Funct. Anal., 257:593–609, 2009. Cited on 85

[37] I. Nourdin, G. Peccati, and A. Reveillac.´ Multivariate normal approximation using stein’s method and Malliavin calculus. Ann. Inst. Henri Poincare´ Probab. Stat., 46:45–58, 2010. Cited on 79, 114

[38] I. Nourdin and G. Poly. Convergence in total variation on wiener chaos. preprint. Cited on 77

167 [39] D. Nualart. The Malliavin calculus and related topics. Springer-Verlag, 2nd edition, 2006. Cited on8, 11, 12, 44, 85, 89, 91, 105, 117, 153, 154

[40] D. Nualart and S. Ortiz-Latorre. Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl., 118:614–628, 2008. Cited on5, 77, 131

[41] D. Nualart and G. Peccati. Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab., 33:177–193, 2005. Cited on5, 77

[42] D. Nualart and W. Schoutens. Backward stochastic differential equations and Feynman-Kac formula for Levy´ processes, with applications in ﬁnance. Bernoulli, 7:761–776, 2001. Cited on 16

[43] D. Ocone and E.´ Pardoux. A stochastic Feynman-Kac formula for anticipating SPDEs, and application to nonlinear smoothing. Stochastics Stochastics Rep., 45:79–126, 1993. Cited on 16

[44] H. Ouerdiane and Jose´ L. Silva. Generalized Feynman-Kac formula with stochastic potential. (English summary) Inﬁn. Dimens. Anal. Quantum Probab. Relat. Top., 5:243–255, 2002. Cited on 16

[45] G. Peccati and M. S. Taqqu. Stable convergence of generalized l2 stochastic integrals and the principle of conditioning. Electron. J. Probab., 12:447–480, 2007. Cited on5

[46] G. Peccati and M. S. Taqqu. Stable convergence of multiple Wiener-Itoˆ integrals. J. Theoret. Probab., 21:527–570, 2008. Cited on5

[47] G. Peccati and C. A. Tudor. Gaussian limits for vector-valued multiple stochastic integrals. Springer, Berlin, 2005. Cited on5

[48] M. Raic. A multivariate CLT for decomposable random vectors with ﬁnite second moments. J. Theoret. Probab., 17:573–603, 2004. Cited on 79

[49] M. Reimers. One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields, 81:319–340, 1989. Cited on 55

[50] G. Reinert and A. Rollin.¨ Multivariate normal approximation with stein’s method of exchangeable pairs under a general linearity condition. Ann. Probab., 37:2150– 2173, 2009. Cited on6, 79

[51] S. G. Samko, A. A. Kilbas, and O. I. Marichev. Fractional Integrals and Deriva- tives. Theory and applications. Edited and with a foreword by S. M. Nikol’ski˘ı. Translated from the 1987 russian original. Revised by the authors. Gordon and Breach Science Publishers, Yverdon, 1993. xxxvi+976 pp. Cited on 21

168 [52] C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, II: Probability theory:583–602, Univ. California Press, Berkeley, Calif. 1972. Cited on 81

[53] H. Wang. State classiﬁcation for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Related Fields, 109:39–55, 1997. Cited on 54

[54] H. Wang. A class of measure-valued branching diffusions in a random medium. Stoch. Anal. Appl., 16:753–786, 1998. Cited on 54

[55]M.Z ahle.¨ Integration with respect to fractional functions and stochastic calculus. I. Probab. Theory Relat. Fields, 111:333–374, 1998. Cited on 22

169