The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications

In this article, we study the hyperbolic Anderson model driven by a space-time colored Gaussian homogeneous noise with spatial dimension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1,2$$\end{document}d=1,2. Under mild assumptions, we provide \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp-estimates of the iterated Malliavin derivative of the solution in terms of the fundamental solution of the wave solution. To achieve this goal, we rely heavily on the Wiener chaos expansion of the solution. Our first application are quantitative central limit theorems for spatial averages of the solution to the hyperbolic Anderson model, where the rates of convergence are described by the total variation distance. These quantitative results have been elusive so far due to the temporal correlation of the noise blocking us from using the Itô calculus. A novel ingredient to overcome this difficulty is the second-order Gaussian Poincaré inequality coupled with the application of the aforementioned \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp-estimates of the first two Malliavin derivatives. Besides, we provide the corresponding functional central limit theorems. As a second application, we establish the absolute continuity of the law for the hyperbolic Anderson model. The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}Lp-estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continuity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2].


Introduction
One of the main tools of modern stochastic analysis is Malliavin calculus. To put it short, this is a differential calculus on a Gaussian space that represents an infinite dimensional generalization of the usual analytical concepts on an Euclidean space. The Malliavin calculus (also known as the stochastic calculus of variations) was initiated by Paul Malliavin [21] to give a probabilistic proof of Hörmander's "sum of squares" theorem. It has been further developed by Stroock, Bismut, Watanabe and others. One of the main applications of Malliavin calculus is the study of regularity properties of probability laws, for example, the laws of the solutions to certain stochastic differential equations and stochastic partial differential equations (SPDEs), see e.g. [27,Chapter 2]. The Malliavin calculus is also useful in formulating and interpreting stochastic (partial) differential equations when the solution is not adapted to a Brownian filtration, which is the case of SPDEs driven by a Gaussian noise that is colored in time.
Recently, the Malliavin calculus has found another important application in the work of Nualart and Ortiz-Latorre [28], which paved the road for Stein to meet Malliavin. The authors of [28] applied the Malliavin calculus (notably the integration by parts formula) to characterize the convergence in law of a sequence of multiple Wiener integrals, and they were able to give new proofs for the fourth moment theorems of Nualart, Peccati and Tudor [30,37]. Soon after the work [28], Nourdin and Peccati combined Malliavin calculus and Stein's method of normal approximation to quantify the fourth moment theorem. Their work [24] marked the birth of the so-called Malliavin-Stein approach. This combination works admirably well, partially because one of the fundamental ingredients in Stein's method-the so-called Stein's lemma (2.6)-that characterizes the normal distribution, is nothing else but a particular case of the integration by parts formula (2.5) in Malliavin calculus. We refer interested readers to [44,Section 1.2] for a friendly introduction to this approach.
The central object of study in this paper is the stochastic wave equation with linear Gaussian multiplicative noise (in Skorokhod sense): where is the Laplacian in space variables and the Gaussian noiseẆ has the following correlation structure E Ẇ (t, x)Ẇ (s, y) = γ 0 (t − s)γ (x − y), with the following standing assumptions: (i) γ 0 : R → [0, ∞] is locally integrable and non-negative definite; (ii) γ is a non-negative and non-negative definite measure on R d whose spectral measure μ 1 satisfies Dalang's condition: where |ξ | denotes the Euclidean norm of ξ ∈ R d . An important example of the temporal correlation is the Riesz kernel γ 0 (t) = |t| −α 0 for some α 0 ∈ (0, 1) (with γ 0 (0) = ∞). Equation (1.1) is also known in the literature as the hyperbolic Anderson model, by analogy with the parabolic Anderson model in which the wave operator is replaced by the heat operator. The noiseẆ can be formally realized as an isonormal Gaussian process W = {W (φ) : φ ∈ H} and here H is a Hilbert space that is the completion of the set C ∞ c R + × R d ) of infinitely differentiable functions with compact support under the inner product where we write γ (x) for the density of γ if it exists and we shall use the definition (1.4) instead of (1.3) when γ is a measure. In (1.4), * denotes the convolution in the space variable and γ 0 (t) = γ 0 (−t) for t < 0. We denote by H ⊗ p the pth tensor product of H for p ∈ N * , see Sect. 2 for more details. As mentioned before, the existence of a temporal correlation γ 0 prevents us from defining equation (1.1) in the Itô sense due to a lack of the martingale structure. In the recent work [3] by Balan and Song, the following results are established using Malliavin calculus. Let G t denote the fundamental solution to the corresponding deterministic wave equation, that is, for (t, z) ∈ (0, ∞) × R d , (1.5) To ease the notation, we will stick to the convention that G t (z) = 0 when t ≤ 0.
(1.6) Definition 1.1 Fix d ∈ {1, 2}. We say that a square-integrable process u = {u(t, x) : (t, x) ∈ R + × R d } is a mild Skorokhod solution to the hyperbolic Anderson model (1.1) if u has a jointly measurable modification (still denoted by u) such that sup{E[u(t, x) 2 ] : (t, x) ∈ [0, T ] × R d } < ∞ for any finite T ; and for any t > 0 and x ∈ R d , the following equality holds in L 2 ( ): where the above stochastic integral is understood in the Skorokhod sense and the process (s, y) ∈ R + × R d −→ 1 (0,t) (s)G t−s (x − y)u(s, y) is Skorokhod integrable. See Definition 5.1 in [3] and Definition 1.1 in [2].
It has been proved in [3,Section 5] that equation (1.1) admits a unique mild Skorokhod solution u with the following Wiener chaos expansion: where I n denotes the nth multiple Wiener integral associated to the isonormal Gaussian process W (see Sect. 2 for more details), f t,x,n ∈ H ⊗n is defined by (with the convention (1.6) in mind) (1.8) and f t,x,n is the canonical symmetrization of f t,x,n ∈ H ⊗n given by f t,x,n (t 1 , x 1 , . . . , t n , x n ) := 1 n! σ ∈S n f t,x,n (t σ (1) , x σ (1) , . . . , t σ (n) , x σ (n) ), (1.9) where the sum in (1.9) runs over S n , the set of permutations on {1, 2, . . . , n}. For example, f t,x,1 (t 1 , x 1 ) = G t−t 1 (x − x 1 ) and We would like to point out that in the presence of temporal correlation, there is no developed solution theory for the nonlinear wave equation (replacing uẆ in (1.1) by σ (u)Ẇ for some deterministic Lipschitz function σ : R → R). We regard this as a totally different problem. Now let us introduce the following hypothesis when d = 2: γ (x) = |x| −β for some β ∈ (0, 2), (c)γ (x 1 , for some 0 < β i < 1 < i < +∞, i = 1, 2.
Under Hypothesis (H1), we will state our first main result -the L p ( ) estimates of the Malliavin derivatives of u(t, x). The first Malliavin derivative Du(t, x) is a random element in the Hilbert space H, the completion of C ∞ c R + × R d ) under the inner product (1.3); as the space H contains generalized functions, it is not clear at first sight whether (s, y) −→ D s,y u(t, x) is a (random) function. The higher-order Malliavin derivative D m u(t, x) is a random element in H ⊗m for m ≥ 1, see Sect. 2 for more details.
Let us first fix some notation.
Notation A (1) We write a b to mean a ≤ K b for some immaterial constant K > 0. (2) We write X p = E[|X | p ] 1/ p to denote the L p ( )-norm of X for p ∈ [1, ∞).
(3) When p is a positive integer, we often write z p z p z p = (z 1 , . . . , z p ) for points in R p + or R dp , and dz p z p z p = dz 1 · · · dz p , μ(dz p z p z p ) = μ(dz 1 ) · · · μ(dz p ). For a function h : where the constant in the upper bound only depends on ( p, t, γ 0 , γ, m) and is increasing in t. Moreover, D m u(t, x) has a measurable modification.
Throughout this paper, we will work with the measurable modifications of Du(t, x) and D 2 u(t, x) given by Theorem 1.3, which are still denoted by Du(t, x), D 2 u(t, x) respectively.
In this paper, we will present two applications of Theorem 1.3. Our first application are quantitative central limit theorems (CLTs) for the spatial averages of the solution to (1.1), which have been elusive so far due to the temporal correlation of the noise preventing the use of Itô calculus approach. A novel ingredient to overcome this difficulty is the so-called second-order Gaussian Poincaré inequality in an improved form. We will address these CLT results in Sect. 1.1. While in Sect. 1.2, as the second application, we establish the absolute continuity of the law of the solution to equation (1.1) using the L p -estimates of Malliavin derivatives that are crucial to establish a local version of Bouleau-Hirsch criterion [5].

Gaussian fluctuation of spatial averages
Spatial averages of SPDEs have recently attracted considerable interest. It was Huang, Nualart and Viitasaari who first studied the fluctuation of spatial statistics and established a central limit theorem for a nonlinear SPDE in [15]. More precisely, they considered the following one-dimensional stochastic heat equation on R + × R, whereẆ is a space-time Gaussian white noise, with constant initial condition u(0, •) = 1 and the nonlinearity σ : R → R is a Lipschitz function. In view of the localization property of its mild formulation (in the Walsh sense [43]), with p t denoting the heat kernel, 3 one can regard u(t, x) and u(t, y) as weakly dependent random variables for x, y far apart so that the integral can be roughly understood as a sum of weakly dependent random variables. Therefore, it is very natural to expect Gaussian fluctuations when R tends to infinity. Let us stop now to briefly fix some notation to facilitate our discussion. (1.14) (2) We write f (R) ∼ g(R) to mean that f (R)/g(R) converges to some positive constant as R → ∞.
(3) For two real random variables X , Y with distribution measures μ, ν respectively, the total variation distance between X , Y (or μ, ν) is defined to be (1.15) where the supremum runs over all Borel set B ⊂ R. The total variation distance is well known to induce a stronger topology than that of convergence in distribution, see [25,Appendix C].
(4) We define the following quantities for future reference: • The authors first rewrite F R (t) = δ(V t,R ) with the random kernel where δ denotes the Skorokhod integral, the adjoint of the Malliavin derivative D. • By standard computations, they obtained σ 2 The above general strategy has been adapted to various settings, see [9,10,16,19,20,38] for the study of stochastic heat equations and see [4,12,35] for the study of stochastic wave equations. All these references consider a Gaussian noise that is white in time. Nevertheless, when the Gaussian noise is colored in time, the mild formulation (1.13) cannot be interpreted in the Walsh-Itô sense. In this situation, only in the case σ (u) = u the stochastic heat equation (1.12) (also known as the parabolic Anderson model) can be properly solved using Wiener chaos expansions, so that F R (t), defined in (1.14), can be expressed as an infinite sum of multiple Wiener integrals. With this well-known fact in mind, Nualart and Zheng [33] considered the parabolic Anderson model (i.e. (1.12) with σ (u) = u) on R + × R d such that d ≥ 1, the initial condition is constant and the assumptions (i)-(ii) hold (see page 2). The main result of [33] is the chaotic CLT that is based on the fourth moment theorems [30,37]. When, additionally, γ is a finite measure, the authors of [33] established σ R (t) ∼ R d/2 and a functional CLT for the process R −d/2 F R ; they also considered the case where γ (x) = |x| −β , for some β ∈ (0, 2 ∧ d), is the Riesz kernel, and obtain the corresponding CLT results. As pointed out in the paper [33], due to the homogeneity of the underlying Gaussian noise, the solution u to (1.12) can be regarded as the functional of a stationary Gaussian random field so that, with the Breuer-Major theorem [6] in mind, it is natural to study Gaussian fluctuations for the problems (1.12) and (1.1). Note that the constant initial condition makes the solution stationary in space and, in fact it is spatially ergodic (see [10,36]). At last, let us mention the paper [32] in which chaotic CLT was used to study the parabolic Anderson model driven by a colored Gaussian noise that is rough in space. However, let us point out that the aforementioned methods fail to provide the rate of convergence when the noise is colored in time.
In this paper, we bring in a novel ingredient-the second-order Gaussian Poincaré inequality 4 -to reach quantitative CLT results for the hyperbolic Anderson model (1.1). Let us first state our main result. (1.1) and recall the definition of F R (t) and σ R (t) from (1.14). Let Z ∼ N (0, 1) be the standard normal random variable. We assume that γ 0 is not identically zero meaning

Theorem 1.4 Let u denote the solution to the hyperbolic Anderson model
(1.17) Then the following statements hold true: Moreover, as R → ∞, the process R −d/2 F R (t) : t ∈ R + converges weakly in the space of continuous functions C(R + ) to a centered Gaussian process G with covariance structure Moreover, as R → ∞, the process R −d+ β 2 F R (t) : t ∈ R + converges weakly in the space C(R + ) to a centered Gaussian process G β with the covariance structure for t, s ∈ R + . Here the quantity κ β,d is introduced in (1.16). 4 The use of second-order Gaussian Poincaré inequality for obtaining CLT on a Gaussian space is one of the central techniques in the Malliavin-Stein approach; for example, in the recent paper [13], Dunlap et al. have used this Poincaré inequality to investigate the Gaussian fluctuation of the KPZ in dimension three and higher. We remark here that we can not directly apply this inequality because of the complicated correlation structure of the underlying Gaussian homogeneous noise, while the underlying Gaussian noise in [13] is white in time and smooth in space so that they can directly apply the version from [26]. In this article, we have established a quite involved variant of second-order Poincaré inequality, which is tailor-made for our applications.
(3) Suppose d = 2 and γ (x 1 , x 2 ) = γ 1 (x 1 )γ 2 (x 2 ) such that one of the following two conditions holds: (1.20) Then, Moreover, as R → ∞, in case (a ) , the process R −2+ β 1 +β 2 2 F R (t) : t ∈ R + converges weakly in the space C(R + ) to a centered Gaussian process G β 1 ,β 2 with the covariance structure (1. 22) and in case (b ) , the process R β−3 2 F R (t) : t ∈ R + converges weakly in the space C(R + ) to a centered Gaussian process G β with the covariance structure for t, s ∈ R + , where For the above functional convergences, we specify that the space C(R + ) is equipped with the topology of uniform convergence on compact sets.
The variance orders and the rates in parts (1) and (2) of Theorem 1.4 are consistent with previous work on stochastic wave equations, see [4,12,35]. The setting in part (3) is new. As we will see shortly, our strategy is quite different from that in these papers.
Now, let us briefly explain our strategy and begin with the Gaussian Poincaré inequality. For F ∈ D 1,2 , the Gaussian Poincaré inequality (see e.g. [14] or (2.12)) ensures that where D 2 F ⊗ 1 D 2 F denotes the 1-contraction between D 2 F and itself (see 2.10).
It has been known that this inequality usually gives sub-optimal rate. In the recent work [42] by Vidotto, she provided an improved version of the above inequality, where she considered an L 2 -based Hilbert space H = L 2 (A, ν) with ν a diffusive measure (nonnegative, σ -finite and non-atomic) on some measurable space A. Let us state this result for the convenience of readers.
The proof of the above inequality follows from the general Malliavin-Stein bound Recall that our Hilbert space H is the completion of C ∞ c (R + × R d ) under the inner product (1.3). The Hilbert space H contains generalized functions, but fortunately the objects D 2 u(t, x), Du(t, x) are random functions in view of Theorem 1.3. By adapting Vidotto's proof to our setting, we have the following version of second-order Gaussian Poincaré inequality. Note we write f ∈ |H ⊗ p | to mean f is a real valued function and • → | f (•)| belongs to H ⊗ p . Proposition 1.8 If F ∈ D 2,4 has mean zero and variance σ 2 ∈ (0, ∞) such that with probability 1, D F ∈ |H| and D 2 F ∈ |H ⊗2 |, then where Z ∼ N (0, σ 2 ) and As mentioned before, Proposition 1.8 will follow from the Malliavin-Stein bound (1.26) and Cauchy-Schwarz inequality, taking into account that, by the duality relation A by Proposition 1.9 below. Proposition 1.9 If F, G ∈ D 2,4 have mean zero such that with probability one, D F, DG ∈ |H| and D 2 F, D 2 G ∈ |H ⊗2 |, then and A 2 is defined by switching the positions of F, G in the definition of A 1 .
For the sake of completeness, we sketch the proof of Proposition

Absolute continuity of the law of the solution to Eq. (1.1)
In this part, we fix the following extra hypothesis on the correlation kernels γ 0 , γ . The following is the main result of this section. Let us sketch the proof of Theorem 1.10. In view of the Bouleau-Hirsch criterion for absolute continuity (see [5]), it suffices to prove that for each m ≥ 1, where m = {|u(t, x)| ≥ 1/m}. Notice that where P 0 is the completion of C ∞ c (R d ) with respect to the inner product ·, · 0 introduced in (2.1). The usual approach to show the positivity of this norm is to get a lower bound for this integral by integrating on a small interval [t − δ, t] 2 and use that, for r close to t, D r ,y u(t, x) behaves as G t−r (x − y)u(s, y) (see, e.g., [31]). However, for r = s, the inner product D r ,• u(t, x), D s,• u(t, x) 0 is not necessarily non-negative. Our strategy to overcome this difficulty consists in making use of Hypothesis (H2) in order to show that This allows us to reduce the problem to the non-degeneracy of t t−δ D r ,• u(t, x) 2 0 dr for δ small enough, which can be handled by the usual arguments. At this point, we will make use of the estimates provided in Theorem 1.3.
For d = 1, Theorem 1.10 was proved in [2] under stronger assumptions on the covariance structure. The result in Theorem 1.10 for d = 2 is new. Indeed, the study of the existence (and smoothness) of the density for the stochastic wave equation has been extensively revisited over the last three decades. We refer the readers to [7,22,23,31,[39][40][41]. In all these articles, the authors considered a stochastic wave equation of the form Here,Ẋ denotes a space-time white noise in the case d = 1, or a Gaussian noise that is white in time and has a spatially homogeneous correlation (slightly more general than that of W ) in the case d ≥ 2. The functions b, σ are usually assumed to be globally Lipschitz, and such that the following non-degeneracy condition is fulfilled: |σ (z)| ≥ C > 0, for all z ∈ R. The temporal nature of the noisė X made possible to interpret the solution in the classical Dalang-Walsh sense, making use of all needed martingale techniques. The first attempt to consider a Gaussian noise that is colored in time was in the paper [2], where the hyperbolic Anderson model with spatial dimension one was considered. As mentioned above, in that paper the existence of density was proved under a slightly stronger assumption than Hypothesis (H2).
The rest of this paper is organized as follows. Section 2 contains preliminary results and the proofs of our main results-Theorems 1.3, 1.4 and 1.10-are given in Sects. 3, 4 and 5 , respectively.

Preliminary results
This section is devoted to presenting some basic elements of the Malliavin calculus and collecting some preliminary results that will be needed in the sequel.

Basic Malliavin calculus
Recall that the Hilbert space H is the completion of C ∞ c (R + × R d ) under the inner product (1.3) that can be written as we can express (2.1) using the Fourier transform: (2. 3) The Parseval-type relation (2.3) also holds for functions h, g ∈ L 1 (R d ) ∩ |P 0 |. For every integer p ≥ 1, H ⊗ p and H p denote the pth tensor product of H and its symmetric subspace, respectively. For example, f t,x,n in (1.8) belongs to H ⊗n and f t,x,n ∈ H n ; we also have f ⊗ g ∈ H ⊗(n+m) , provided f ∈ H ⊗m and g ∈ H ⊗n ; see [25, Appendix B] for more details.
Fix a probability space ( , B, P), on which we can construct the isonormal Gaussian process associated to the Gaussian noiseẆ in (1.1) that we denote by In the sequel, we recall some basics on Malliavin calculus from the books [25,27]. Let C ∞ poly (R n ) denote the space of smooth functions with all their partial derivatives having at most polynomial growth at infinity and let S denote the set of simple smooth functionals of the form For such a random variable F, its Malliavin derivative D F is the H-valued random variable given by And similarly its mth Malliavin derivative D m F is the H ⊗m -valued random variable given by is closable for any p ∈ [1, ∞); see e.g. Lemma 2.3.1 and Proposition 2.3.4 in [25]. Let D m, p be the closure of S under the norm Now, let us introduce the adjoint of the derivative operator D m . Let Dom(δ m ) be the set of random variables v ∈ L 2 ( ; H ⊗m ) such that there is a constant C v > 0 for which By Riesz representation theorem, there is a unique random variable, denoted by δ m (v), such that the following duality relationship holds: Equality (2.5) holds for all v ∈ Dom(δ m ) and all F ∈ D m,2 . In the simplest case when which is exactly part of the Stein's lemma recalled below: For σ ∈ (0, ∞) and an integrable random variable Z , Stein's lemma (see e.g. [25, Lemma 3.1.2]) asserts that for any differentiable function f : R → R such that the above expectations are finite. The operator δ is often called the Skorokhod integral since in the case of the Brownian motion, it coincides with an extension of the Itô integral introduced by Skorokhod, see e.g. [29]. Then we can say Dom(δ m ) is the space of Skorokhod integrable random variables with values in H ⊗m . The Wiener-Itô chaos decomposition theorem asserts that L 2 ( , σ {W }, P) can be written as a direct sum of mutually orthogonal subspaces: where C W 0 , identified as R, is the space of constant random variables and C W n = {δ n (h) : h ∈ H ⊗n is deterministic}, for n ≥ 1, is called the nth Wiener chaos associated to W . Note that the first Wiener chaos consists of centered Gaussian random variables. When h ∈ H ⊗n is deterministic, we write I n (h) = δ n (h) and we call it the nth multiple integral of h with respect to W . By the symmetry in (2.4) and the duality relation (2.5), δ n (h) = δ n ( h) with h the canonical symmetrization of h, so that we have I n (h) = I n ( h) for any h ∈ H ⊗n . The above decomposition can be rephrased as follows. For any F ∈ L 2 ( , σ {W }, P), with f n ∈ H n uniquely determined for each n ≥ 1. Moreover, the (modified) isometry property holds for any f ∈ H ⊗ p and g ∈ H ⊗q . We have the following product formula: For f ∈ H p and g ∈ H q , where f ⊗ r g is the r -contraction between f and g, which is an element in H ⊗( p+q−2r ) defined as follows. Fix an orthonormal basis ..,r } e i r +1 ⊗ · · · ⊗ e i p ⊗ e j r +1 ⊗ · · · ⊗ e j q . (2.10) In the particular case when f , g are real-valued functions, we can write ds r s r s r ds r s r s r dy r y r y r dy r y r y r provided the above integral exists. For F ∈ D m,2 with the representation (2.7) and m ≥ 1, we have whenever the above series makes sense and converges in L 2 ( ). With the decomposition (2.11) in mind, we have the following Gaussian Poincaré inequality: For F ∈ D 1,2 , it holds that In fact, if F has the representation (2.7), then which gives us (2.12) and, moreover, indicates that the equality in (2.12) holds only when F ∈ C W 0 ⊕ C W 1 , that is, only when F is a real Gaussian random variable. Now let us mention the particular case when the Gaussian noise is white in time, which is used in the reduction step in Sect. 3.2. First, let us denote H 0 := L 2 R + ; P 0 and point out that the following inequality reduces many calculations to the case of the white noise in time. For any nonnegative function f ∈ H ⊗n 0 that vanishes outside whenever no ambiguity arises, we write LetẊ denote the Gaussian noise that is white in time and has the same spatial correlation as W . More precisely, Denote by I X p the p-th multiple stochastic integral with respect to X. The product formula (2.9) still holds with W replaced by the noise X. Moreover, if f ∈ H ⊗ p and g ∈ H ⊗q have disjoint temporal supports, 8 then we have f ⊗ r g = 0 for r = 1, . . . , p ∧ q and the product formula (2.9) reduces to (2.14) In this case, the random variables I X p ( f ) and I X q (g) are independent by the Üstünel-Zakai-Kallenberg criterion (see Exercise 5.4.8 of [25]) and note that we do not need to assume f , g to be symmetric in (2.14). Now let us introduce the Ornstein-Uhlenbeck operator L that can be defined as follows. We say that F belongs to the Dom(L) if F ∈ D 1,2 and D F ∈ Dom(δ); in this case, we let L F = −δ D F. For F ∈ L 2 ( ) of the form (2.7), F ∈ Dom(L) if and only if n≥1 n 2 n! f n 2 H ⊗n < ∞. In this case, we have L F = n≥1 −n I n ( f n ). Using the chaos expansion, we can also define the Ornstein-Uhlenbeck semigroup {P t = e t L , t ∈ R + } and the pseudo-inverse L −1 of the Ornstein-Uhlenbeck operator L as follows. For F ∈ L 2 ( ) having the chaos expansion (2.7), 7 For the sake of completeness, we sketch a proof of (2.13) here: Given such a function f ∈ H ⊗n 0 , ds n s n s n dt n t n t n f (s n s n s n , ds n s n s n dt n t n t n 1 2 f (s n s n s n , •) Observe that for any centered random variable F ∈ L 2 ( , σ {W }, P), L L −1 F = F and for any G ∈ Dom(L), The above expression and the modified isometry property (2.8) give us the contraction property of P t on L 2 ( ), that is, for F ∈ L 2 ( , σ {W }, P), P t F 2 ≤ F 2 . Moreover, P t is a contraction operator on L q ( ) for any q ∈ [1, ∞); see [25,Proposition 2.8.6].
Finally, let us recall Nelson's hypercontractivity property of the Ornstein-Uhlenbeck semigroup: In this paper, we need one of its consequences -a moment inequality comparing L q ( )-norms on a fixed chaos: [25, Corollary 2.8.14].

Inequalities
Let us first present a few inequalities, which will be used in Sect. 3.

Lemma 2.1 Fix an integer d ≥ 1.
Suppose that either one of the following conditions hold: Then, for any f , g ∈ L 2q (R d ), Proof In the case d = 2, this result was essentially proved on page 15 of [35] in case (a), and on page 6 of [4] in case (b). We reproduce the arguments here for the sake of completeness.
To deal with case (c) in (H1), we need the following modification of Lemma 2.1.
The constants C γ i are defined as in Lemma 2.1.
Proof By Lemma 2.1, inequality (2.17) holds for d = 1 with ν = 0. Now let us consider d ≥ 2 and prove inequality (2.17) by induction. Suppose (2.17) holds for d ≤ k − 1 (k ≥ 2). We use the notation x k x k . Without loss of any generality we assume q 1 ≥ q 2 ≥ · · · ≥ q k , so that q = q 1 . Applying the initial step (d = 1) yields (2.18) By the induction hypothesis, we can bound the right-hand side of (2.18) by . By Hölder's inequality, We will need the following generalization of Lemmas 2.1 and 2.2.

Lemma 2.3 (1) Under the conditions of Lemma
(2) Let γ, C γ and q be given as in Lemma Proof The proof will be done by induction on m simultaneously for both cases (1) and (2). Let C = C γ in case (1) and C = ν C γ in case (2). The results are true for m = 1 by Lemmas 2.1 and 2.2. Assume that the results hold for m − 1. Applying the inequality for m = 1 yields By the induction hypothesis, the latter term can be bounded by , which completes the proof.
Let us return to the three cases of Hypothesis (H1).
Recall that P 0 has been defined at the beginning of Sect. 2.1. Moreover, for any For case (c) of Hypothesis (H1), we consider three sub-cases: , (2.23) such that f , g vanish outside a box with side lengths bounded by , then inequality (2.21) still holds with where the constants C 1,β i are given as in Lemma 2.1. From Lemma 2.3, we deduce that in cases (a) and (b), for any measurable function f : x 1 , . . . , t n , x n ) = f (t n t n t n , x n x n x n ) = 0 for t n t n t n / ∈ [0, t] n and x n x n

inequality (2.25) holds true for any measurable function
Let us present a few facts on the fundamental solution G. When d = 2, and (2.28) We will use also the following estimate.
where A q > 0 is a constant depending on q.
Finally, we record the expression of the Fourier transform of G t for d ∈ {1, 2}: Note that (see e.g. (3.4) of [3]) (2.30) In Sect. 4, we need the following two results.

Lemma 2.5 For d ∈ {1
, 2}, let γ 0 satisfy the assumption (i) on page 2 and let μ p be a symmetric measure on (R d ) p , for some integer p ≥ 1. Then, with 0 < s ≤ t and for any measurable function g : (R + × R d ) p → R + for which the above integral is finite.
Proof After applying |ab| ≤ a 2 +b 2 2 and using the symmetry of μ p , we have that the left-hand side quantity is bounded by As a consequence, for any integer p ≥ 1 and w 1 , . . . , w p ∈ [0, t], In particular, by (2.30), which is finite due to Dalang's condition (1.2). Applying this inequality several times yields R dp which is a uniform bound over (z p z p z p , w p w p w p ) ∈ R dp × [0, t] p .

L p estimates for Malliavin derivatives
This section is mainly devoted to the proof of Theorem 1.3. The proof will be done in several steps organized in Sects. 3.1, 3.2, 3.3, 3.4 and 3.5. In Sect. 3.6, we record a few consequences of Theorem 1.3 that will be used in the proof of Theorem 1.10 in Sect. 5.

Step 1: Preliminaries
Let us first introduce some handy notation. Recall that for t n t n t n := (t 1 , . . . , t n ) and x n x n x n := (x 1 , . . . , x n ), we defined in (1.8) with the convention (1.6), and we denote by f t,x,n the symmetrization of f t,x,n ; see (1.9). We treat the time-space variables (t i , x i ) as one coordinate and we write as in Notation A-(3). Recall that the solution u(t, x) has the Wiener chaos expansion where the kernel f t,x,n is not symmetric and in this case, by definition, I n ( f t,x,n ) = I n f t,x,n .
Our first goal is to show that, for any fixed (r , z) ∈ [0, t] × R d and for any p ∈ [2, ∞), the series n≥1 n I n−1 f t,x,n (r , z; •) (3.1) converges in L p ( ), and the sum, denoted by D r ,z u(t, x), satisfies the L p estimates (1.11).
The first term of the series (3.1) is f t, x,1 (r , z) = G t−r (x − z). In general, for any n ≥ 1, , which is obtained from f t,x,n by placing r on position j among the time instants, and z on position j among the space points: With the convention (1.6), That is, with f r ,z,1 = 1. For example, f t,x,1 (r , z; •) = G t−r (x −z) and f (1) t,x,n (r , z; t n−1 t n−1 t n−1 , . By the definition of the symmetrization, t,x,n (r , z; t n−1 t n−1 t n−1 , x n−1 x n−1 x n−1 ) is symmetric, meaning that for any σ ∈ S m , D s 1 ,y 1 D s 2 ,y 2 · · · D s m ,y m u(t, x) = D s σ (1) ,y σ (1) D s σ (2) ,y σ (2) · · · D s σ (m) ,y σ (m) u(t, x).
From now on, we assume t > s 1 > ... > s m > 0 without losing any generality. Note that like (3.2), we can write where i m i m i m ∈ n,m means 1 ≤ i 1 < i 2 < · · · < i m ≤ n and h which is a generalization of (3.4).

Step 2: Reduction to white noise in time
LetẊ denote the Gaussian noise that is white in time and has the same spatial correlation as The product formula (2.14) and the decomposition (3.8) yield, with (i 0 , s 0 , y 0 ) = (0, t, x), where the last equality is obtained by using the independence among the random variables inside the expectation. It remains to estimate two typical terms: t,x, j (r , z; •) 2 2 for 1 ≤ j ≤ n and t > r . (3.13) The first term in (3.13) can be estimated as follows. Using Fourier transform in space (see (2.29)), we have, with t 0 = r , (3.14) By Lemma 2.6,

Remark 3.1
By the arguments that lead to (3.9), we can also get, for any p ∈ [2, ∞), In what follows, we estimate the second term in (3.13) separately for the cases d = 1 and d = 2. As usual, we will use C to denote an immaterial constant that may vary from line to line.

Estimation of I
t,x,j (r, z; •) 2 2 when d = 1 t,x, j (r , z; •) = G t−r (x − z) with the convention (1.6). For j ≥ 2, it follows from the (modified) isometry property (2.8) that t,x, j (r , z; •); see (3.5). Then, taking advantage of the simple form of G t (x) for d = 1, we get from which we further get where the last inequality follows from (3.15) and (3.14).

Estimation of I
Let q be defined as in (2.20) and (2.23) and we fix such a q throughout this subsection.
with the convention (1.6). For j ≥ 2, we begin with where we applied Lemma 2.3 for the inequality above 10 and we denote Note that we can choose C to depend only on (t, γ, q) and be increasing in t. Case j = 2. In this case, we deduce from Lemma 2.4 and (2.27) that 10 The function x j−1 Case j ≥ 3. In this case, we use Minkowski inequality with respect to the norm in L 1/q ([t 2 , t], dt 1 ) in order to get

Applying Lemma 2.4 yields
(3.20) If j = 3, we have Owing to (2.27), we can bound G 2q−1 , and then we apply again Lemma 2.4 and (2.27) to conclude that For j ≥ 4, we continue with the estimate (3.20). We can first apply Minkowski inequality with respect to the norm L 1/q [t 4 , t 2 ], dt 3 and then apply Lemma 2.4 to obtain

Then, by Cauchy-Schwarz inequality and (2.26), we can infer that
where c 1 = (2π) 3−4q 4−4q . Thus, substituting this estimate into (3.22), we end up with Focusing on the indicators, the right-hand side of this estimate can be bounded by For j = 4, using (2.28), we have Now for j ≥ 5, we just integrate in each of the variables x 4 , . . . , x j−1 (with this order) so that, thanks to (2.26), we end up with where we used the rough estimate a ν ≤ (b + 1) ν for 0 < a ≤ b and ν > 0. Thus, using (2.28) we obtain: Hence, combining the estimates (3.19), (3.21), (3.23) and (3.24) and taking into account that I X 0 ( f (1) t,x,1 (r , z; •) = G r −s (z − y), we can write where the constant C > 1 depends on (t, γ, q) and is increasing in t. For 1 ≤ j ≤ n, we obtain the following bound From this, it follows that we can find a measurable modification of the process Finally, by standard arguments we deduce the existence of a measurable modification of the series (3.6).

Step 5: Proof of u(t, x) ∈ D ∞
We have already seen in Remark 3.1 that u(t, x) ∈ L p ( ) for any p ∈ [2, ∞).
This shows u(t, x) ∈ D ∞ and completes the proof of Theorem 1.3.

Consequences of Theorem 1.3
We will establish two estimates that will be useful in Sect. 5.
The space |H⊗P 0 | appearing in the next corollary is defined as the set of measurable functions h : R + × R 2d → R such that Then, |H ⊗ P 0 | ⊂ H ⊗ P 0 .
Proof Using Theorem 1.3, Cauchy-Schwarz inequality and the estimate (1.11), we can write As a consequence, By the arguments used in the proof of Theorem 1.3, it follows that Therefore, and the same argument as in the proof of Corollary 3.3 ends our proof.

Gaussian fluctuation: Proof of Theorem 1.4
Recall that . First, we need to obtain the limiting covariance structure, which is the content of Proposition 4.1. It will give us the growth order of σ R (t). Then, in Sect. 4.2, we apply the second-order Gaussian Poincaré inequality to establish the quantitative CLT for F R (t)/σ R (t). Finally, we will prove the functional CLT by showing the convergence of the finite-dimensional distributions and the tightness. (1) Suppose d ∈ {1, 2} and γ (R d ) ∈ (0, ∞). Then, for any t, s ∈ (0, ∞),

Proof of part (1) in Proposition 4.1
Preparation. In the following, we will denote by ϕ the density of μ. For 0 < s ≤ t < ∞ and x, y ∈ R d , we have where f t,x, p ∈ H ⊗ p is defined as in (1.8)-(1.9) and p (t, s; x − y), defined in the obvious manner, depends only on the difference x − y. To see this dependency and to prepare for the future computations, we rewrite p (t, s; x − y) using Fourier transform in space: where is introduced in (2.29) and we have used again the convention G t (z) = 0 for t ≤ 0.

Proof of (4.1) Let us begin with
where ω 1 = 2, ω 2 = π and Leb(A) stands for the Lebesgue measure of A ⊂ R d . We claim that from which and the dominated convergence theorem we can deduce that We remark that, by the monotone convergence theorem and the fact that p (t, s; z) ≥ 0 for all z ∈ R d , the claim (4.10) is equivalent to Let us show the claim (4.12).
For p = 1, by direct computations, we can perform integration with respect to z, y,ỹ (one by one in this order) to obtain where R d 1 (t, s; z)dz > 0 due to the non-degeneracy assumption (1.17) on γ 0 . This implies in particular that σ R (t) > 0 for large enough R. Next we consider p ≥ 2. Using the expression (4.7) and applying Fubini's theorem with the dominance condition (4.9), we can write Now, plugging the above estimate and (4.17) into (4.12), and using (4.13) for p = 1, we have sup ε>0 p≥1 This shows the claim (4.12) and the claim (4.10), which confirm the limiting covariance structure (4.11). Hence the proof of (4.1) is completed.

Proof of part (2) in Proposition 4.1
In this case, the corresponding spectral density is given by ϕ(ξ ) = c d,β |ξ | β−d , for some constant c d,β that only depends on d and β. Now, let us recall the chaos expansion (1.7) of u(t, x), from which we can obtain the following chaos expansion of F R (t): where J p,R (t) := I p |x|≤R f t,x, p dx is the projection of F R (t) onto the pth Wiener chaos, with f t,x, p given as in (1.9).
Using the orthogonality of Wiener chaoses with different order, we have Let us first consider the variance of J 1,R (t). With B R = {x ∈ R d : |x| ≤ R}, we can write Then, making the change of variables (x, x , ξ) → (Rx, Rx , ξ/R), we get Note that G t (ξ/R) is uniformly bounded and convergent to t as R → ∞; observe also that Thus we deduce from the dominated convergence theorem that, with κ β,d := In the same way, we can get In what follows, we will show that as R → ∞, In view of the orthogonality again, the above claim (4.22) and the results (4.20)-(4.21) imply that the first chaos of F R (t) is dominant and which gives us the desired limiting covariance structure. Moreover, we obtain immediately that the process R −d+ β 2 F R (t) : t ∈ R + converges in finite-dimensional distributions to the centered Gaussian process G β , whose covariance structure is given by (1.19).
The rest of Sect. 4.1.2 is then devoted to proving (4.22). We point out that the strategy in Sect. 4.1.1 can not be directly used, because ϕ is not uniformly bounded here.

Proof of Claim (4.22)
We begin by writing (with s 0 =s σ (0) = t and B R = {x : |x| ≤ R}) where in the last inequality we used | G t | ≤ t and the following Fourier transform: Note that the integral B 2 1 dxdx |x − x | −β e −i(x−x )·η 2 R is uniformly bounded by κ β,d and it converges to zero as R → ∞ for η 2 = 0. This convergence is a consequence of the Riemann-Lebesgue's lemma. Taking into account the definition (4.16) of Q p−1 , then we have which is summable over p ≥ 2 by the arguments in the previous section. Hence by the dominated convergence theorem, we get This proves the claim (4.22).

Proof of part (3) in Proposition 4.1
Recall the two cases from (4.3): .
where c 1,β is a constant that only depends on β. Now, using the notation from Sect. 4.1.2, we write where the last equality is obtained by the change of variables (x, x , ξ 1 , ξ 2 ) to (Rx, Rx , ξ 1 /R, ξ 2 /R). Thus, by the exactly same arguments that lead to (4.20), we can get with K β 1 ,β 2 introduced in (1.22). Similar to (4.21), we also have To obtain the result (r 1 ), it remains to show Its proof can be done verbatim as for the result (4.22), so we omit the details here. Finally, let us look at the more interesting case (c 2 ) where γ 1 ∈ L 1 (R) and γ 2 (x) = |x| −β for some fixed β ∈ (0, 1). In this case, the corresponding spectral density is ϕ(ξ 1 , ξ 2 ) = ϕ 1 (ξ 1 )ϕ 2 (ξ 2 ), where (i) γ 1 = Fϕ 1 and ϕ 1 is uniformly continuous and bounded, (ii) ϕ 2 (ξ 2 ) = c 1,β |ξ 2 | β−1 for some constant c 1,β that only depends on β. (4.26) Let us begin with (4.18) and make the usual change of variables (x, x , ξ) → (Rx, Rx , ξ/R) to obtain Recall that ϕ 1 , G t−s and G t−s are uniformly bounded and continuous. Note that, applying Plancherel's theorem and the Parseval-type relation (2.3), we have Therefore, by the dominated convergence theorem and the fact that ϕ 1 (0) = 1 2π γ 1 (R), we get where L β is defined in (1.24). In the same way, we get for s, t ∈ (0, ∞), Note that the desired limiting covariance structure follows from (4.27) and the above claim (4.28). The rest of this section is devoted to proving claim (4.28).

Remark 4.2
Under the assumptions of Proposition 4.1, we point out that σ R (t) > 0 for large enough R so that the renormalized random variable F R (t)/σ R (t) is well-defined for large R.

Quantitative central limit theorems (QCLT) and f.d.d. convergence
In this section, we prove the quantitative CLTs that are stated in Theorem 1.4 and, as an easy consequence, we are also able to show the convergence of finite-dimensional distributions in Theorem 1.4. We consider first the part (1) and later we treat parts (2) and (3).

Part (1)
We will first show the estimate where Z ∼ N (0, 1). By Proposition 1.8 applied to 1 where drdr dsds dθ dθ dzdz dydy dwdw Recall from Sect. 4.1.1 that σ 2 R (t) ∼ R d . Therefore, in order to show (4.29) it suffices to prove the estimate Using Minkowski's inequality, we can write Then, it follows from our fundamental estimates in Theorem 1.3 that with f t,x,2 (r , z, θ, w) and, in the same way, we have The four terms A R,1 , . . . , A R,4 are defined according to whether r > θ or r < θ, and whether s > θ or s < θ . For example, the term A R,1 corresponds to r > θ and s > θ : (4.34) The term A R,2 corresponds to r > θ and s < θ , the term A R,3 corresponds to r < θ and s > θ and the term A R,4 corresponds to r < θ and s < θ . In the following, we estimate A R, j for j = 1, 2, 3, 4 by a constant times R d , which yields (4.31).
To get the bound for A R,1 , it suffices to perform the integration with respect to dx 1 , dx 2 , dx 4 , dy , dy, dw , dw, dz, dz , dx 3 one by one, by taking into account the following facts: To get the bound for A R,2 , it suffices to perform the integration with respect to dx 1 , dx 3 , dz , dz, dx 2 , dw, dw , dy, dy , dx 4 . To get the bound for A R, 3 , it suffices to perform the integration with respect to dx 4 , dy , dx 2 , dy, dw , dx 1 , dw, dz, dz , dx 3 one by one. To get the bound for A R, 4 , it suffices to perform the integration with respect to dx 1 , dx 3 , dx 2 , dz , dz, dw, dw , dy, dy , dx 4 one by one. This completes the proof of (4.29).
In the second part of this subsection, we show the f.d.d. convergence in Theorem 1.4-(1).
Fix an integer m ≥ 1 and choose t 1 , . . . , t m ∈ (0, ∞). Put F R = F R (t 1 ), . . . , F R (t m ) . Then, by the result on limiting covariance structure from Sect. 4.1.1, we have that the covariance matrix of R −d/2 F R , denoted by C R , converges to the matrix C = (C i j : It is clear that the second term in (4.35) tends to zero as R → ∞. For the variance term in (4.35), taking advantage of Proposition 1.9 applied to F = F R (t i ) and G = F R (t j ) and using arguments analogous to those employed to derive (4.31), we obtain Thus, the first term in (4.35) is converges to zero as R → ∞. This shows the convergence of the finite-dimensional distributions of {R −d/2 F R (t) : t ∈ R + } to those of the centered Gaussian process G, whose covariance structure is given by This concludes the proof of part (1) in Theorem 1.4.

Proofs in parts (2) and (3)
In part (2), in view of the dominance of the first chaos, we have already obtained in Sect. 4.1.2 that the finite-dimensional distributions of the process R −d+ β 2 F R (t) : t ∈ R + converge to those of a centered Gaussian process {G β (t)} t∈R + , whose covariance structure is given by (1.19). By the same reason, the convergence of the finite-dimensional distributions in part (3) follows from (4.24), (4.25), (4.27) and (4.28).
In this section, we show that: (2), (4.37) Since the total-variation distance is always bounded by one, the bound (4.36) still holds for R < t by choosing the implicit constant large enough. The rest of this section is then devoted to proving (4.37) for R ≥ t and for j ∈ {1, 2, 3, 4}.
Proof of (4.37) Let us first consider the term A R,1 , which can be expressed as From now on, when d = 2, we write (w, w , y, y , z, z ) = (w 1 , w 2 , w 1 , w 2 , y 1 , y 2 , y 1 , y 2 , z 1 , z 2 , z 1 , z 2 ) and then dy = dy 1 dy 2 ; note also that x 1 , . . . , x 4 denote the dummy variables in R d . By making the following change of variables (z, z , y, y , w, w , and using the scaling property G t (Rz) = R 1−d G t R −1 (z) for d ∈ {1, 2}, we get Note that we have replaced the integral domain R 6d by [−2, 2] 6d in (4.39) without changing the value of S 1,R , because, for example, In view of the expression of γ in part (2) and part (3), we write, for z ∈ R d (z = (z 1 , z 2 ) ∈ R 2 when d = 2), in part (2), and it is easy to see that (2), (3), To ease the notation, we just rewrite the above estimates as with α = β in part (2), α = β 1 + β 2 in case (a ) of part (3), and α = 1 + β in case (b ) of part (3).
To estimate A R,1 , we can use (4.40) to perform integration with respect to dx 1 , dx 2 , dx 4 , dy , dy, dw , dw, dz, dz , dx 3 successively. More precisely, perform-ing the integration with respect to dx 1 , dx 2 , dx 4 and using the fact gives us by integrating out dy and using (4.40) by integrating out dy and using (4.41) by integrating out dw and using (4.40); then, using (4.41) to integrate out dw where the last inequality is obtained by integrating out dz, dz , dx 3 one by one and using (4.40) and (4.41). The bound (3) is uniform over (r , r , s, s ,  . This concludes the proof of (4.37) and hence completes the proof of (4.36).

Tightness
This section is devoted to establishing the tightness in Theorem  and we will show, for any fixed T > 0, that the following inequality holds for any integer k ≥ 2 and any 0 < s < t ≤ T ≤ R: where the implicit constant does not depend on R, s or t. This moment estimate (4.43) ensures the tightness of σ −1 R F R (t) : t ∈ [0, T ] for any fixed T > 0 and, therefore, the desired tightness on R + holds.
To show the above moment estimate (4.43) for the increment F R (t) − F R (s), we begin with the chaos expansion where s, t are fixed, so we leave them out of the subscript of the kernel g n,R and g n,R (s n s n s n , y n y n y n ) = ϕ t,R (s 1 , y 1 ) − ϕ s,R (s 1 , y 1 ) with 0 j=1 = 1 and ϕ t,R (r , y) := B R G t−r (x − y)dx. The rest of this section is then devoted to proving (4.43).
Recall the expression (2.29) G t (ξ ) = sin(t|ξ |) |ξ | and note that it is a 1-Lipschitz function in the variable t, uniformly over ξ ∈ R d . Then Therefore, plugging this inequality into (4.45) and then applying Lemma 2.6 yields n! g n,R 2 H ⊗n 0 ≤ (t − s) 2 t>s 1 >···>s n >0 ds n s n s n which is finite since 1 B R ∈ P 0 . Using Fourier transform, we can write Now let us consider the cases in (4.42).
As a consequence, we get and therefore, which leads to (4.43).

Proof of Theorem 1.10
We argue as in the proof of Theorem 1.2 of [2]. As we explained in the introduction, it suffices to show that for each m ≥ 1, where m = {|u(t, x)| ≥ 1/m}. We claim that, almost surely, the function (s, y) → D s,y u(t, x) satisfies the assumptions of Lemma A.1. Indeed, for d = 2, by Minkowski's inequality and the estimate (1.11), we have belongs to the space D 1,2 (H ⊗ P 0 ). This is because, using Corollary 3.3, we can write γ 0 (s − s )γ (y − y )dydy dsds < ∞, and in the same way, using Corollary 3.4 we can show that E DK (r ) 2 H⊗H⊗P 0 < ∞. Therefore, the process K (r ) belongs to the domain of the P 0 -valued Skorokhod integral, denoted by δ. Then, using the same arguments as in the proof of Proposition 5.2 of [2], replacing L 2 (R) by P 0 , we can show that for any r ∈ [0, t], the following equation holds in L 2 ( ; P 0 ): • u(s, y)W (δs, δ y).