Stoch PDE: Anal Comp (2022) 10:757–827 https://doi.org/10.1007/s40072-021-00227-5 The hyperbolic Anderson model: moment estimates of the Malliavin derivatives and applications Raluca M. Balan1 · David Nualart2 · Lluís Quer-Sardanyons3 · Guangqu Zheng4 This article is dedicated to István Gyöngy on the occasion of his 70th birthday. Received: 23 September 2021 / Revised: 27 October 2021 / Accepted: 9 November 2021 / Published online: 18 January 2022 © The Author(s) 2022 Abstract In this article, we study the hyperbolic Anderson model driven by a space-time col- ored Gaussian homogeneous noise with spatial dimension d = 1, 2. Under mild assumptions, we provide L p -estimates of the iterated Malliavin derivative of the solu- tion 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 applica- tion 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 inequal- ity coupled with the application of the aforementioned L p -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 L p -estimates of Malliavin derivatives are crucial ingredients to verify a local version of Bouleau-Hirsch criterion for absolute continu- ity. Our approach substantially simplifies the arguments for the one-dimensional case, which has been studied in the recent work by [2]. Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Supported by NSF Grant DMS 1811181. Supported by the Grant PGC2018-097848-B-I00 (Ministerio de Economía y Competitividad). B Guangqu Zheng zhengguangqu@gmail.com Extended author information available on the last page of the article 123 758 Stoch PDE: Anal Comp (2022) 10:757–827 Keywords Hyperbolic Anderson model · Wiener chaos expansion · Malliavin calculus · Second-order Poincaré inequality · Quantitative central limit theorem · Riesz kernel · Dalang’s condition Mathematics Subject Classification 60H15 · 60H07 · 60G15 · 60F05 1 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): ⎧ 2 ⎨ ∂ u = u + u Ẇ ⎪ ∂t 2 on R+ × Rd for d ∈ {1, 2}, (1.1) ⎩u(0, x) = 1, ∂u (0, x) = 0 ⎪ ∂t where  is the Laplacian in space variables and the Gaussian noise Ẇ has the following correlation structure   E Ẇ (t, x)Ẇ (s, y) = γ0 (t − s)γ (x − y), 123 Stoch PDE: Anal Comp (2022) 10:757–827 759 with the following standing assumptions: (i) γ0 : R → [0, ∞] is locally integrable and non-negative definite; (ii) γ is a non-negative and non-negative definite measure on Rd whose spectral measure μ1 satisfies Dalang’s condition:  1 μ(dξ ) < ∞, (1.2) Rd 1 + |ξ |2 where |ξ | denotes the Euclidean norm of ξ ∈ Rd . An important example of the temporal correlation is the Riesz kernel γ0 (t) = |t|−α0 for some α0 ∈ (0, 1) (with γ0 (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 noise Ẇ 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 Cc∞ R+ × Rd ) of infinitely differentiable functions with compact support under the inner product  φ, ψH = φ(t, x)ψ(s, y)γ0 (t − s)γ (x − y)dtd xdsdy (1.3) R2+ ×R2d     = dtdsγ0 (t − s) d xφ(t, x) ψ(s, •) ∗ γ (x), (1.4) R2+ Rd 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 γ0 (t) = γ0 (−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 γ0 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 deter- ministic wave equation, that is, for (t, z) ∈ (0, ∞) × Rd , ⎧1 ⎪ ⎨ 1{|z| 0 and x ∈ Rd , the following equality holds in L 2 ( ):  t u(t, x) = 1 + G t−s (x − y)u(s, y)W (ds, dy), 0 Rd where the above stochastic integral is understood in the Skorokhod sense and the process (s, y) ∈ R+ × Rd −→ 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 Sko- rokhod solution u with the following Wiener chaos expansion: u(t, x) = 1 + In f t,x,n , (1.7) n≥1 where In denotes the nth multiple Wiener integral associated to the isonormal Gaus- sian process W (see Sect. 2 for more details), f t,x,n ∈ H⊗n is defined by (with the convention (1.6) in mind) f t,x,n (t1 , x1 , . . . , tn , xn ) := G t−t1 (x − x1 )G t1 −t2 (x1 − x2 ) · · · G tn−1 −tn (xn−1 −xn ), (1.8) and f t,x,n is the canonical symmetrization of f t,x,n ∈ H⊗n given by 1 f t,x,n (t1 , x1 , . . . , tn , xn ) := f t,x,n (tσ (1) , xσ (1) , . . . , tσ (n) , xσ (n) ), (1.9) n! σ ∈Sn where the sum in (1.9) runs over Sn , the set of permutations on {1, 2, . . . , n}. For example, f t,x,1 (t1 , x1 ) = G t−t1 (x − x1 ) and f t,x,2 (t1 , x1 , t2 , x2 ) 1  = G t−t1 (x − x1 )G t1 −t2 (x1 −x2 )+G t−t2 (x − x2 )G t2 −t1 (x2 − x1 ) . 2 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 u Ẇ 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: ⎧ ⎪ ⎪ (a)γ ∈ L (R2 ) for some ∈ (1, ∞), ⎪ ⎨ (b)γ (x) = |x|−β for some β ∈ (0, 2), (H1) ⎪ ⎪ ⎪ (c)γ (x1 , x2 ) = γ1 (x1 )γ2 (x2 ), where γi (xi ) = |xi |−βi or γi ∈ L i (R) ⎩ for some 0 < βi < 1 < i < +∞, i = 1, 2. 123 Stoch PDE: Anal Comp (2022) 10:757–827 761 Remark 1.2 (i) Note that condition (a) for d = 2 is slightly stronger than Dalang’s condition (1.2). In fact, when d = 2, the paper [18] pointed out that Dalang’s condition (1.2) is equivalent to  ln(|x|−1 )γ (x)d x < ∞; (1.10) |x|≤1 let  = −1 and 0 < ε < 1/  , then there is some δ ∈ (0, 1) and a constant Cε such that ln(|x|−1 ) ≤ Cε |x|−ε for any |x| ≤ δ, from which we deduce that    −1 −1 ln(|x| )γ (x)d x ≤ ln(δ ) γ (x)d x + Cε |x|−ε γ (x)d x |x|≤1 δ<|x|≤1 |x|≤δ  −1 ≤ ln(δ ) γ (x)d x δ<|x|≤1  1/   + Cε γ L (R2 ) |x|−ε d x < ∞. |x|≤δ (ii) The case (c) in Hypothesis (H1) is a mixture of cases (a) and (b). Accordingly, more examples of the noise Ẇ arise. In the space variables, W can behave like a fractional Brownian sheet with Hurst indices greater than 1/2 in both directions, i.e. γ (x1 , x2 ) = |x1 |2H1 −2 |x2 |2H2 −2 for some H1 , H2 ∈ (1/2, 1). (iii) For d = 1 we just assume that γ is a non-negative and non-negative definite mea- sure on R. In this case (see, for instance, Remark 10 of [11]) Dalang’s condition is always satisfied. 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 Cc∞ R+ × Rd ) under the inner product (1.3); as the space H contains generalized functions, it is not clear at first sight whether (s, y) −→ Ds,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. 1/ p (2) We write X p = E[|X | 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 1 , . . . , z p ) for points in p R+ or Rd p , and dzz p = dz 1 · · · dz p , μ(dzz p ) = μ(dz 1 ) · · · μ(dz p ). For a function h : (R+ × Rd ) p → R with p ≥ 2, we often write h(ss p , y p ) = h(s1 , . . . , s p , y1 , . . . , y p ) = h(s1 , y1 , . . . , s p , y p ), which shall not cause any confusion. For m ∈ {1, . . . , p−1} and (ssm , ym ) ∈ Rm + ×R , dm the expression h(ssm , ym ; •) stands for the function 123 762 Stoch PDE: Anal Comp (2022) 10:757–827 (t1 , x1 , . . . , t p−m , x p−m ) → h(s1 , y1 , . . . , sm , ym , t1 , x1 , . . . , t p−m , x p−m ) = h(ssm , ym ; t p−m , x p−m ). Now, with the above notation in mind, we are in the position to state the first main result.2 Theorem 1.3 Let d ∈ {1, 2} and suppose that Hypothesis (H1) holds if d = 2. Then, for any (t, x) ∈ R+ × Rd , the random variable u(t, x) belongs to D∞ (see Sect. 2.1). Moreover, for any integer m ≥ 1, the mth Malliavin derivative D m u(t, x) is a random symmetric function denoted by (ssm , ym ) = (s1 , y1 , . . . , sm , ym ) −→ Ds1 ,y1 Ds2 ,y2 . . . Dsm ,ym u(t, x) = Dsmm ,yym u(t, x), and for any p ∈ [2, ∞), we have, for almost all (ssm , ym ) ∈ [0, t]m × Rmd ,   m! f t,x,m (ssm , ym ) ≤  Dsmm ,yym u(t, x) p  f t,x,m (ssm , ym ), (1.11) where the constant in the upper bound only depends on ( p, t, γ0 , γ , m) and is increas- ing 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]. 1.1 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 estab- lished a central limit theorem for a nonlinear SPDE in [15]. More precisely, they considered the following one-dimensional stochastic heat equation 2 In higher dimension (d ≥ 3), the fundamental wave solution is a uniform measure supported on certain surfaces, then the Malliavin derivative Du(t, x) is expected to be merely a random measure instead of being a random function. In this case, the expression Ds,y u(t, x) does not make sense; see also the recent article [34] for related discussions. 123 Stoch PDE: Anal Comp (2022) 10:757–827 763 ∂u 1 = u + σ (u)Ẇ (1.12) ∂t 2 on R+ × R, where Ẇ 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]),  t u(t, x) = 1 + pt−s (x − y)σ u(s, y) W (ds, dy), (1.13) 0 R with pt denoting the heat kernel,3 one can regard u(t, x) and u(t, y) as weakly depen- dent random variables for x, y far apart so that the integral  R   u(t, x) − 1 d x −R 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. Notation B. (1) For t > 0, we define, with B R := {x ∈ Rd : |x| ≤ R},     FR (t) := u(t, x) − 1 d x and σ R (t) = Var FR (t) . (1.14) BR (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  dTV (X , Y ) = sup μ(B) − ν(B)|, (1.15) B 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:  ω1 = 2, ω2 = π, and κβ,d := d xd y|x − y|−β 1 B1 (x)1 B1 (y) for β ∈ (0, d). R2d (1.16) (5) For an integer m ≥ 1 and p ∈ [1, ∞), we say F ∈ Dm, p if F is m-times p Malliavin differentiable random variable in L p ( ) and E D j F H⊗ j < ∞ for every j = 1, . . . , m; see Sect. 2.1 for more details. Now let us illustrate the strategy in [15]: (For this reference, d = 1) 3 p (x) = (2π t)−d/2 e−|x|2 /(2t) for t > 0 and x ∈ Rd ; in (1.13), d = 1. t 123 764 Stoch PDE: Anal Comp (2022) 10:757–827 • The authors first rewrite FR (t) = δ(Vt,R ) with the random kernel  Vt,R (s, y) = σ (u(s, y)) pt−s (x − y)d x, BR where δ denotes the Skorokhod integral, the adjoint of the Malliavin derivative D. • By standard computations, they obtained σ R2 (t) ∼ R. • If F = δ(v) ∈ D1,2 is a centered random variable with variance one, for some v in the domain of δ, the (univariate) √ Malliavin-Stein bound (see [15, Proposition 2.2]) ensures that dTV (F, Z ) ≤ 2 Var(D F, vH ) for Z ∼ N (0, 1). • Combining the above points, one can see that the obtention of a quantitative CLT is reduced to the computation of Var(D FR (t), Vt,R H ). Because the driving noise is white in time as considered in [15], tools from Itô calculus (Clark-Ocone formula, Burkholder’s inequality, etc.) are used to estimate the above variance term. It is proved in [15] that dTV (FR (t)/σ R (t), Z )  R −1/2 . Meanwhile, a multivariate Malliavin-Stein bound and similar computations lead to the convergence of the finite-dimensional distributions, which coupled with the tightness property gives a functional CLT for {R −1/2 FR (t) : t ∈ R+ }. 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 FR (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+ × Rd 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 FR ; 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. 123 Stoch PDE: Anal Comp (2022) 10:757–827 765 In this paper, we bring in a novel ingredient—the second-order Gaussian Poincaré inequality4 —to reach quantitative CLT results for the hyperbolic Anderson model (1.1). Let us first state our main result. Theorem 1.4 Let u denote the solution to the hyperbolic Anderson model (1.1) and recall the definition of FR (t) and σ R (t) from (1.14). Let Z ∼ N (0, 1) be the standard normal random variable. We assume that γ0 is not identically zero meaning γ0 L 1 ([0,ε]) > 0 for any ε ∈ (0, 1). (1.17) Then the following statements hold true: (1) Suppose that 0 < γ (Rd ) < ∞ if d = 1 and γ ∈ L 1 (Rd ) ∩ L (Rd ) for some > 1 if d = 2. Then, σ R (t) ∼ R d/2 and dTV FR (t)/σ R (t), Z  R −d/2 .   Moreover, as R → ∞, the process R −d/2 FR (t) : t ∈ R+ converges weakly in the space of continuous functions C(R+ ) to a centered Gaussian process G with covariance structure      E G(t)G(s) = ωd p! f t,x, p , f s,0, p H⊗ p d x, (1.18) p≥1 Rd for t, s ∈ R+ . Here ω1 = 2, ω2 = π and f t,x, p are introduced in (1.16) and (1.9), respectively. The convergence of the series in (1.18) is part of the conclusion. (2) Suppose d ∈ {1, 2} and γ (x) = |x|−β for some β ∈ (0, 2 ∧ d). Then, β σ R (t) ∼ R d− 2 and dTV FR (t)/σ R (t), Z  R −β/2 .  β  Moreover, as R → ∞, the process R −d+ 2 FR (t) : t ∈ R+ converges weakly in the space C(R+ ) to a centered Gaussian process Gβ with the covariance structure  t  s   E Gβ (t)Gβ (s) = κβ,d dr dr  γ0 (r − r  )(t − r )(s − r  ), (1.19) 0 0 for t, s ∈ R+ . Here the quantity κβ,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. 123 766 Stoch PDE: Anal Comp (2022) 10:757–827 (3) Suppose d = 2 and γ (x1 , x2 ) = γ1 (x1 )γ2 (x2 ) such that one of the following two conditions holds:  (a ) γi (xi ) = |xi |−βi for some βi ∈ (0, 1), i = 1, 2; (b ) γ1 ∈ L (R) ∩ L 1 (R) and γ2 (x2 ) = |x2 |−β for some 0 < β < 1 < < ∞. (1.20) Then,  1 σ R (t) ∼ R 2− 2 (β1 +β2 ) and dTV FR (t)/σ R (t), Z  R −(β1+β2 )/2 in case (a ), σ R (t) ∼ R (3−β)/2 and dTV FR (t)/σ R (t), Z  R −(β+1)/2 in case (b ).  β1 +β2  Moreover, as R → ∞, in case (a  ) , the process R −2+ 2 FR (t) : t ∈ R+ converges weakly in the space C(R+ ) to a centered Gaussian process Gβ1 ,β2 with the covariance structure  t  s   E Gβ1 ,β2 (t)Gβ1 ,β2 (s) = K β1 ,β2 dr dr  γ0 (r − r  )(t − r )(s − r  ), (1.21) 0 0 for t, s ∈ R+ , where  K β1 ,β2 : = 1{x 2 +x 2 ≤1} 1{y 2 +y 2 ≤1} |x1 − y1 |−β1 |x2 − y2 |−β2 d x1 d x2 dy1 dy2 ; 1 2 1 2 R4 (1.22)  β−3  and in case (b ) , the process R 2 FR (t) : t ∈ R+ converges weakly in the space C(R+ ) to a centered Gaussian process G β with the covariance structure     t s EG β (s) = γ1 (R)Lβ β (t)G dr dr  γ0 (r − r  )(t − r )(s − r  ) (1.23) 0 0 for t, s ∈ R+ , where  Lβ := d x1 d x2 d x3 1{x 2 +x 2 ≤1} 1{x 2 +x 2 ≤1} |x2 − x3 |−β . (1.24) 1 2 1 3 R3 For the above functional convergences, we specify that the space C(R+ ) is equipped with the topology of uniform convergence on compact sets. Remark 1.5 (1) Note that the case when γ (x) = γ1 (x1 )γ2 (x2 ) with γi ∈ L i (R) ∩ L 1 (R) for some i > 1, i = 1, 2, is covered in part (1). Indeed, suppose that 1 ≥ 2 , then by Hölder’s inequality, γ1 ∈ L (R)∩ L (R) implies γ1 ∈ L (R)∩ L (R) 1 1 2 1 and hence γ ∈ L (R ) ∩ L (R ). 2 2 1 2 (ii) The rate of convergence can also be described using other common distances such as the Wasserstein distance and the Kolmogorov distance; see [25, Appendix C]. 123 Stoch PDE: Anal Comp (2022) 10:757–827 767 (iii) 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 ∈ D1,2 , the Gaussian Poincaré inequality (see e.g. [14] or (2.12)) ensures that   Var(F) ≤ E D F 2H with equality if and only if F is Gaussian, that is, if D F is small, then the random variable F has necessarily small fluctua- tions. In the paper [8], Chatterjee pointed out that for F = f (X 1 , . . . , X d ) with X 1 , . . . , X d i.i.d. N (0, 1) and f twice differentiable, F is close in total variation dis- tance to a normal distribution with matched mean and variance if the Hessian matrix Hess f (X 1 , . . . , X d ) is negligible, roughly speaking. This is known as the second- order Gaussian Poincaré inequality. In what follows, we state the infinite-dimensional version of this inequality due to Nourdin, Peccati and Reinert; see the paper [26] as well as the book [25].5 Proposition 1.6 Let F be a centered element of D2,4 such that E[F 2 ] = σ 2 > 0 and let Z ∼ N (0, σ 2 ). Then, 3   2 1/4   1/4 dTV (F, Z ) ≤ 2 E  D 2 F ⊗1 D 2 F H⊗2 E D F 4H , (1.25) σ 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. Theorem 1.7 (Theorem 2.1 in [42]) Let F ∈ D2,4 with mean zero and variance σ 2 > 0 and let Z ∼ N (0, σ 2 ). Suppose H = L 2 (A, ν) with ν a diffusive measure on some measurable space A. Then, dTV F, Z   1 4  2    2 ≤ 2 E D F ⊗1 D F (x, y) ×E (D F) (x)(D F) (y) ν(d x)ν(dy) . 2 2 2 2 σ A×A The proof of the above inequality follows from the general Malliavin-Stein bound 2   dTV F, Z ≤ 2 E σ 2 − D F, −DL −1 FH  (1.26) σ 5 Note that there is a typo in equation (5.3.2) of [25]: We have E[ D F 4 ]1/4 instead of E[ D 2 F 4 ]1/4 . H H 123 768 Stoch PDE: Anal Comp (2022) 10:757–827 (see [25, equation (5.1.4)])6 and Vidotto’s new bound of   E (Cov(F, G) − D F, −DL −1 GH )2 for centered F, G ∈ D2,4 (see [42, Proposition 3.2]), where L −1 is the pseudo-inverse of the Ornstein-Uhlenbeck operator L; see Sect. 2.1 for the definitions. Recall that our Hilbert space H is the completion of Cc∞ (R+ × Rd ) 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 ∈ D2,4 has mean zero and variance σ 2 ∈ (0, ∞) such that with probability 1, D F ∈ |H| and D 2 F ∈ |H⊗2 |, then 4 √ dTV F, Z ≤ A, σ2 where Z ∼ N (0, σ 2 ) and  A:= dr dr  dsds  dθ dθ  dzdz  dydy  dwdw  γ0 (θ − θ  )γ0 (s − s  )γ0 (r − r  ) R6+ ×R6d × γ (z − z  )γ (w − w  )γ (y − y  ) Dr ,z Dθ,w F 4 Ds,y Dθ  ,w F 4 × Dr  ,z  F 4 Ds  ,y  F 4. 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 (2.5), we have that E D F, −DL −1 FH = E[F 2 ] = σ 2 . Indeed, we can write  2  2 −1   ≤ 2 Var D F, −DL −1 FH dTV (F, Z ) ≤ E σ − D F, −DL FH σ2 σ2 4 √ ≤ 2 A by Proposition 1.9 below. σ Proposition 1.9 If F, G ∈ D2,4 have mean zero such that with probability one, D F, DG ∈ |H| and D 2 F, D 2 G ∈ |H⊗2 |, then     Var D F, −DL −1 GH = E (Cov(F, G) − D F, −DL −1 GH )2 ≤ 2 A1 + 2 A2 , (1.27) 6 Unlike in [25], we do not assume F to have a density; in fact, it suffices to use [44, Proposition 2.1.1] and [25, (5.1.1)] to establish [25, Eq. (5.1.4)]. 123 Stoch PDE: Anal Comp (2022) 10:757–827 769 where  A1 : = dr dr  dsds  dθ dθ  dzdz  dydy  dwdw  R6+ ×R6d × γ0 (θ − θ  )γ0 (s − s  )γ0 (r − r  )γ (z −z  )γ (w−w  )γ (y − y  ) × Dr ,z Dθ,w F 4 Ds,y Dθ  ,w F 4 Dr  ,z  G 4 Ds  ,y  G 4 and A2 is defined by switching the positions of F, G in the definition of A1 . For the sake of completeness, we sketch the proof of Proposition 1.9 in Appendix A.2. Once we have the information on the growth order of σ R (t), we can apply Theorem 1.3 and Proposition 1.9 to obtain the error bounds in Theorem 1.4. The proof of Theorem 1.4 will be given in Sect. 4: In Sect. 4.1, we will establish the limiting covariance struc- ture, which will be used to obtain the quantitative CLTs in Sect. 4.2; Proposition 1.9, combined with a multivariate Malliavin-Stein bound (see e.g. [25, Theorem 6.1.2]), also gives us easy access to the convergence of finite-dimensional distributions (f.d.d. convergence) for part (1), while in the other parts, the f.d.d. convergence follows easily from the dominance of the first chaotic component of FR (t); finally in Sect. 4.3, we establish the functional CLT by showing the required tightness, which will follow by verifying the well-known criterion of Kolmogorov-Chentsov (see e.g. [17, Corollary 16.9]). 1.2 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 γ0 , γ .  γ0 = Fμ0 and γ = Fμ, where μ0 , μ are nonnegative tempered measures (H2) and have strictly positive densities with respect to the Lebesgue measure. The following is the main result of this section. Theorem 1.10 Let d ∈ {1, 2} and assume that Hypothesis (H2) holds. In addition, assume that Hypothesis (H1) holds if d = 2. Let u be the solution to (1.1). For any t > 0 and x ∈ Rd , the law of u(t, x) restricted to the set R\{0} is absolutely continuous with respect to the Lebesgue measure on R\{0}. 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, Du(t, x) H > 0 a.s. on m, (1.28) where m = {|u(t, x)| ≥ 1/m}. Notice that  t t Du(t, x) 2H = γ0 (r − s)Dr ,• u(t, x), Ds,• u(t, x)0 dr ds, 0 0 123 770 Stoch PDE: Anal Comp (2022) 10:757–827 where P0 is the completion of Cc∞ (Rd ) with respect to the inner product ·, ·0 intro- duced 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, Dr ,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 Dr ,• u(t, x), Ds,• 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  t Dr ,• u(t, x) 20 dr > 0 implies Du(t, x) H > 0 (see Lemma A.1). 0 t This allows us to reduce the problem to the non-degeneracy of t−δ Dr ,• u(t, x) 20 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 [7,22,23,31,39–41]. In all these articles, the authors considered a stochastic wave equation of the form ∂ 2u (t, x) = u(t, x) + b(u(t, x)) + σ (u(t, x))Ẋ(t, x), ∂t 2 on R+ × Rd , with d ≥ 1. 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 noise Ẋ 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. 2 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. 123 Stoch PDE: Anal Comp (2022) 10:757–827 771 2.1 Basic Malliavin calculus Recall that the Hilbert space H is the completion of Cc∞ (R+ × Rd ) under the inner product (1.3) that can be written as      ψ, φ H = dsdtγ0 (t − s) ψ(t, •), φ(s, •) 0 for ψ, φ ∈ Cc∞ (R+ × Rd ), R2+ where  h, g0 = dzdz  γ (z − z  )h(z)g(z  ). (2.1) R2d As defined in Sect. 1.2, we denote by P0 the completion of Cc∞ (Rd ) with respect to the inner product h, g0 . Let |P0 | be the set of measurable functions h : Rd → R such that  dzdz  γ (z − z  )|h|(z)|h|(z  ) < ∞. (2.2) R2d  Then |P0 | ⊂ P0 and for h ∈ |P0 |, h 20 = R2d dzdz  γ (z − z  )h(z)h(z  ). We define the space |H| in a similar way. For h, g ∈ Cc∞ (Rd ) we can express (2.1) using the Fourier transform:  h, g0 = μ(dξ )F h(ξ )F g(ξ ). (2.3) Rd The Parseval-type relation (2.3) also holds for functions h, g ∈ L 1 (Rd ) ∩ |P0 |. 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 ∈ Hn ; 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 noise Ẇ in (1.1) that we denote by {W (φ) : φ ∈ H}. That is, {W (φ) : φ ∈ H} is a centered Gaussian family of real- valued random variables defined on ( , B, P) such that E[W (ψ)W (φ)] = ψ, φH for any ψ, φ ∈ H. We will take B to be the σ -algebra σ {W } generated by the family of random variables {W (h) : h ∈ Cc∞ (R+ × Rd )}. In the sequel, we recall some basics on Malliavin calculus from the books [25,27]. ∞ (Rn ) denote the space of smooth functions with all their partial derivatives Let Cpoly having at most polynomial growth at infinity and let S denote the set of simple smooth functionals of the form ∞ F = f W (h 1 ), . . . , W (h n ) for f ∈ Cpoly (Rn ) and h i ∈ H, 1 ≤ i ≤ n. 123 772 Stoch PDE: Anal Comp (2022) 10:757–827 For such a random variable F, its Malliavin derivative D F is the H-valued random variable given by n ∂f DF = W (h 1 ), . . . , W (h n ) h i . ∂ xi i=1 And similarly its mth Malliavin derivative D m F is the H⊗m -valued random variable given by n ∂m f Dm F = W (h 1 ), . . . , W (h n ) h i1 ⊗ · · · ⊗ h im , (2.4) ∂ xi1 · · · ∂ xim i 1 ,...,i m =1 which is an element in L p ( ; Hm ) for any p ∈ [1, ∞). It is known that the space S is dense in L p ( , σ {W }, P) and D m : S −→ L p ( ; Hm ) is closable for any p ∈ [1, ∞); see e.g. Lemma 2.3.1 and Proposition 2.3.4 in [25]. Let Dm, p be the closure of S under the norm       1/ p F  p   p Dm, p = E |F| p + E D F H + · · · + E D m F H⊗m and let D∞ := Dm, p . m, p≥1 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 Cv > 0 for which     E D m F, vH⊗m  ≤ Cv F 2 for all F ∈ S. By Riesz representation theorem, there is a unique random variable, denoted by δ m (v), such that the following duality relationship holds:     E Fδ m (v) = E D m F, vH⊗m . (2.5) Equality (2.5) holds for all v ∈ Dom(δ m ) and all F ∈ Dm,2 . In the simplest case when F = f (W (h)) with h ∈ H and f ∈ Cpoly 1 (R), we have δ(h) = W (h) ∼ N (0, h 2 ) H and equality (2.5) reduces to     E f (W (h))W (h) = E f  (W (h)) h 2H , 123 Stoch PDE: Anal Comp (2022) 10:757–827 773 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 Z ∼ N (0, σ 2 ) if and only if E[Z f (Z )] = σ 2 E[ f  (Z )], (2.6) 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: ! L 2 ( , σ {W }, P) = CnW , n≥0 where C0W , identified as R, is the space of constant random variables and CnW = {δ n (h) : h ∈ H⊗n is deterministic}, for n ≥ 1, is called the nth Wiener chaos asso- ciated to W . Note that the first Wiener chaos consists of centered Gaussian random variables. When h ∈ H⊗n is deterministic, we write In (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 In (h) = In (h) for any h ∈ H⊗n . The above decomposition can be rephrased as follows. For any F ∈ L 2 ( , σ {W }, P), F = E[F] + In ( f n ), (2.7) n≥1 with f n ∈ Hn uniquely determined for each n ≥ 1. Moreover, the (modified) isometry property holds     E I p ( f )Iq (g) = p!1{ p=q} f , g H⊗ p , (2.8) for any f ∈ H⊗ p and g ∈ H⊗q . We have the following product formula: For f ∈ H p and g ∈ Hq , p∧q    p q I p ( f )Iq (g) = r! I p+q−2r ( f ⊗r g), (2.9) r r r =0 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 {ei , i ∈ O} of H. Then, for 1 ≤ r ≤ p∧q, f ⊗r g :=  f , ei1 ⊗ · · · ⊗ ei p H⊗ p g, e j1 ⊗ · · · ⊗ e jq H⊗ p i 1 ,...,i p , j1 ,..., jq ∈O × 1{ik = jk ,∀k=1,...,r } eir +1 ⊗ · · · ⊗ ei p ⊗ e jr +1 ⊗ · · · ⊗ e jq . (2.10) 123 774 Stoch PDE: Anal Comp (2022) 10:757–827 In the particular case when f , g are real-valued functions, we can write   ( f ⊗r g)(tt p−r , x p−r , tq−r , xq−r ) ⎛ ⎞  $ r = dssr dssr dyyr dyyr ⎝ γ0 (s j − s j )γ (y j − y j )⎠ R2r + ×R 2r d j=1 × f (ssr , t p−r , yr , x p−r )g(ssr , tq−r  , yr , xq−r  ), provided the above integral exists. For F ∈ Dm,2 with the representation (2.7) and m ≥ 1, we have n! D•m F = In−m f n (•, ∗) with convergence in L 2 ( ; H⊗m ), (2.11) n≥m (n − m)! where In−m f n (•, ∗) is understood as the (n − m)th multiple integral of f n (•, ∗) ∈ H⊗(n−m) for fixed •. We can write n! Dsmm ,yym F = In−m f n (ssm , ym ; ∗) , n≥m (n − m)! whenever the above series makes sense and converges in L 2 ( ). With the decomposi- tion (2.11) in mind, we have the following Gaussian Poincaré inequality: For F ∈ D1,2 , it holds that   Var(F) ≤ E D F 2H . (2.12) In fact, if F has the representation (2.7), then   Var(F) = n! f n 2H⊗n and E D F 2H = nn! f n 2H⊗n , n≥1 n≥1 which gives us (2.12) and, moreover, indicates that the equality in (2.12) holds only when F ∈ C0W ⊕ C1W , 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 H0 := L 2 R+ ; P0 and point out that the following inequality reduces many calculations to the case of the white noise in time. For any nonnegative function f ∈ H0⊗n that vanishes outside ([0, t] × Rd )n , f 2H⊗n ≤ tn f 2H⊗n , (2.13) 0 123 Stoch PDE: Anal Comp (2022) 10:757–827 775 where7  t  t = 2 γ0 (s)ds and f H⊗n = 2 f (t1 , ·, . . . , tn , ·) 2P ⊗n dt1 · · · dtn ; 0 0 [0,t]n 0 whenever no ambiguity arises, we write f 0 := f P ⊗n so that f 2H⊗n =  0 0 [0,t]n f (t t n , •) 2 dtt . 0 n Let Ẋ denote the Gaussian noise that is white in time and has the same spatial correlation as W . More precisely, {X( f ) : f ∈ H0 } is a centered Gaussian family with covariance E[X( f )X(g)] =  f , gH0 , for any f , g ∈ H0 . Denote by I pX 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 I pX ( f )IqX (g) = I p+q X ( f ⊗ g). (2.14) In this case, the random variables I pX ( f ) and IqX (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 ∈ D1,2 and D F ∈ Dom(δ); in 'let L F = −δ D F. For F ∈ L ( ) of the form (2.7),' F ∈ Dom(L) if this case, we 2 and only if n≥1 n 2 n! f n 2H⊗n < ∞. In this case, we have L F = n≥1 −n In ( f n ). Using the chaos expansion, we can also define the Ornstein-Uhlenbeck semigroup {Pt = et 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), 1 Pt F := e−nt In ( f n ) and L −1 F = − In ( f n ). n n≥0 n≥1 7 For the sake of completeness, we sketch a proof of (2.13) here: Given such a function f ∈ H⊗n , 0  $ n   f 2H⊗n = dssn dttn f (ssn , •), f (ttn , •) P ⊗n γ0 (s j − t j ) [0,t]2n 0 j=1  1  2  $ 2  n ≤ dssn dttn  f (ssn , •)P ⊗n +  f (ttn , •)P ⊗n γ0 (s j − t j ) ≤ tn f 2 ⊗n . [0,t]2n 2 0 0 H0 j=1 8 This means f = 0 outside (J × Rd ) p and g = 0 outside (J c × Rd )q for some set J ⊂ R . We will + ( j) apply this formula to functions f = f t,x, j (r , z; •) and g = fr ,z,n− j given in Sect. 3.1, in which case J = (r , t). 123 776 Stoch PDE: Anal Comp (2022) 10:757–827 Observe that for any centered random variable F ∈ L 2 ( , σ {W }, P), L L −1 F = F and for any G ∈ Dom(L), L −1 LG = G − E[G]. The above expression and the modified isometry property (2.8) give us the contraction property of Pt on L 2 ( ), that is, for F ∈ L 2 ( , σ {W }, P), Pt F 2 ≤ F 2 . Moreover, Pt 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: For F ∈ L q ( , σ {W }, P) with q ∈ (1, ∞), it holds for each t ≥ 0 that Pt F qt ≤ F q with qt = 1 + (q − 1)e2t . In this paper, we need one of its consequences – a moment inequality comparing L q ( )-norms on a fixed chaos: If F ∈ CnW and p ∈ [2, ∞), then F p ≤ ( p − 1)n/2 F 2; (2.15) see e.g. [25, Corollary 2.8.14]. 2.2 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: (a)γ ∈ L (Rd ) f or some ∈ (1, ∞) (b)γ (x) = |x|−β f or someβ ∈ (0, d). Define  /(2 − 1) in case (a) q= d/(2d − β) in case (b). Then, for any f , g ∈ L 2q (Rd ),   f (x)g(y)γ (x − y)d xd y ≤ Cγ f L 2q (Rd ) g L 2q (Rd ) , Rd Rd where Cγ = γ L (Rd ) in case (a), and Cγ = Cd,β is the constant (depending on d, β) that appears in the Hardy–Littlewood–Sobolev inequality (2.16) below, in case (b). 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. In case (a), we apply Hölder’s inequality and Young’s convolution inequality:  f (x)(g ∗ γ )(x)d x ≤ f 2 g∗γ L 2 (Rd ) L2 −1 (Rd ) Rd ≤ f 2 g 2 γ L (Rd ) . L2 −1 (Rd ) L2 −1 (Rd ) 123 Stoch PDE: Anal Comp (2022) 10:757–827 777 In case (b), we apply Hölder’s inequality and Hardy-Littlewood-Sobolev inequality:  f (x)(g ∗ γ )(x)d x ≤ f 2d g∗γ L 2d/β (Rd ) Rd L 2d−β (Rd ) ≤ Cd,β f 2d g 2d . (2.16) L 2d−β (Rd ) L 2d−β (Rd ) This concludes the proof.   To deal with case (c) in (H1), we need the following modification of Lemma 2.1. (d Lemma 2.2 Suppose that γ (x1 , . . . , xd ) = i=1 γi (x i ), where for each i ∈ {1, . . . , d}, (M1)γi ∈ L i (R) for some i ∈ (1, ∞) or (M2)γi (x) = |x|−βi for some βi ∈ (0, 1) Let qi = i /(2 i − 1) in case (M1) and qi = 1/(2 − βi ) in case (M2). Let q = max{qi : i = 1, . . . , d}. (d If f , g ∈ L 2q (Rd ) satisfy f (x) = g(x) = 0 for x ∈ / i=1 [ai , bi ] for some real numbers ai < bi ,9 then   f (x)g(y)γ (x − y)d xd y ≤ ν Cγ f L 2q (Rd ) g L 2q (Rd ) , (2.17) Rd Rd (d 'd with  = max{bi − ai ; i = 1, . . . , d}, Cγ = i=1 Cγi and ν = i=1 (qi−1 − q −1 ). In particular, when qi = q for all i ∈ {1, . . . , d}, we have   f (x)g(y)γ (x − y)d xd y ≤ Cγ f L 2q (Rd ) g L 2q (Rd ) . Rd Rd The constants Cγi 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 = (x1 , . . . , xk ) =: xk . Without loss of any generality we assume q1 ≥ q2 ≥ · · · ≥ qk , so that q = q1 . Applying the initial step (d = 1) yields  $ k dxxk dyyk f (xxk )g(yyk ) γi (xi − yi ) R2k i=1  $     k−1 ≤ Cγk dxxk−1 dyyk−1  f (xxk−1 , •) L 2qk (R) g(yyk−1 , •) L 2qk (R) γi (xi − yi ). R2(k−1) i=1 (2.18) 9 We can apply this lemma to the function y ∈ R2 → G t−s (x − y) whose support is contained in {y ∈ R2 ; |x − y| < t − s}, so we can choose  = 2t − 2s. 123 778 Stoch PDE: Anal Comp (2022) 10:757–827 By the induction hypothesis, we can bound the right-hand side of (2.18) by ) k *  1 $   2q ν∗  f (xxk−1 , •)2q2q Cγi  L k (R) dxxk−1 i=1 Rk−1  1   2q × g(yyk−1 , •)2q2q dyyk−1 , L k (R) Rk−1 'k−1 −1 with ν ∗ = i=1 (qi − q −1 ). By Hölder’s inequality,  1   2q  f (xxk−1 , •)2q2q dxxk−1 L k (R) Rk−1 ⎛ ⎞1    2q2q 2q bk   k =⎝  f (xxk−1 , xk )2qk d xk dxxk−1 ⎠ Rk−1 ak    2q1 1 1 − 2q bk   ≤ 2qk  f (xxk−1 , xk )2q d xk dxxk−1 . Rk−1 ak 'k A similar inequality holds for g. Since ν ∗+(qk−1−q −1 ) = −1 −1 i=1 (qi −q ), inequality (2.17) holds for d = k.   We will need the following generalization of Lemmas 2.1 and 2.2. Lemma 2.3 (1) Under the conditions of Lemma 2.1, for any f , g ∈ L 2q (Rmd )  $ m f (xxm )g(yym ) γ (x j − y j )dxxm dyym ≤ Cγm f L 2q (Rmd ) g L 2q (Rmd ) , R2md j=1 (2.19) where Cγ is the same constant as in Lemma 2.1. Here xm = (x1 , . . . , xm ) with xi ∈ Rd . (2) Let γ , Cγ and q be given as in Lemma 2.2. If f , g ∈ L 2q (Rmd ) satisfy f (xxmd ) = (md g(xxmd ) = 0 for xmd ∈/ i=1 [ai , bi ] for some real numbers ai < bi , then inequality (2.19) holds with Cγ replaced by ν Cγ , where  = max{bi −ai : i = 1, . . . , md} 'd and ν = i=1 (qi−1 − q −1 ). Here x md = (x1 , . . . , xmd ) with xi ∈ R. Proof The proof will be done by induction on m simultaneously for both cases (1) and (2). Let C = Cγ in case (1) and C = ν Cγ 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 123 Stoch PDE: Anal Comp (2022) 10:757–827 779  $ m f (xxm )g(yym ) γ (x j − y j )dxxm dyym R2dm j=1  ≤C f (xxm−1 , •) L 2q (Rd ) g(yym−1 , •) L 2q (Rd ) R2d(m−1) $ m−1 × γ (x j − y j )dxxm−1 dyym−1 . j=1 By the induction hypothesis, the latter term can be bounded by  1 2q 2q C m f (xxm−1 , •) dxx L 2q (Rd ) m−1 Rd(m−1)  1 2q 2q × g(xxm−1 , •) dxx L 2q (Rd ) m−1 , Rd(m−1) which completes the proof.   Let us return to the three cases of Hypothesis (H1). Lemma 2.1 indicates that L 2q (R2 ) is continuously embedded into P0 , with q ∈ (1/2, 1) given by  /(2 − 1) in case (a), q= (2.20) 2/(4 − β) in case (b). Recall that P0 has been defined at the beginning of Sect. 2.1. Moreover, for any f , g ∈ L 2q (R2 ),     f (x)g(x)γ (x − y)d xd y ≤ Dγ f L 2q (R2 ) g L 2q (R2 ) , (2.21) R4 where  γ L (R2 ) in case (a), Dγ = (2.22) C2,β in case (b). For case (c) of Hypothesis (H1), we consider three sub-cases: ⎧ ⎪ ⎨ (i) γi ∈ L i (R) for some i > 1, i = 1, 2; (ii) γi (xi ) = |xi |−βi for some βi ∈ (0, 1), i = 1, 2; ⎪ ⎩ (iii) γ1 ∈ L (R) for some ∈ (1, ∞) and γ2 (x2 ) = |x2 |−β for some β ∈ (0, 1). 123 780 Stoch PDE: Anal Comp (2022) 10:757–827 Lemma 2.2 implies that, for any f , g ∈ L 2q (R2 ) with ⎧ ⎪ ⎨max{ i /(2 i − 1) : i = 1, 2} in case (i) q = max{1/(2 − βi ) : i = 1, 2} in case (ii) , (2.23) ⎪ ⎩ max{ /(2 − 1), 1/(2 − β)} in case (iii) such that f , g vanish outside a box with side lengths bounded by , then inequality (2.21) still holds with ⎧ ⎪ ⎪ |1−1| ⎨ γ1 L 1 (R) γ2 L 2 (R)  1 2 in case (i) Dγ = C1,β1 C1,β2 |β1 −β2 | in case (ii) , (2.24) ⎪ ⎪ ⎩C γ  | 1 −β| in case (iii) 1,β 1 L (R) where the constants C1,βi are given as in Lemma 2.1. From Lemma 2.3, we deduce that in cases (a) and (b),  f 2H⊗n ≤ Dγn f (ttn , •) 2L 2q (R2n ) dttn , (2.25) 0 [0,t]n for any measurable function f : (R+ × R2 )n → R such that f vanishes outside ([0, t] × R2 )n ; in case (c), inequality (2.25) holds true for any measurable function f : (R+ × R2 )n → R such that $ 2n f (t1 , x1 , . . . , tn , xn ) = f (ttn , xn ) = 0 for tn ∈ / [0, t]n and xn ∈ / [ai , bi ] i=1 with  := max{bi − ai : i = 1, . . . , 2n} < ∞. Let us present a few facts on the fundamental solution G. When d = 2,  1/ p (2π )1− p 2 −1 Gt L p (R2 ) = tp for all p ∈ (0, 2), (2.26) 2− p p q G t (x) ≤ (2π t)q− p G t (x) for all p < q, (2.27) and 1{|x| 0 is a constant depending on q. 123 Stoch PDE: Anal Comp (2022) 10:757–827 781 Finally, we record the expression of the Fourier transform of G t for d ∈ {1, 2}:  sin(t|ξ |) t (ξ ). F G t (ξ ) = e−iξ ·x G t (x)d x = =: G (2.29) Rd |ξ | Note that (see e.g. (3.4) of [3])  2 1 G t (ξ ) ≤ 2(t 2 ∨ 1) . (2.30) 1 + |ξ |2 In Sect. 4, we need the following two results. Lemma 2.5 For d ∈ {1, 2}, let γ0 satisfy the assumption (i) on page 2 and let μ p be a symmetric measure on (Rd ) p , for some integer p ≥ 1. Then, with 0 < s ≤ t and p  p (t) = {ss p ∈ R+ : t = s0 > s1 > · · · > s p > 0}, ⎛ ⎞   $ p  dss p d s̃ p 1{s>s̃σ (1)>···>s̃ } ⎝ γ0 (s j − s̃ j )⎠ μ p (dξξ p ) σ ( p) >0 σ ∈S p  p (t) [0,s] p j=1 R pd × g(s1 , ξ1 , . . . , s p , ξ p )g(s̃σ (1) , ξσ (1) , . . . , s̃σ ( p) , ξσ ( p) )    t p ≤ t s ds p ξ μ p (dξ p )g(s1 , ξ1 , . . . , s p , ξ p ) , with t := 2 γ0 (a)da,  p (t) R pd −t for any measurable function g : (R+ × Rd ) p → R+ for which the above integral is finite. Proof After applying |ab| ≤ a +b 2 2 2 and using the symmetry of μ p , we have that the left-hand side quantity is bounded by   $ p 1 dss p d s̃ p 1{s>s̃σ (1)>···>s̃ >0 } h(ss p ) γ0 (s j − s̃ j ) (2.31) 2 σ ( p) σ ∈S p  p (t) [0,s] p j=1   $ p 1 + dss p d s̃ p 1{s>s̃σ (1)>···>s̃ }h s̃σ (1) , ..., s̃σ ( p) γ0 (s j − s̃ j ) 2 σ ( p) >0 σ ∈S p  p (t) [0,s] p j=1 (2.32) with ⎧ ⎨ μ p (dξξ p )g(s1 , ξ1 , . . . , s p , ξ p )2 , for s p ∈  p (t) h(s1 , . . . , s p ) := R pd ⎩ 0, otherwise. 123 782 Stoch PDE: Anal Comp (2022) 10:757–827 Putting Is (s1 , . . . , s p ) := 1{s>s1 >···>s p >0} and letting Is (s1 , . . . , s p ) be its canonical   symmetrization (so that Is  ≤ ( p!)−1 ), we can rewrite the term in (2.31) as   $ p p! dss p d s̃ p h(ss p )Is (s̃s̃ p ) γ0 (s j − s̃ j ) 2  p (t) [0,s] p j=1   $ p 1 ≤ dss p d s̃ p h(ss p ) γ0 (s j − s̃ j ) 2  p (t) [0,s] p j=1  1 p ≤  dss p h(ss p ), 2 t  p (t) s using also the bound sup{ 0 γ0 (r −r  )dr  : r ∈ [0, t]} ≤ t . For the other term (2.32), we argue in the same way: With (Is · h)(s1 , ..., s p ) = Is (s1 , . . . , s p )h(s1 , ..., s p ), we rewrite the term (2.32) as   $ p p!  p!   dss p d s̃ p It (ss p ) × (Is · h)(s p ) γ0 (s j − s̃ j ) = It , I s · h H⊗ p 2 [0,t] p [0,s] p 2 j=1   $ p p!   1 = It , Is · h H⊗ p ≤ dtt p h(s p ) γ0 (s j − s j ) 2 2 [0,t] p  p (s) j=1  1 p ≤ t dss p h(ss p ), 2  p (s)   since h ≥ 0 and It  ≤ ( p!)−1 . This concludes the proof.   Lemma 2.6 For d ∈ {1, 2} let γ , μ satisfy the assumption (ii) on page 2. Then, for any nonnegative function h ∈ P0 ∩ L 1 (Rd ),   sup μ(dξ )|F h(ξ + z)| ≤ 2 μ(dξ )|F h(ξ )|2 . z∈Rd Rd Rd As a consequence, for any integer p ≥ 1 and w1 , . . . , w p ∈ [0, t],  $ p   p  2 μ(dξ ) sup sup μ(dξξ p )   G w j (ξ j + z j ) ≤ 2(t ∨ 1) 2 . w p ∈[0,t] p z p ∈Rdp Rdp Rd 1 + |ξ |2 j=1 (2.33) Proof Since h ≥ 0, using the fact that F h(ξ + z) = F(e−i z· h)(ξ ) together with |e−i z(x+y) | = 1, we get    2 μ(dξ )F h(ξ + z) = e−i z(x+y) h(x)h(y)γ (x − y)d xd y Rd R2d 123 Stoch PDE: Anal Comp (2022) 10:757–827 783  ≤ h(x)h(y)γ (x − y)d xd y, R2d   2 which is exactly Rd μ(dξ )F h(ξ ) . In particular, by (2.30),     2  2 μ(dξ ) sup μ(dξ )G s (ξ + z) ≤ μ(dξ )G s (ξ ) ≤ 2(s 2 ∨ 1) , z∈Rd Rd Rd Rd 1 + |ξ |2 which is finite due to Dalang’s condition (1.2). Applying this inequality several times yields  $ p   p  2 μ(dξ ) μ(dξξ p ) G w j (ξ j + z j ) ≤ 2(t 2 ∨ 1) , Rdp Rd 1 + |ξ |2 j=1 which is a uniform bound over (zz p , w p ) ∈ Rd p × [0, t] p .   3 Lp 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. 3.1 Step 1: Preliminaries Let us first introduce some handy notation. Recall that for tn := (t1 , . . . , tn ) and xn := (x1 , . . . , xn ), we defined in (1.8) f t,x,n (ttn , xn ) = G t−t1 (x − x1 )G t1 −t2 (x1 − x2 ) · · · G tn−1 −tn (xn−1 − xn ), 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 (ti , xi ) as one coordinate and we write f t,x,n (r , z; tn−1 , xn−1 ) := f t,x,n (r , z, t1 , x1 , . . . , tn−1 , xn−1 ) as in Notation A-(3). Recall that the solution u(t, x) has the Wiener chaos expansion ∞ u(t, x) = 1 + In ( f t,x,n ), n=1 where the kernel f t,x,n is not symmetric and in this case, by definition, In ( f t,x,n ) = In f t,x,n . 123 784 Stoch PDE: Anal Comp (2022) 10:757–827 Our first goal is to show that, for any fixed (r , z) ∈ [0, t] × Rd and for any p ∈ [2, ∞), the series n In−1 f t,x,n (r , z; •) (3.1) n≥1 converges in L p ( ), and the sum, denoted by Dr ,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, n 1 ( j) f t,x,n (r , z; •) = h t,x,n (r , z; •), (3.2) n j=1 ( j) ( j) where h t,x,n (r , z; •) is the symmetrization of the function (ttn−1 , xn−1 ) → f t,x,n (r , z; tn−1 , xn−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), ( j) f t,x,n (r , z; tn−1 , xn−1 ) = G t−t1 (x − x1 ) · · · G t j−1 −r (x j−1 −z)G r −t j (z − x j ) · · · G tn−2−tn−1 (xn−2 −xn−1 ). (3.3) That is, ( j) ( j) f t,x,n (r , z; •) = f t,x, j (r , z; •) ⊗ fr ,z,n− j , (3.4) (1) (1) with fr ,z,1 = 1. For example, f t,x,1 (r , z; •) = G t−r (x −z) and f t,x,n (r , z; tn−1 , xn−1 ) = G t−r (x − z) fr ,z,n−1 (ttn−1 , xn−1 ). By the definition of the symmetrization, ( j) h t,x,n (r , z; tn−1 , xn−1 ) 1 ( j) = f t,x,n (r , z; tσ (1) , xσ (1) , . . . , tσ (n−1) , xσ (n−1) ). (3.5) (n − 1)! σ ∈Sn−1 Similarly, for sm ∈ [0, t]m and ym ∈ Rdm , and for any p ∈ [2, ∞), we will show that n! Dsmm ,yym u(t, x) := In−m f t,x,n (ssm , ym ; •) (3.6) n≥m (n − m)! converges in L p ( ). Note that if the series (3.6) converges in L p ( ), we can see that almost surely, the function (ssm , ym ) → Dsmm ,yym u(t, x) 123 Stoch PDE: Anal Comp (2022) 10:757–827 785 is symmetric, meaning that for any σ ∈ Sm , Ds1 ,y1 Ds2 ,y2 · · · Dsm ,ym u(t, x) = Dsσ (1) ,yσ (1) Dsσ (2) ,yσ (2) · · · Dsσ (m) ,yσ (m) u(t, x). From now on, we assume t > s1 > ... > sm > 0 without losing any generality. Note that like (3.2), we can write n! (ii ) f t,x,n (ssm , ym ; •) = m h t,x,n (ssm , ym ; •), (3.7) (n − m)! i m ∈n,m (ii ) where i m ∈ n,m means 1 ≤ i 1 < i 2 < · · · < i m ≤ n and h t,x,n m (ssm , ym ; •) is the (iim ) symmetrization of the function f t,x,n (ssm , ym ; •) that is defined by (ii ) (i ) (i −i ) m f t,x,n (ssm , ym ; •) = f t,x,i 1 (s , y1 ; •) ⊗ f s1 ,y 1 1 2 1 (s , y2 ; •) ⊗ · · · ⊗ 1 ,i 2 −i 1 2 (i −i ) ,ym−1 ,i m −i m−1 (sm , ym ; •) ⊗ f sm ,ym ,n−i m , m m−1 f sm−1 (3.8) which is a generalization of (3.4). 3.2 Step 2: Reduction to white noise in time Let Ẋ denote the Gaussian noise that is white in time and has the same spatial correla- tion as W and let {X( f ) : f ∈ H0 } denote the resulting isonormal Gaussian process; see Sect. 2.1. For any p ∈ [2, ∞), we deduce from (3.6) and (3.7) that  ⎛ ⎞    m   (iim )   Ds   In−m ⎝ h t,x,n (ssm , ym ; •)  ⎠ sm ,yym u(t, x) p ≤   by triangle inequality n≥m  i m ∈n,m  p  ⎛ ⎞    (iim )  ( p − 1) 2  ⎝ ⎠  by (2.15). n−m ≤ I  n−m h (s s t,x,n m m , y ; •)  n≥m  i m ∈n,m  2 ' (iim ) n−m The function im ∈n,m h t,x,n (ssm , ym ; •) vanishes outside [0, t] × Rd , thus we deduce from (2.13) that  ⎛ ⎞ 2  2      (iim )   (iim )   In−m ⎝ h (s s , y ; •) ⎠  = (n−m)!  h (s s , y ; •)   t,x,n m m   t,x,n m m   i m ∈n,m  i m ∈n,m  ⊗(n−m) 2 H  2    (iim )  ≤ tn−m (n − m)!   h t,x,n (ssm , ym ; •)  im ∈n,m  ⊗(n−m) H0 123 786 Stoch PDE: Anal Comp (2022) 10:757–827  ⎛ ⎞ 2    n−m  X ⎝ (iim )  = t  In−m h t,x,n (ssm , ym ; •)⎠ .  i m ∈n,m  2 Therefore, we get      m    n−m  (iim )   Ds  ( p − 1)t 2  X  . sm ,yym u(t, x) p ≤  In−m f t,x,n (ssm , ym ; •)  n≥m im ∈n,m  2 (3.9) This leads to  m    n−m  Ds  ≤ ( p − 1)t 2 Qm,n , sm ,yym u(t, x) p (3.10) n≥m with ⎡⎛ ⎞2 ⎤ ⎢ (iim ) ⎥ Qm,n : = E ⎣⎝ X In−m f t,x,n (ssm , ym ; •) ⎠ ⎦ i m ∈n,m     n X (iim ) 2 ≤ E In−m f t,x,n (ssm , ym ; •) . (3.11) m i m ∈n,m The product formula (2.14) and the decomposition (3.8) yield, with (i 0 , s0 , y0 ) = (0, t, x),   n Qm,n ≤ m ⎛ ⎞ $ m  (i j −i j−1 ) 2 E ⎝ In−i X IiXj −i j−1 −1 y j ; •) ⎠ 2 m f sm ,ym ,n−im f s j−1 ,y j−1 ,i j −i j−1 (s j , i m ∈n,m j=1   n  X 2 = I f sm ,ym ,n−im 2 n−i m m i m ∈n,m m  $  (i −i ) 2  X  × Ii j −i j−1 −1 f s j−1 ,y j−1 ,i j −i j−1 (s j , y j ; •)  , j j−1 (3.12) 2 j=1 where the last equality is obtained by using the independence among the random variables inside the expectation. It remains to estimate two typical terms:  X  2  I ( fr ,z, j )  ( j)  j 2 2 and  I X j−1 ( f t,x, j (r , z; •)  for 1 ≤ j ≤ n and t > r . (3.13) 2 123 Stoch PDE: Anal Comp (2022) 10:757–827 787 The first term in (3.13) can be estimated as follows. Using Fourier transform in space (see (2.29)), we have, with t0 = r ,   X  2    I ( fr ,z, j ) 2 = j! fr ,z, j H⊗ j =  fr ,z, j (tt j , •)2 dtt j j 2 0 0 [0,r ] j     = F fr ,z, j (tt j , ξ j )2 μ(dξξ j )dtt j r >t1 >···>t j >0 Rd j ⎛ ⎞   $ j−1  2 = ⎝ F G t −t (ξk+1 +· · ·+ξ j ) μ(dξk )⎠ dtt j . k k+1 r >t1 >···>t j >0 Rd j k=0 (3.14) By Lemma 2.6,  X Cj  I ( fr ,z, j ) 2 ≤ , (3.15) j 2 j!  2 )−1 μ(dξ ). where C = 2(t 2 + 1) Rd (1 + |ξ | Remark 3.1 By the arguments that lead to (3.9), we can also get, for any p ∈ [2, ∞),      n/2  X  u(t, x) ≤ 1 +  In ( f t,x,n ) ≤ 1 + ( p − 1)t  I ( f t,x,n ) p p n 2 n≥1 n≥1 and then the estimate (3.15) implies u(t, x) ∈ L p ( ). Moreover, sup u(s, y) p < +∞ for any t ∈ R+ . (3.16) (s,y)∈[0,t]×Rd This is done under the Dalang’s condition (1.2) only and the case p = 2 provides another proof of [3, Theorem 4.4] when d = 1, 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.    X (j) 2 3.2.1 Estimation of Ij−1 (ft,x,j (r, z; •)  when d = 1 2 ( j) When d = 1, G t (x) = 21 1{|x|t1 >···>t j−1 >r =C j−1 Tj, where we applied Lemma 2.3 for the inequality above10 and we denote   1/q 2q 2q T j := dtt j−1 G t−t1 (x − x1 )· · ·G t j−1 −r (x j−1 −z)dxx j−1 . t>t1 >···>t j−1 >r R2( j−1) (3.18) 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  t 2q 2q 2− 1 T2 = dt1 (G t−t1 ∗ G t1 −r )1/q (x −z) ≤ C G t−rq (x − z) ≤ C G 2t−r (x − z). (3.19) r 10 The function x ( j) j−1 → f t,x, j (tt j−1 , x j−1 ) = G t−t1 (x − x 1 )G t1 −t2 (x 1 − x 2 ) . . . G t j−1 −r (x j−1 − z) has support contained in {xx j−1 ∈ R2( j−1) ; |xi − x| < t − ti , for all i = 1, . . . , j − 1}. 123 Stoch PDE: Anal Comp (2022) 10:757–827 789 Case j ≥ 3. In this case, we use Minkowski inequality with respect to the norm in L 1/q ([t2 , t], dt1 ) in order to get    t q 2q 2q 1/q Tj ≤ G t−t1 ∗ G t1 −t2 (x − x2 )dt1 t>t2 >···>t j−1 >r R2( j−2) t2 1/q 2q 2q × G t2 −t3 (x2 − x3 ) · · · G t j−1 −r (x j−1 − z)d x2 · · · d x j−1 dt2 · · · dt j−1 . Applying Lemma 2.4 yields   1 T j ≤ Aq (t − t2 ) q −1 2q−1 G t−t2 (x − x2 ) t>t2 >···>t j−1 >r R2( j−2) 1/q 2q 2q × G t2 −t3 (x2 − x3 ) · · · G t j−1 −r (x j−1 − z)d x2 · · · d x j−1 dt2 · · · dt j−1 . (3.20) If j = 3, we have  t  1/q 1 T3 ≤ Aq (t − t2 ) q −1 2q−1 G t−t2 (x − 2q x2 )G t2 −r (x2 − z)d x2 dt2 . r R2 2q−1 2q Owing to (2.27), we can bound G t−t2 (x − x2 ) by (2π )(t − t2 )G t−t2 (x − x2 ), and then we apply again Lemma 2.4 and (2.27) to conclude that 1 3 −2 2− 1 T3 ≤ Aq2 (2π ) q (t − r ) q G t−rq (x − z) ≤ C G 2t−r (x − z). (3.21) For j ≥ 4, we continue with the estimate (3.20). We can first apply Minkowski inequality with respect to the norm L 1/q [t4 , t2 ], dt3 and then apply Lemma 2.4 to obtain  1 1 −1 −1 T j ≤ Aq2 dt2 dt4 · · · dt j−1 (t − t2 ) q (t2 − t4 ) q t>t2 >t4 >···>t j−1 >r  2q−1 G t−t2 (x − x2 ) R2( j−3) 1/q 2q−1 2q 2q × G t2 −t4 (x2 − x4 )G t4 −t5 (x4 − x5 ) · · · G t j−1 −r (x j−1 − z)d x2 d x4 · · · d x j−1 . (3.22) Note that 2q−1 2q−1 2q−1 2q−1 G t−t2 (x − x2 )G t2 −t4 (x2 − x4 ) ≤ 1{|x−x4 |≤t−t4 } G t−t2 (x − x2 )G t2 −t4 (x2 − x4 ). 123 790 Stoch PDE: Anal Comp (2022) 10:757–827 Then, by Cauchy-Schwarz inequality and (2.26), we can infer that  2q−1 2q−1 2q−1 2q−1 G t−t2 (x −x2 )G t2 −t4 (x2 − x4 )d x2 ≤ 1{|x−x4 |≤t−t4 } G t−t2 L 2 (R2 ) G t2 −t4 L 2 (R2 ) R2 = c1 (t − t2 )2−2q (t2 − t4 )2−2q 1{|x−x4 |≤t−t4 } , (2π )3−4q where c1 = 4−4q . Thus, substituting this estimate into (3.22), we end up with  3 3 1/q −3 −3 T j ≤ Aq2 c1 dt2 dt4 · · · dt j−1 (t − t2 ) q (t2 − t4 ) q t>t2 >t4 >···>t j−1 >r  1/q 2q 2q × 1{|x−x4 |≤t−t4 } G t4 −t5 (x4 − x5 ) · · · G t j−1 −r (x j−1 − z)d x4 · · · d x j−1 . R2( j−4) Focusing on the indicators, the right-hand side of this estimate can be bounded by  3 3 1/q −3 −3 Aq2 c1 1{|x−z|≤t−r } dt2 dt4 · · · dt j−1 (t − t2 ) q (t2 − t4 ) q t>t2 >t4 >···>t j−1 >r  1/q 2q 2q × G t4 −t5 (x4 − x5 ) · · · G t j−1 −r (x j−1 − z)d x4 · · · d x j−1 . R2( j−4) For j = 4, using (2.28), we have 6 1/q −6 T4 ≤ Aq2 c1 (t − r ) q 1{|x−z|≤t−r } ≤ C G 2t−r (x − z). (3.23) Now for j ≥ 5, we just integrate in each of the variables x4 , . . . , x j−1 (with this order) so that, thanks to (2.26), we end up with  1/q j−4 Tj ≤ Aq2 c1 c2 1{|x−z|≤t−r } dt2 dt4 · · · dt j−1 t>t2 >t4 >···>t j−1 >r 3 3 2 2 −3 −3 −2 −2 ×(t − t2 ) q (t2 − t4 ) q (t4 − t5 ) q · · · (t j−1 − r ) q with c2  2 (2π ) 1−2q 1/q j−4 (t − r ) j−3 j( 2 −2) = ≤ Aq2 c1 c2 (t − r + 1) q 1{|x−z|≤t−r } , 2 − 2q ( j − 3)! where we used the rough estimate a ν ≤ (b + 1)ν for 0 < a ≤ b and ν > 0. Thus, using (2.28) we obtain: C j−3 Tj ≤ G 2 (x − z) for any j ≥ 5. (3.24) ( j − 3)! t−r 123 Stoch PDE: Anal Comp (2022) 10:757–827 791 Hence, combining the estimates (3.19), (3.21), (3.23) and (3.24) and taking into (1) account that I0X ( f t,x,1 (r , z; •) = G r −s (z − y), we can write ⎧  2 ⎨C G 2t−r (x − z) for j = 1, 2, 3, 4  X ( j)   I j−1 ( f t,x, j (r , z; •)  ≤ Cj , 2 ⎩ G 2t−r (x − z) for j ≥ 5 ( j − 3)! where the constant C > 1 depends on (t, γ , q) and is increasing in t. For 1 ≤ j ≤ n, we obtain the following bound  2 Cj 3 2  X ( j)  I j−1 ( f t,x, j (r , z; •)  ≤ n G t−r (x − z). (3.25) 2 j! 3.3 Step 3: Proof of (1.11) Let us first consider the lower bound in (1.11) for d ∈ {1, 2}. For p ∈ [2, ∞), we deduce from the modified isometry (2.8) that  m     Ds  ≥  D m y u(t, x) ≥ m! f t,x,m (ssm , ym ). sm ,yym u(t, x) p sm ,ym 2 Now let us establish the upper bound in (1.11). By symmetry, we can assume t > s1 > · · · > sm > 0. First we consider the case where d = 2. Recall the definition of Qm,n from (3.11), and then plugging the estimates (3.15) and (3.25) into (3.12) yields, with (i 0 , s0 , y0 ) = (0, t, x),   $ n 3 C i j −i j−1 m n C n−im Qm,n ≤ × G2 (y j−1 − y j ) m (n − i m )! (i j − i j−1 )! s j−1 −s j i m ∈n,m j=1 ⎛ ⎞ n 3m ⎝ 1 ⎠ f t,x,m ≤ (2C) n 2 (ssm , ym ), i 1 !(i 2 − i 1 )! · · · (i m − i m−1 )!(n − i m )! i m ∈n,m n where we used the rough bound m ≤ 2n . The sum in the above display is equal to   1 n (m + 1)n = , n! a a1 , ..., am+1 n! 1 +...+am+1 =n ai ∈N,∀i by multinomial formula. That is, we can get  n C(m + 1) n 3m 2 Qm,n ≤ f t,x,m (ssm , ym ), n! which, together with the estimate (3.10), implies the upper bound in (1.11), when d = 2. 123 792 Stoch PDE: Anal Comp (2022) 10:757–827 The case d = 1 can be done in the same way by noticing that the bound in (3.17) j can be replaced by n Cj! G 2t−r (x − z) for 1 ≤ j ≤ n. Then, like the estimate for d = 2, we can get, for t > s1 > · · · > sm > 0,  n C(m + 1) n m 2 Qm,n ≤ f t,x,m (ssm , ym ), n! which together with the estimate (3.10) implies the upper bound in (1.11), when d = 1. This completes the proof of the estimate (1.11). Notice that the upper bound also shows the convergence in L p for any p ∈ [2, ∞) of the series (3.6), for any fixed sm ∈ [0, t]m and ym ∈ Rdm . 3.4 Step 4: Existence of a measurable version We claim that there is a random field Y such that Y (ssm , ym ) = Dsmm ,yym u(t, x) almost surely for almost all (ssm , ym ) ∈ [0, t]m × Rmd and the mapping (ω, sm , ym ) ∈ × [0, t]m × Rmd −→ Y (ω, sm , ym ) ∈ R is jointly measurable. This fact is rather standard and we will sketch the proof only in the case d = 2. From the explicit form of the kernels f t,x,n given in (1.8), it follows that the mapping (ssm , ym ) → f t,x,n (ssm , ym ; •) (3.26) is measurable from [0, t]m × R2m to L 2 ([0, t]n−m ; L 2q (R2(n−m) )). Because L 2 ([0, t]n−m ; L 2q (R2(n−m) ))is continuously embedded into H⊗(n−m) (see (2.13) and (2.25)), we deduce that the map (3.26) is measurable from [0, t]m × R2m into H⊗(n−m) . This implies that the mapping (ssm , ym ) → In−m ( f t,x,n (ssm , ym ; •)) (3.27) is measurable from [0, t]m × R2m to L 2 ( ). The upper bound in (1.11) implies that the mapping (3.27) belongs to the space L 2q ([0, t]m × R2m ; L 2 ( )) ⊂ L 2q ([0, t]m × R2m × ). From this, it follows that we can find a measurable modification of the process {In−m ( f t,x,n (ssm , ym ; •))(ω) : (ω, sm , ym ) ∈ × [0, t]m × R2m }. Finally, by standard arguments we deduce the existence of a measurable modification of the series (3.6). 123 Stoch PDE: Anal Comp (2022) 10:757–827 793 3.5 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, ∞). Then, it remains to show that the function Dsmm ,yym u(t, x) defined as the limit of the series (3.6) coincides  with the mth Malliavin  derivative of u(t, x). To do this, it suffices to p show that E D m u(t, x) H⊗m < ∞ for any m ≥ 1. By Fubini’ theorem and using the upper bound (1.11), we write   2/ p p E D m u(t, x) H⊗m   =  dssm dssm dyym dyym Dsmm ,yym u(t, x) Dsm ,yy  u(t, x) [0,t]2m ×R2md m m  $  m  γ0 (s j − s j )γ (y j − y j )  j=1  p/2      ≤ dssm dssm dyym dyym  Dsmm ,yym u(t, x) p  Dsm ,yy u(t, x) p  [0,t]2m ×R2md m m $ m γ0 (s j − s j )γ (y j − y j ) j=1  2   f t,x,m H⊗m < ∞. This shows u(t, x) ∈ D∞ and completes the proof of Theorem 1.3.   Remark 3.2 When d = 2, p = 2, m = 1 and for the cases (a), (b) in Hypothesis (H1), the upper bound in (1.11) can be proved in a much simpler way for almost all (r , z) ∈ [0, t] × R2 . Let vλ be the solution to the stochastic wave equation ⎧ 2 ⎨ ∂ vλ = vλ + λvλ Ẋ ⎪ ∂t 2 ⎩vλ (0, •) = 1, ∂vλ (0, •) = 0, ⎪ ∂t where ' λ > 0 and Ẋ is given as before. This solution has the chaos expansion vλ (t, x) = n X n≥0 λ In ( f t,x,n ) and its Malliavin derivative has the chaos expansion ⎛ ⎞ n X ⎝ ( j) Dr ,z vλ (t, x) = λn In−1 h t,x,n (r , z; •)⎠ ; n≥1 j=1 see (3.1) and (3.2). From this, we infer that for any (λ, t, x) ∈ (0, ∞)2 × R2 and for almost every (r , z) ∈ [0, t] × R2 ,    n 2  Dr ,z vλ (t, x)2 = 2n  ( j)  2 (n − 1)! λ  h t,x,n (r , z; •) ⊗(n−1) ≤ Cλ,t,γ G 2t−r (x − z), H0 n≥1 j=1 (3.28) 123 794 Stoch PDE: Anal Comp (2022) 10:757–827 where Cλ,t,γ > 0 is a constant depending on (λ, t, γ ) and is increasing in t. The inequality above is due to Theorem 1.3 of [35] for case (a), respectively Theorem 1.2 of [4] for case (b). Therefore,    n ( j) 2  Dr ,z u(t, x)2 = (n − 1)!  h t,x,n (r , z; •)H⊗(n−1) 2 n≥1 j=1  n ( j) 2 ≤ (n − 1)! tn−1  h t,x,n (r , z; •)H⊗(n−1) by (2.13). 0 n≥1 j=1 √  2 Thus, using (3.28) with λ = t , we get  Dr ,z u(t, x)2 ≤ Ct ,t,γ G 2t−r (x − z). 3.6 Consequences of Theorem 1.3 We will establish two estimates that will be useful in Sect. 5. Corollary 3.3 Let d = 1, 2. Then, for any finite T > 0,  2  sup sup E |Dr ,• u(t, x)|0 < ∞. (3.29) (t,x)∈[0,T ]×Rd r ∈[0,t] In particular, Dr ,• u(t, x)(ω) ∈ |P0 | for almost every (ω, r ) ∈ × [0, t], where |P0 | is defined in (2.2). Proof We work with a version of {Dr ,z u(t, x) : (r , z) ∈ [0, t] × R2 } that is jointly measurable. By Fubini’s theorem and Cauchy-Schwarz inequality, we have   2  E |Dr ,• u(t, x)|0 ≤ E |Dr ,z u(t, x)||Dr ,z  u(t, x)|γ (z − z  )dzdz   R 2d ≤ Dr ,z u(t, x) 2 Dr ,z  u(t, x) 2 γ (z − z  )dzdz  R2d  ≤C G t−r (x −z)G t−r (x −z  )γ (z −z  )dzdz  by Theorem 1.3 R2d   2 =C μ(dξ ) G t−r (ξ ) using Fourier transform Rd  μ(dξ ) ≤ 2C(t 2 ∨ 1) by (2.33), Rd 1 + |ξ | 2 where C is a constant depending on γ0 , γ , t and is increasing in t. The above (uniform) bound implies (3.29). Hence, Dr ,• u(t, x)(ω) ∈ |P0 | for almost all (ω, r ) ∈ × [0, t].   123 Stoch PDE: Anal Comp (2022) 10:757–827 795 The space |H⊗P0 | appearing in the next corollary is defined as the set of measurable functions h : R+ × R2d → R such that  |h(r , w, z)||h(r  , w  , z  )| R2+ ×R4d γ0 (r − r  )γ (w − w  )γ (z − z  )dwdw  dzdz  dr dr  < ∞. Then, |H ⊗ P0 | ⊂ H ⊗ P0 . Corollary 3.4 Let d = 1, 2. For almost all (ω, r ) ∈ × [0, t], D Dr ,• u(t, x)(ω) ∈ |H ⊗ P0 | and for any finite T > 0,     2 sup sup E  D Dr ,• u(t, x) < +∞. (3.30) (t,x)∈[0,T ]×Rd r ∈[0,t] H⊗P0 Proof Using Theorem 1.3, Cauchy-Schwarz inequality and the estimate (1.11), we can write    2  E  D Dr ,• u(t, x)  H⊗P0   =E |D(θ,w),(r 2 2 ,z) u(t, x)||D(θ  ,w ),(r ,z  ) u(t, x)| [0,t]2 R4d  × γ0 (θ − θ  )γ (w − w  )γ (z − z  )dwdw  dzdz  dθ dθ     2   2  ≤ D    (θ,w),(r ,z) u(t, x) 2 D(θ  ,w ),(r ,z  ) u(t, x) 2 [0,t]2 R4d × γ0 (θ − θ  )γ (w − w  )γ (z − z  )dwdw  dzdz  dθ dθ    ≤C f t,x,2 (r , z, θ, w) f t,x,2 (r , z  , θ  , w  ) [0,t]2 R4d γ0 (θ − θ )γ (w − w  )γ (z − z  )dwdw  dzdz  dθ dθ  .  As a consequence,    2 E  D Dr ,• u(t, x)  H⊗P0  ≤C f t,x,2 (r , z; •) H f t,x,2 (r , z  ; •) H γ (z − z  )dzdz  . R2d By the arguments used in the proof of Theorem 1.3, it follows that f t,x,2 (r , z; •) H ≤ C G t−r (x − z). 123 796 Stoch PDE: Anal Comp (2022) 10:757–827 Therefore,      2 E  D Dr ,• u(t, x) ≤C γ (z − z  )G t−r (x − z)G t−r (x − z  )dzdz  H⊗P0 R2d and the same argument as in the proof of Corollary 3.3 ends our proof.    2 Remark 3.5 Note that for any finite T > 0, E |D 2 u(t, x)|H⊗2 < ∞ for any (t, x) ∈ [0, T ] × Rd . 4 Gaussian fluctuation: Proof of Theorem 1.4 Recall that    FR (t) = u(t, x) − 1 d x BR  and σ R (t) = Var FR (t) . 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 FR (t)/σ R (t). Finally, we will prove the functional CLT by showing the convergence of the finite-dimensional distributions and the tightness. 4.1 Limiting covariance Proposition 4.1 Let u denote the solution to the hyperbolic Anderson model (1.1) and assume that the non-degeneracy condition (1.17) holds. Then, the following results hold true: (1) Suppose d ∈ {1, 2} and γ (Rd ) ∈ (0, ∞). Then, for any t, s ∈ (0, ∞),  −d     lim R E FR (t)FR (s) = ωd p! f t,x, p , f s,0, p H⊗ p d x, (4.1) R→∞ Rd p≥1 see also (1.18). In particular, σ R (t) ∼ R d/2 . (2) Suppose d ∈ {1, 2} and γ (x) = |x|−β for some β ∈ (0, 2 ∧ d). Then, for any t, s ∈ (0, ∞),     t s lim R β−2d E FR (t)FR (s) = κβ,d dr dr  γ0 (r − r  )(t − r )(s − r  ), R→∞ 0 0 (4.2)  where κβ,d = B12 d xd y|x − y|−β is introduced in (1.16). In particular, σ R (t) ∼ β R d− 2 . 123 Stoch PDE: Anal Comp (2022) 10:757–827 797 (3) Suppose d = 2 and γ (x1 , x2 ) = γ1 (x1 )γ2 (x2 ) satisfies one of the following conditions: ⎧ ⎪ −β ⎨(c1 ) γi (xi ) = |xi | i for some βi ∈ (0, 1), i = 1, 2; . (4.3) ⎪ ⎩ (c2 ) γ1 ∈ L 1 (R) and γ2 (x) = |x|−β for some β ∈ (0, 1) For any s, t ∈ (0, ∞), the following results hold true: (r1 ) In (c1 ), we have   lim R β1 −β2 −4 E FR (t)FR (s) R→∞  t  s = K β1 ,β2 dr dr  γ0 (r − r  )(t − r )(s − r  ), (4.4) 0 0 where K β1 ,β2 is defined in (1.22). (r2 ) In (c2 ), we have  t    s lim R β−3E FR (t)FR (s) = γ1 (R)Lβ dr dr  γ0 (r − r  )(t − r )(s − r  ), R→∞ 0 0 (4.5) where Lβ is defined in (1.24). 4.1.1 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 ∈ Rd , we have     E u(t, x)u(s, y) − 1 = p! f t,x, p , f s,y, p H⊗ p p≥1 1 =:  p (t, s; x − y), p! p≥1 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:    p (t, s; x − y) = ( p!)2 f t,x, p , f s,y, p H⊗ p ⎛ ⎞   $ p = p! dss p d s̃ p ⎝ γ0 (s j − s̃ j )⎠ σ ∈S p  p (t) [0,s] p j=1 123 798 Stoch PDE: Anal Comp (2022) 10:757–827 ⎛ ⎞  $ p dyy p d ỹ p ⎝ γ (y j − ỹ j )⎠ R2 pd j=1 ⎛ ⎞⎛ ⎞ $ p−1 $ p−1 ×⎝ G s j −s j+1 (y j − y j+1 )⎠ ⎝ G s̃σ ( j) −s̃σ ( j+1) ( yσ ( j) − yσ ( j+1) )⎠ (4.6) j=0 j=0 ⎛ ⎞   $ p = p! dss p d s̃ p ⎝ γ0 (s j − s̃ j )⎠ σ ∈S p  p (t) [0,s] p j=1 ⎛ ⎞  $ p × dξξ p ⎝ ϕ(ξ j )⎠ e−i(x−y)·(ξ1 +···+ξ p ) R pd j=1 ⎛ ⎞⎛ ⎞ $ p−1 $ p−1 × ⎝  G s j −s j+1 (ξ p + · · · + ξ j+1 ) ⎠⎝ s̃ G (ξσ ( p) +· · ·+ξσ ( j+1) )⎠ , σ ( j)−s̃σ ( j+1) j=0 j=0 (4.7) where  p (t) = {ss p : t > s1 > · · · > s p > 0}, (s0 , y0 , s̃σ (0) , ỹσ (0) ) = (t, x, s, y), t (ξ ) = sin(t|ξ |) is introduced in (2.29) and we have used again the convention G |ξ | G t (z) = 0 for t ≤ 0. Relation (4.6) shows that  p (t, s; x − y) is always nonnegative and equality (4.7) indicates that  p (t, s; x − y) indeed depends only on the difference x − y, so that we can write    p (t, s; z) = ( p!)2 f t,z, p , f s,0, p H⊗ p . (4.8) Note that  p (t, t; 0) coincides with α p (t) given in [3, Equation (4.11)]. Moreover, applying Lemma 2.5 with μ p (dξξ p ) = ϕ(ξ1 ) · · · ϕ(ξ p )dξ1 · · · dξ p and ( p−1 s j −s j+1 (ξ p + · · · + ξ j+1 )|, we get (with s ≤ t) g(s1 , ξ1 , . . . , s p , ξ p ) = j=0 |G   $ p−1 2 p    p (t, s; z) ≤ t p! dss p μ(dξξ p ) G s j −s j+1 (ξ p + · · · + ξ j+1 ) ,  p (t) R pd j=0 (4.9) t where we recall that t = −t γ0 (a)da and point out that the right-hand side of (4.9) is finite by applying Lemma 2.6 with z j = ξ j+1 + · · · + ξ p and z p = 0. Now we are ready to show (4.1). Proof of (4.1) Let us begin with      E FR (t)FR (s) E u(t, x)u(s, y) − 1 = d xd y Rd B R2 Rd 123 Stoch PDE: Anal Comp (2022) 10:757–827 799  ωd Leb B R ∩ B R (−z) =  p (t, s; z)dz, p! Rd Leb(B R ) p≥1 where ω1 = 2, ω2 = π and Leb(A) stands for the Lebesgue measure of A ⊂ Rd . We claim that  1  p (t, s; z)dz < ∞, (4.10) p! Rd p≥1 from which and the dominated convergence theorem we can deduce that    1 lim R −d E FR (t)FR (s) = ωd  p (t, s; z)dz. (4.11) R→∞ p! Rd p≥1 We remark that, by the monotone convergence theorem and the fact that  p (t, s; z) ≥ 0 for all z ∈ Rd , the claim (4.10) is equivalent to  1 ε  p (t, s; z)e− 2 |z| dz < ∞. 2 sup (4.12) ε>0 p≥1 p! Rd 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  1 (t, s; z)dz Rd   t  s   = dr d r̃ γ0 (r − r̃ ) dyd ỹG t−r (y − z)G s−r̃ ( ỹ)γ (y − ỹ) dz Rd 0 0 R2d  t s = γ (Rd ) γ0 (r − r̃ )(t − r )(s − r̃ )d r̃ dr ≤ γ (Rd )t 3 t , (4.13) 0 0  where Rd 1 (t, s; z)dz > 0 due to the non-degeneracy assumption (1.17) on γ0 . 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  ε −d  p (t, s; z)e− 2 |z| dz 2 T p,ε := (2π ) Rd   $ p  = p! dss p d s̃ p γ0 (s j − s̃ j ) dξξ p σ ∈S p  p (t) [0,s] p j=1 R pd $ p−1 × pε (ξ1 + · · · + ξ p ) s j −s j+1 (ξ p + · · · + ξ j+1 ) ϕ(ξ j+1 )G j=0 123 800 Stoch PDE: Anal Comp (2022) 10:757–827 s̃ −s̃ G (ξ + · · · + ξσ ( j+1) ) σ ( j) σ ( j+1) σ ( p) ⎛ ⎞ ⎛ ⎞   $p p dξξ p ⎝ ϕ(ξ j )⎠ pε ⎝ ξj⎠ p ≤ t p! dss p  p (t) R pd j=1 j=1 $ p−1  2  G s j −s j+1 (ξ p + · · · + ξ j+1 ) , (4.14) j=0 where pε (ξ ) = (2π ε)−d/2 e−|ξ | /(2ε) for ξ ∈ Rd and we applied Lemma 2.5 with 2 μ p (dξξ p ) = ϕ(ξ1 ) · · · ϕ(ξ p ) pε (ξ1 + · · · + ξ p )dξ1 · · · dξ p . Next, we make the change of variables η j = ξ p + · · · + ξ j with the conventionη p+1 = 0, and the bound (4.14) becomes ⎛ ⎞   $ p $ p−1 2   dηη p ⎝ ϕ(η j − η j+1 )⎠ pε (η1 ) p T p,ε ≤ t p! dss p G s j −s j+1 (η j+1 )  p (t) R pd j=1 j=0    p ≤ t p! ϕ ∞t 2 dη1 pε (η1 ) dss p dη2 · · · dη p Rd  p (t) R pd−d ⎛ ⎞ $ p  2 ⎝     ϕ(η j − η j+1 )⎠ G (η s1 −s2 2 ) G (η s2 −s3 3 ) · · · G (η ) s p−1 −s p p  j=2  p = t p! ϕ ∞t 2 dη1 pε (η1 )Q p−1 , (4.15) Rd where we used |Gt−s1 (ξ )| ≤ t, and ϕ(η1 − η2 ) ≤ ϕ ∞ (which is finite because γ (R ) < ∞) to obtain (4.15), and d   $ p  2 Q p−1 := dss p ϕ(η j − η j+1 )G s j−1 −s j (η j ) dη j . (4.16)  p (t) R pd−d j=2 Observe that Q p−1 does not depend on η1 , thus for any p ≥ 2 p T p,ε ≤ t p! ϕ ∞t 2 Q p−1 . (4.17) By Lemma 2.6, we have for any p ≥ 2    p−1 μ(dξ ) tp Cp Q p−1 ≤ 2(t 2 ∨ 1) ≤ . Rd 1 + |ξ |2 p! p! 123 Stoch PDE: Anal Comp (2022) 10:757–827 801 Now, plugging the above estimate and (4.17) into (4.12), and using (4.13) for p = 1, we have  p 1 − 2ε |z|2 t C p sup  p (t, s; z)e dz ≤ γ (R )t t + (2π ) ϕ d 3 d ∞t 2 < +∞. ε>0 p≥1 p! Rd p! p≥2 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.   4.1.2 Proof of part (2) in Proposition 4.1 In this case, the corresponding spectral density is given by ϕ(ξ ) = cd,β |ξ |β−d , for some constant cd,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 FR (t): FR (t) = J p,R (t), p≥1   where J p,R (t) := I p |x|≤R f t,x, p d x is the projection of FR (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 σ R2 (t) = Var FR (t) = Var J p,R (t) . p≥1 Let us first consider the variance of J1,R (t). With B R = {x ∈ Rd : |x| ≤ R}, we can write  Var J1,R (t) = d xd x  G t−• (x − ∗), G t−• (x  − ∗)H B R2     = d xd x   dsds γ0 (s − s )  dξ ϕ(ξ )e−i(x−x )·ξ G t−s  (ξ ). t−s (ξ )G B R2 [0,t]2 Rd (4.18) Then, making the change of variables (x, x  , ξ ) → (Rx, Rx  , ξ/R), we get   Var J1,R (t) = R 2d−β dsds  γ0 (s − s  ) d xd x  [0,t]2 B12   t−s (ξ/R)G dξ ϕ(ξ )e−i(x−x )·ξ G t−s  (ξ/R). Rd 123 802 Stoch PDE: Anal Comp (2022) 10:757–827 t (ξ/R) is uniformly bounded and convergent to t as R → ∞; observe Note that G also that    2 R (ξ ) := d xd x  e−i(x−x )·ξ = F1 B R  (ξ ) ∈ [0, ∞). (4.19) B R2 Thus  we deduce from the dominated convergence theorem that, with κβ,d := B 2 d xd x  |x − x  |−β , 1   Var J1,R (t) R→∞  2 −−−→    dsds γ0 (s − s )(t − s)(t − s ) dξ ϕ(ξ )F1 B1  (ξ ) R 2d−β [0,t]2 R d  = κβ,d dsds  γ0 (s − s  )ss  . (4.20) [0,t]2 In the same way, we can get    t  s E J1,R (t)J1,R (s) R→∞ −−−→κβ,d dr dr  γ0 (r − r  )(t − r )(s − r  ) (4.21) R 2d−β 0 0 In what follows, we will show that as R → ∞, Var J p,R (t) = o(R 2d−β ). (4.22) p≥2 In view of the orthogonality again, the above claim (4.22) and the results (4.20)–(4.21) imply that the first chaos of FR (t) is dominant and    t  s E FR (t)FR (s) R→∞ −−−→ κβ,d dr dr  γ0 (r − r  )(t − r )(s − r  ), R 2d−β 0 0 which gives us the desired limiting covariance structure. Moreover, we obtain imme-  β  diately that the process R −d+ 2 FR (t) : t ∈ R+ converges in finite-dimensional distributions to the centered Gaussian process Gβ , 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 s0 = s̃σ (0) = t and B R = {x : |x| ≤ R})        Var J p,R (t) = p! d xd x f t,x, p , f t,x  , p H⊗ p = p! d xd x  f t,x, p , f t,x  , p H⊗ p B R2 B R2 ⎛ ⎞   $ p  $ p = p cd,β d xd x  dss p d s̃ p γ0 (sk − s̃k ) ⎝ dξ j |ξ j |β−d ⎠ σ ∈S p B R2 [0,t]2 p k=1 R pd j=1 123 Stoch PDE: Anal Comp (2022) 10:757–827 803  $ p−1 × e−i(x−x )·(ξ p +···+ξ1 ) G s̃ −s̃ s j −s j+1 (ξ p + · · · + ξ j+1 )G σ ( j) σ ( j+1) j=0 × (ξσ ( p) + · · · + ξσ ( j+1) ), where we recall the convention that G t (z) = 0 for t ≤ 0. Then, recalling definition (4.19) of R (ξ ), we can apply Lemma 2.5 with μ(dξξ p ) = ϕ(ξ1 ) · · · ϕ(ξ p ) R (ξ1 + · · · + ξ p )dξ1 · · · dξ p to get Var J p,R (t) bounded by ⎛ ⎞   $ p p cd,β t p dss p ⎝ dξ j |ξ j |β−d ⎠ R (ξ1 + · · · + ξp)  p (t) R pd j=1 $ p−1  2  × G s j −s j+1 (ξ p + · · · + ξ j+1 ) . (4.23) j=0 Making change of variables (i) η j = ξ p + · · · + ξ j with η p+1 = 0 (ii) (x, x  , η1 ) → (Rx, Rx  , η1 R −1 ), we obtain ⎛ ⎞   $ p Var J p,R (t) ≤ cd,β t p p dss p ⎝ dη j |η j − η j+1 |β−d ⎠  p (t) R pd j=1 ) * p−1 $  2 ×  d xd x  e−i(x−x )·η1 s j −s j+1 (η j+1 ) G B R2 j=0   dη1 |η1 − η2 R|β−d p p = cd,β t R 2d−β dss p  p (t) R pd ⎛ ⎞ $ p ⎝ dη j |η j − η j+1 |β−d ⎠ j=2 ) *  2 p−1  2  −i(x−x  )·η1   $   × d xd x e (η  t−s1 1 G /R)  G s j −s j+1 (η j+1 ) B12 j=1 ⎛ ⎞   $ p p−1 ≤ t 2 cd,β t R 2d−β p dss p ⎝ dη j |η j − η j+1 |β−d ⎠  p (t) R pd−d j=2 ) * p−1 $  2 ×  d xd x |x − x |  −β −i(x−x  )·η2 R e s j −s j+1 (η j+1 ) , G B12 j=1 123 804 Stoch PDE: Anal Comp (2022) 10:757–827 t | ≤ t and the following Fourier transform: where in the last inequality we used |G    d xd x  cd,β dη1 |η1 − η2 R|β−d e−i(x−x )·η1 B12 Rd   2 = cd,β dη1 |η1 − η2 R|β−d F1 B1  (η1 ) Rd   = d xd x  |x − x  |−β e−i(x−x )·η2 R . B12   Note that the integral B 2 d xd x  |x − x  |−β e−i(x−x )·η2 R is uniformly bounded by κβ,d 1 and it converges to zero as R → ∞ for η2 = 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 R β−2d Var J p,R (t) ≤ t 2 κβ,d t Q p−1 , p which is summable over p ≥ 2 by the arguments in the previous section. Hence by the dominated convergence theorem, we get R→∞ R β−2d Var J p,R (t) −−−→ 0. p≥2 This proves the claim (4.22).   4.1.3 Proof of part (3) in Proposition 4.1 Recall the two cases from (4.3):  (c1 ) γi (xi ) = |xi |−βi for some βi ∈ (0, 1), i = 1, 2, . (c2 ) γ1 ∈ L 1 (R) and γ2 (x) = |x|−β for some β ∈ (0, 1). In (c1 ), the spectral density is ϕ(ξ1 , ξ2 ) = c1,β1 c1,β2 |ξ1 |β1 −1 |ξ2 |β2 −1 for (ξ1 , ξ2 ) ∈ R2 ,where c1,β is a constant that only depends on β. Now, using the notation from Sect. 4.1.2, we write    Var J1,R (t) = d xd x dsds  γ0 (s − s  ) B R2 [0,t]2   × dξ ϕ(ξ )e−i(x−x )·ξ G t−s  (ξ ) see (4.18) t−s (ξ )G Rd   4−β1 −β2   =R dsds γ0 (s − s ) dξ ϕ(ξ1 , ξ2 ) [0,t]2 Rd   × t−s (ξ/R)G d xd x  e−i(x−x )·ξ G t−s  (ξ/R), B12 123 Stoch PDE: Anal Comp (2022) 10:757–827 805 where the last equality is obtained by the change of variables (x, x  , ξ1 , ξ2 ) to (Rx, Rx  , ξ1 /R, ξ2 /R). Thus, by the exactly same arguments that lead to (4.20), we can get  Var J1,R (t) R→∞ −−−→ K β1 ,β2 dsds  γ0 (s − s  )ss  , R 4−β1 −β2 [0,t]2 with K β1 ,β2 introduced in (1.22). Similar to (4.21), we also have    t  s E J1,R (t)J1,R (s) R→∞ −−−→ K β1 ,β2 dr dr  γ0 (r − r  )(t − r )(s − r  ). (4.24) R 4−β1 −β2 0 0 To obtain the result (r1 ), it remains to show Var J p,R (t) = o R 4−β1 −β2 . (4.25) p≥2 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 (c2 ) where γ1 ∈ L 1 (R) and γ2 (x) = |x|−β for some fixed β ∈ (0, 1). In this case, the corresponding spectral density is ϕ(ξ1 , ξ2 ) = ϕ1 (ξ1 )ϕ2 (ξ2 ), where  (i) γ1 = Fϕ1 and ϕ1 is uniformly continuous and bounded, (4.26) (ii) ϕ2 (ξ2 ) = c1,β |ξ2 |β−1 for some constant c1,β that only depends on β. Let us begin with (4.18) and make the usual change of variables (x, x  , ξ ) → (Rx, Rx  , ξ/R) to obtain   Var J1,R (t) = d xd x  dsds  γ0 (s − s  ) B R2 [0,t]2   dξ ϕ1 (ξ1 )ϕ2 (ξ2 )e−i(x−x )·ξ G t−s (ξ )Gt−s  (ξ ) R2   = R 3−β dsds  γ0 (s − s  ) dξ ϕ1 (ξ1 /R)ϕ2 (ξ2 ) [0,t]2 R2 ) *  −i(x−x  )·ξ  t−s  (ξ/R) d xd x e G t−s (ξ/R)G B12  = R 3−β dsds  γ0 (s − s  ) [0,t]2   2 dξ ϕ1 (ξ1 /R)ϕ2 (ξ2 )F1 B1  (ξ )Gt−s (ξ/R)G t−s  (ξ/R). R2 123 806 Stoch PDE: Anal Comp (2022) 10:757–827 t−s and G Recall that ϕ1 , G t−s  are uniformly bounded and continuous. Note that, applying Plancherel’s theorem and the Parseval-type relation (2.3), we have    2  2 dξ ϕ2 (ξ2 )F1 B1  (ξ ) = 2π d x1 dξ2 ϕ2 (ξ2 ) F1 B1 (x1 , •)(ξ2 ) R2 R2  = 2π d x1 d x2 d x3 1{x 2+x 2≤1} 1{x 2 +x 2 ≤1} |x2 −x3 |−β < ∞. 1 2 1 3 R3 Therefore, by the dominated convergence theorem and the fact that ϕ1 (0) = 2π γ1 (R), 1 we get  Var J1,R (t) R→∞ −−−→ ϕ1 (0) dsds  γ0 (s − s  )(t − s)(t − s  ) R 3−β [0,t]2   2 × dξ ϕ2 (ξ2 )F1 B1  (ξ ) R 2  = γ1 (R)Lβ dsds  γ0 (s − s  )ss  , [0,t]2 where Lβ is defined in (1.24). In the same way, we get for s, t ∈ (0, ∞),    t  s E J1,R (t)J1,R (s) R→∞ −−−→ γ 1 (R)Lβ dr dr  γ0 (r − r  )(t − r )(s − r  ). R 3−β 0 0 (4.27) Now we claim that the other chaoses are negligible, that is, as R → ∞, Var J p,R (t) = o(R 3−β ). (4.28) p≥2 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). Proof of Claim (4.28) By the same arguments that lead to the estimate (4.23), we can obtain   p Var J p,R (t)) ≤ t dss p dξξ p ϕ p (ξξ p )  p (t) R2 p $ p−1  2  × G s j −s j+1 (ξ p + · · · + ξ j+1 ) with s0 = t, j=0 (1) (2) where ϕ p (ξξ p ) = ϕ(ξ1 ) · · · ϕ(ξ p ) R (ξ1 + · · · + ξ p ) for ξ j = (ξ j , ξ j ) ∈ R2 , j = 1, . . . , p and R is defined in (4.19). Recall that in the current case, ϕ(ξ ) = 123 Stoch PDE: Anal Comp (2022) 10:757–827 807 ϕ1 (ξ (1) )ϕ2 (ξ (2) ) for ξ = (ξ (1) , ξ (2) ) ∈ R2 and ϕ1 , ϕ2 satisfy the conditions in (4.26). Then, the following change of variables η j = ξ j + ξ j+1 + · · · + ξ p with η p+1 = 0 yields   $ p−1  2 p   Var J p,R (t)) ≤ t dss p dηη p R (η1 ) ϕ(η j+1 −η j+2 )G s j −s j+1 (η j+1 ) .  p (t) R2 p j=0 In view of (4.19), we have R (η1 /R) = R 4 1 (η1 ). Thus, by changing η1 to η1 /R, we write    2   dηη p 1 (η1 )ϕ(η1 R −1 − η2 )G p Var J p,R (t)) ≤ R 2 t dss p (η t−s1 1 /R)   p (t) R2 p $ p−1  2   × ϕ(η j+1 − η j+2 )G s j −s j+1 (η j+1 ) j=1   p ≤ R 3−β t ϕ1 ∞t 2 dss p dη2 ...dη p  p (t) R2 p−2    β−1 dη1 1 (η1 )c1,β η1(2) − η2(2) R  R2 $ p−1  2   × ϕ(η j+1 − η j+2 )G s j −s j+1 (η j+1 ) , j=1 where we used |G t−s1 (η1 /R)|2 ≤ t 2 . Observe that with η = (η(1) , η(2) ), we deduce  2 from the fact 1 (η) = F1 B1  (η(1) , η(2) ) that    2 dη 1 (η)ϕ2 (η(2) − x R) = dη(1) dη(2) F1 B1  (η(1) , η(2) + x R)ϕ2 (η(2) ) R2 R2  = 2π 1{x 2 +x 2 ≤1} 1{x 2 +x 2 ≤1} 1 2 1 3 R3 −i(x2 −x3 )x R −β e |x2 − x3 | d x1 d x2 d x3 , by inverting the Fourier transform. The above quantity is uniformly bounded by 2π Lβ with Lβ given in (1.24) and convergent to zero as R → ∞ for every x = 0 in view of the Riemann-Lebesgue lemma. Thus, R β−3 Var J p,R (t)) is uniformly bounded p by 2π Lβ t ϕ1 ∞ t 2 Q p−1 , with Q p−1 given by (4.16) and it converges to zero as R → ∞. Since Q p ≤ C p / p!, we have p t Q p−1 < ∞, p≥2 123 808 Stoch PDE: Anal Comp (2022) 10:757–827 and the dominated convergence theorem implies (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 FR (t)/σ R (t) is well-defined for large R. 4.2 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). 4.2.1 Part (1) We will first show the estimate dTV FR (t)/σ R (t), Z  R −d/2 , (4.29) where Z ∼ N (0, 1). By Proposition 1.8 applied to σ R (t) FR (t), 1 we have 4 dTV FR (t)/σ R (t), Z ≤ AR , (4.30) σ R2 (t) where  AR = dr dr  dsds  dθ dθ  dzdz  dydy  dwdw  R6+ ×R6d γ0 (θ − θ )γ0 (s − s  )γ0 (r − r  )γ (z − z  )γ (w − w  )  × γ (y − y  ) Dr ,z Dθ,w FR (t) 4 Ds,y Dθ  ,w FR (t) 4 Dr  ,z  FR (t) 4 Ds  ,y  FR (t) 4 . Recall from Sect. 4.1.1 that σ R2 (t) ∼ R d . Therefore, in order to show (4.29) it suffices to prove the estimate AR  Rd . (4.31) Using Minkowski’s inequality, we can write         Dr ,z Dθ,w FR (t) 4 =  Dr ,z Dθ,w u(t, x)d x  ≤  Dr ,z Dθ,w u(t, x) d x.  4 BR 4 BR Then, it follows from our fundamental estimates in Theorem 1.3 that  Dr ,z Dθ,w FR (t) 4  f t,x,2 (r , z, θ, w)d x, (4.32) BR 123 Stoch PDE: Anal Comp (2022) 10:757–827 809 with f t,x,2 (r , z, θ, w) 1  = G t−r (x − z)G r −θ (z − w)1{r >θ} + G t−θ (x − w)G θ−r (z − w)1{r <θ} ; 2 and, in the same way, we have  Dr ,z FR (t) 4  G t−r (x − z)d x, (4.33) BR where the implicit constants in (4.32)–(4.33) do not depend on (R, r , z, θ, w) and are increasing in t. Now, plugging (4.32)–(4.33) into the expression of A R , we get  AR  dr dr  dsds  dθ dθ  dzdz  dydy  dwdw  [0,t]6 ×R6d × γ0 (r − r  )γ0 (s − s  )γ0 (θ − θ  )γ (z − z  )γ (w − w  )  × γ (y − y  ) f t,x1 ,2 (r , z, θ, w) f t,x2 ,2 (s, y, θ  , w  )G t−r  (x3 − z  ) B R4 4  ×G t−s  (x4 − y )dxx4 =: A R, j . j=1 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 :  1 A R,1 = dr dr  dsds  dθ dθ  dzdz  dydy  dwdw  4 [0,t]6 ×R6d × γ0 (r − r  )γ0 (s − s  )γ0 (θ − θ  ) × γ (w − w )γ (y − y  )γ (z − z  )G r −θ (z − w)G s−θ  (y − w  )  × dxx4 G t−r (x1 − z)G t−s (x2 − y)G t−r  (x3 − z  )G t−s  (x4 − y  ). B R4 (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 d x1 , d x2 , d x4 , dy  , dy, dw  , dw, dz, dz  , d x3 one by one, by taking into account the following facts:   sup G t−r (x − z)d x ≤ t and sup γ (y − y  )dy = γ L 1 (Rd ) . z∈Rd BR y  ∈Rd Rd 123 810 Stoch PDE: Anal Comp (2022) 10:757–827 To get the bound for A R,2 , it suffices to perform the integration with respect to d x1 , d x3 , dz  , dz, d x2 , dw, dw  , dy, dy  , d x4 . To get the bound for A R,3 , it suffices to perform the integration with respect to d x4 , dy  , d x2 , dy, dw  , d x1 , dw, dz, dz  , d x3 one by one. To get the bound for A R,4 , it suffices to perform the integration with respect to d x1 , d x3 , d x2 , dz  , dz, dw, dw , dy, dy  , d x4 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 t1 , . . . , tm ∈ (0, ∞). Put F R = FR (t1 ), . . . , FR (tm ) . 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 = (Ci j : 1 ≤ i, j ≤ m), with    Ci j = ωd p! f ti ,x, p , f t j ,0, p H⊗ p d x. p≥1 Rd Since FR (t) = δ(−DL −1 FR (t)), according to [25, Theorem 6.1.2],11 for any twice differentiable function h : Rm → R with bounded second partial derivatives,           E h(R −d/2 F R ) − h(Z)  ≤ E h(R −d/2 F R ) − h(Z R )  + E h(Z) − h(Z R )  1 2 m m 2       ∞3  ≤ d h Var D FR (ti ), −DL −1 FR (t j ) H + E h(Z) − h(Z R ) , 2R i, j=1 (4.35)  2  with Z R ∼ N 0, C R , Z ∼ N 0, C and h  ∞ = sup  ∂ x∂i ∂ x j h(x) : x ∈ Rm , i, j =  1, . . . , m . 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 = FR (ti ) and G = FR (t j ) and using arguments analogous to those employed to derive (4.31), we obtain    Var D FR (ti ), −DL −1 FR (t j ) H  R d .   Thus, the first term in (4.35) is O(R −d/2 ), implying that E h(R −d/2 F R ) − h(Z) converges to zero as R → ∞. This shows the convergence of the finite-dimensional distributions of {R −d/2 FR (t) : t ∈ R+ } to those of the centered Gaussian process G, whose covariance structure is given by      E G(t)G(s) = ωd p! f t,x, p , f s,0, p H⊗ p d x, for s, t ∈ [0, ∞). p≥1 Rd This concludes the proof of part (1) in Theorem 1.4.  11 Note that there is a typo in Theorem 6.1.2 of [25]: In (6.1.3) of [25], one has d/2 instead of 1/2. 123 Stoch PDE: Anal Comp (2022) 10:757–827 811 4.2.2 Proofs in parts (2) and (3) In part (2), in view of the dominance of the first chaos, we have already obtained in  −d+ β Sect. 4.1.2 2 FR (t) :  that the finite-dimensional distributions of the process R t ∈ R+ converge to those of a centered Gaussian process {Gβ (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: ⎧ ⎪ −β/2 ⎨R in part (2), 1 − (β +β ) dTV FR (t)/σ R (t), Z  R 2 1 2 in part (3) case (a  ), (4.36) ⎪ ⎩ −(1+β)/2 R in part (3) case (b ), where Z ∼ N (0, 1). Taking into account (4.30) and the variance estimates in Sects. 4.1.2 and 4.1.3, in order to get (4.36) it suffices to show that, for j ∈ {1, 2, 3, 4} and for R ≥ t, ⎧ ⎪ ⎨R 4d−3β in part (2), A R, j  R 8−3(β1 +β2 ) in case (a  ) of part (3), (4.37) ⎪ ⎩ 5−3β R in case (b ) of part (3). 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  A R,1 = dr dr  dsds  dθ dθ  γ0 (r − r  )γ0 (s − s  )γ0 (θ − θ  )S1,R . [0,t]6 with   S1,R : = R6d dzdz  dydy  dwdw  γ (w − w  )γ (y − y  )γ (z − z  ) x4 G t−r (x1 − z) B R4 dx ×G r −θ (z − w)G t−s (x2 − y)G s−θ  (y − w  )G   t−r  (x 3 − z )G t−s  (x 4 − y ). From now on, when d = 2, we write (w, w , y, y  , z, z  ) = (w1 , w2 , w1 , w2 , y1 , y2 , y1 , y2 , z 1 , z 2 , z 1 , z 2 ) and then dy = dy1 dy2 ; note also that x1 , . . . , x4 denote the dummy variables in Rd . By making the following change of variables (z, z  , y, y  , w, w  , x1 , x2 , x3 , x4 ) → R(z, z  , y, y  , w, w  , x1 , x2 , x3 , x4 ) (4.38) 123 812 Stoch PDE: Anal Comp (2022) 10:757–827 and using the scaling property G t (Rz) = R 1−d G t R −1 (z) for d ∈ {1, 2}, we get  S1,R = R 6+4d dzdz  dydy  dwdw  γ (Rw − Rw  )γ (Ry − Ry  )γ (Rz − Rz  ) [−2,2]6d  dxx4 × G t−r (x1 − z)G r −θ (z − w)G t−s (x2 − y)G s−θ  (y − w  ) R R R B14 R   G t−r  (x3 − z )G t−s  (x4 − y ). (4.39) R R Note that we have replaced the integral domain R6d by [−2, 2]6d in (4.39) without changing the value of S1,R , because, for example, x1 ∈ B1 and |x1 − z| ≤ (t − r )/R implies |z| ≤ 1 + t R −1 ≤ 2 while |z − w| ≤ (r − θ )/R and |x1 − z| ≤ (t − r )/R imply |w| ≤ (t − θ )R −1 + 1 ≤ 2. In view of the expression of γ in part (2) and part (3), we write, for z ∈ Rd (z = (z 1 , z 2 ) ∈ R2 when d = 2), ⎧ ⎪ −β ⎨ R γ (z) in part (2), γ (Rz) = R −β1 −β2 γ (z) in case (a  ) of part (3), ⎪ ⎩ −β R γ1 (Rz 1 )γ2 (z 2 ) in case (b ) of part (3), and it is easy to see that  sup γ (Rz − Rz  )dz z  ∈[−2,2]d [−2,2]d ⎧  ⎪ ⎪ −β γ (z)dz < ∞ ⎪ ⎪ R in part (2), ⎪ ⎪ [−4,4]d ⎪ ⎪ ⎪ ⎪ ⎪ ⎨  −β1 −β2 ≤ R γ (z)dz < ∞ in case (a  ) of part (3), ⎪ ⎪ [−4,4]d ⎪ ⎪ ⎪ ⎪ ⎪ ⎪  4 ⎪ ⎪ ⎪ ⎩ R −β−1 γ1 (R) γ2 (s)ds < ∞ in case (b ) of part (3). −4 To ease the notation, we just rewrite the above estimates as  sup γ (Rz − Rz  )dz  R −α (4.40) z  ∈[−2,2]d [−2,2]d with α = β in part (2), α = β1 + β2 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 d x1 , d x2 , d x4 , dy  , dy, dw  , dw, dz, dz  , d x3 successively. More precisely, perform- 123 Stoch PDE: Anal Comp (2022) 10:757–827 813 ing the integration with respect to d x1 , d x2 , d x4 and using the fact  sup G s/R (z − z  )dz = t/R (4.41) (s,z  )∈[0,t]×Rd Rd gives us  S1,R ≤ R 3+4d 3 t dzdz  dydy  dwdw  γ (Rw − Rw  ) [−2,2]6d    γ (Ry − Ry )γ (Rz − Rz ) d x3 B1 × G r −θ (z − w)G s−θ  (y − w  )G t−r  (x3 − z  ) R  R R  3+4d −α R R dzdz dydwdw  γ (Rw − Rw  )γ (Rz − Rz  )  d x3 [−2,2]5d B1 × G r −θ (z − w)G s−θ  (y − w  )G t−r  (x3 − z  ) R R R by integrating out dy  and using (4.40)    R 2+4d−α dzdz  dwdw  γ (Rw − Rw  )γ (Rz − Rz  ) d x3 [−2,2]4d B1 × G r −θ (z − w)G t−r  (x3 − z  ) by integrating out dy and using (4.41) R  R    R 2+4d−2α dzdz dwγ (Rz − Rz ) d x3G r −θ (z − w)G t−r  (x3 − z  ) [−2,2]3d B1 R R by integrating out dw and using (4.40); then, using (4.41) to integrate out dw     R 1+4d−2α dzdz γ (Rz − Rz ) d x3 G t−r  (x3 − z  )  R 4d−3α [−2,2]2d B1 R where the last inequality is obtained by integrating out dz, dz  , d x3 one by one and using (4.40) and (4.41). The bound ⎧ ⎪ ⎨R 4d−3β in part (2), S1,R  R 4d−3α = R 8−3β1 −3β2 in cae (a  ) of part (3), ⎪ ⎩ 5−3β R in cae (b ) of part (3) is uniform over (r , r  , s, s  , θ, θ  ) ∈ [0, t]6 , and hence we obtain (4.37) for j = 1. For the other terms A R,2 , A R,3 and A R,4 , the arguments are the same: We first go through the same change of variables (4.38) to obtain terms S j,R similar to S1,R in (4.39), and then use the facts (4.40) and (4.41) to perform one-by-one integration with respect to 123 814 Stoch PDE: Anal Comp (2022) 10:757–827 the variables ⎧ ⎪    ⎨d x1 , d x3 , dz , dz, d x2 , dw, dw , dy, dy , d x4 for estimating A R,2 d x4 , dy  , d x2 , dy, dw  , d x1 , dw, dz, dz  , d x3 for estimating A R,3 . ⎪ ⎩ d x1 , d x3 , d x2 , dz  , dz, dw, dw , dy, dy  , d x4 for estimating A R,4 This concludes the proof of (4.37) and hence completes the proof of (4.36).   4.3 Tightness This section is devoted to establishing the tightness in Theorem 1.4. This, together with the results in Sects. 4.1 and 4.2 will conclude the proof of Theorem 1.4. To get the tightness, we appeal to the criterion of Kolmogorov-Chentsov (see e.g. [17, Corollary 16.9]). Put ⎧ ⎪ ⎪ R d/2 in part (1) of Theorem 1.4 ⎪ ⎪ ⎨ R d− β2 in part (2) of Theorem 1.4 σR = (4.42) ⎪ ⎪ 1 R 2− 2 (β1 +β2 ) in part .(3) − (a  ) of Theorem 1.4 ⎪ ⎪ ⎩ R (3−β)/2 in part (3) − (b ) of Theorem 1.4 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:    FR (t) − FR (s)  (t − s)σ R , (4.43) k where the implicit constant  does not depend on R,  s or t. This moment estimate (4.43) ensures the tightness of σ R−1 FR (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 FR (t) − FR (s), we begin with the chaos expansion   FR (t) − FR (s) = In d x[ f t,x,n − f s,x,n ] = In gn,R , n≥1 BR n≥1 where s, t are fixed, so we leave them out of the subscript of the kernel gn,R and   n−1 $ gn,R (ssn , yn ) = ϕt,R (s1 , y1 ) − ϕs,R (s1 , y1 ) G s j −s j+1 (y j − y j+1 ) (4.44) j=1 (  with 0j=1 = 1 and ϕt,R (r , y) := B R G t−r (x − y)d x. The rest of this section is then devoted to proving (4.43). 123 Stoch PDE: Anal Comp (2022) 10:757–827 815 Proof of (4.43) By the triangle inequality and using the moment estimate (2.15), we get, for any k ∈ [2, ∞),      FR (t) − FR (s) ≤ (k − 1)n/2  In gn,R 2 . k n≥1 Note that the kernel gn,R = 0 outside [0, t]n × Rdn . Then, using (2.8) and (2.13), we can write    n/2  1/2  FR (t) − FR (s) ≤ t (k − 1) n! gn,R 2H⊗n , k 0 n≥1 where gn,R is the canonical symmetrization of gn,R : 1   gn,R (ssn , yn ) = ϕt,R (sσ (1) , yσ (1) ) − ϕs,R (sσ (1) , yσ (1) ) n! σ ∈Sn $ n−1 × G sσ ( j) −sσ ( j+1) (yσ ( j) − yσ ( j+1) ). j=1 With the convention (1.6) in mind, we can write     n! gn,R H⊗n =2 dssn ϕt,R (s1 , y1 ) − ϕs,R (s1 , y1 ) 0 t>s1 >···>sn >0 R2nd ⎛ ⎞ $ n−1   ×⎝ G s j −s j+1 (y j − y j+1 )⎠ × ϕt,R (s1 , y1 ) − ϕs,R (s1 , y1 ) j=1 ⎛ ⎞ $ n−1 $ n ×⎝ G s j −s j+1 (y j − y j+1 )⎠ γ (y j − y j )dy j dy j . j=1 j=1 Then, using Fourier transform, we can rewrite n! gn,R 2H⊗n as follows: 0    2 n! gn,R 2H⊗n = dssn μ(dξξ p )F1 B R  (ξ1 + · · · + ξ p ) 0 t>s1 >···>sn >0 Rnd   ×  s−t1 (ξ1 + · · · + ξ p )2 G t−t1 (ξ1 + · · · + ξ p ) − G $ n−1  2 × G s j −s j+1  (ξ j+1 + · · · + ξ p ). j=1 (4.45) 123 816 Stoch PDE: Anal Comp (2022) 10:757–827 Recall the expression (2.29) Gt (ξ ) = sin(t|ξ |) and note that it is a 1-Lipschitz function |ξ | in the variable t, uniformly over ξ ∈ Rd . Then    s−t1 (ξ1 + · · · + ξ p )2 ≤ (t − s)2 . G t−t1 (ξ1 + · · · + ξ p ) − G Therefore, plugging this inequality into (4.45) and then applying Lemma 2.6 yields     2 n! gn,R 2H⊗n ≤ (t − s)2 dssn μ(dξ )F1 B R  (ξ ) 0 t>s1 >···>sn >0 Rd $  n−1  2 × μ(dξ j ) G s j −s j+1  (ξ j ) j=1 R d   n−1  2t n μ(dξ )  2 ≤ (t − s) 2(t ∨ 1) 2 μ(dξ )F1 B R  (ξ ), n! Rd 1 + |ξ |2 Rd which is finite since 1 B R ∈ P0 . Using Fourier transform, we can write    2 μ(dξ )F1 B R  (ξ ) = 1 B R (x)1 B R (y)γ (x − y)d xd y. Rd R2d Now let us consider the cases in (4.42). In part (1) where γ ∈ L 1 (Rd ),  1 B R (x)1 B R (y)γ (x − y)d xd y ≤ γ (Rd )ωd R d  σ R2 . R2d In the other cases, we can make the change of variables (x, y) → R(x, y) to obtain   1 B R (x)1 B R (y)γ (x − y)d xd y = R 2d 1 B1 (x)1 B1 (y)γ (Rx − Ry)d xd y R2d R2d R 2d−α = σ R2 , using (4.40) with α = β in part (2), α = β1 + β2 in case (a  ), and α = 1 + β in case (b ). As a consequence, we get Cn 2 n! gn,R 2H⊗n ≤ σ (t − s)2 , 0 n! R and therefore,    n/2 1  FR (t) − FR (s) ≤ |t − s|σ R Ct (k − 1) √ , k n≥1 n! which leads to (4.43).   123 Stoch PDE: Anal Comp (2022) 10:757–827 817 5 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, Du(t, x) H > 0 a.s. on m, where m = {|u(t, x)| ≥ 1/m}. We claim that, almost surely, the function (s, y) → Ds,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 )  1/q *  t   1/q t   q 2  E ds |Ds,y u(t, x)| dy 2q ≤ ds E |Ds,y u(t, x)|  dy 0 R2 0 R2  t  1/q 2q ≤C ds G t−s (x − y)dy < ∞. 0 R2 For d = 1, again by the estimate (1.11),  t    t  E ds |Ds,y u(t, x)| dy 2 ≤C ds G 2t−s (x − y)dy < ∞. 0 R 0 R Moreover, (s, y) → Ds,y u(t, x) has compact support on [0, t]× B M for some M > 0. As a consequence, by Lemma A.1, it suffices to prove that  t Dr ,• u(t, x) 20 dr 0  t = Dr ,z u(t, x)Dr ,z  u(t, x)γ (z − z  )dzdz  dr > 0 a.s. on m. 0 R2d (5.1) As in the proof of Lemma 5.1 of [2], Corollaries 3.3 and 3.4 allow us to infer that the H ⊗ P0 -valued process K (r ) defined by K (r ) (s, y, z) = G t−s (x − y)Dr ,z u(s, y) belongs to the space D1,2 (H ⊗ P0 ). This is because, using Corollary 3.3, we can write E K (r ) 2H⊗P0      = G t−s (x − y)G t−s  (x − y  )E Dr ,• u(s, y), Dr ,• u(s  , y  ) 0 [r ,t]2 R2d × γ0 (s − s )γ (y − y  )dydy  dsds     ≤C G t−s (x − y)G t−s  (x − y  ) [r ,t]2 R2d 123 818 Stoch PDE: Anal Comp (2022) 10:757–827 γ0 (s − s  )γ (y − y  )dydy  dsds  < ∞, and in the same way, using Corollary 3.4 we can show that E D K (r ) 2H⊗H⊗P0 < ∞. Therefore, the process K (r ) belongs to the domain of the P0 -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 P0 , we can show that for any r ∈ [0, t], the following equation holds in L 2 ( ; P0 ):  t Dr ,• u(t, x) = G t−r (x − •)u(r , •) + G t−s (x − y)Dr ,• u(s, y)W (δs, δ y). r Rd (5.2) Let δ ∈ (0, t ∧ 1) be arbitrary. Due to relation (5.2) we have, almost surely,  t  t Dr ,• u(t, x) 2 0 dr ≥ Dr ,• u(t, x) 2 0 dr 0 t−δ  t 1 ≥ G t−r (x − •)u(r , •) 2 0 dr − I (δ), (5.3) 2 t−δ where   t  2 t  I (δ) =  G t−s (x − y)Dr ,• u(s, y)W (δs, δ y)   dr t−δ r Rd 0  t  t  2   =  G t−s (x − y)Dr ,• u(s, y)W (δs, δ y)   dr . t−δ d t−δ R 0 On the event m = {|u(t, x)| ≥ 1/m}, we have  t G t−r (x − •)u(r , •) 20 dr t−δ  t  = G t−r (x − z)G t−r (x − z  )u(r , z)u(r , z  )γ (z − z  )dzdz  dr t−δ R2d  t  = G t−r (x − z)G t−r (x − z  )u(t, x)2 γ (z − z  )dzdz  dr t−δ R 2d  t    − G t−r (x −z)G t−r (x −z  ) u(t, x)2 −u(r , z)u(r , z  ) γ (z − z  )dzdz  dr t−δ R2d 1 ≥ 2 ψ0 (δ) − J (δ), m where  t  ψ0 (δ) := G t−r (x − z)G t−r (x − z  )γ (z − z  )dzdz  dr t−δ R2d 123 Stoch PDE: Anal Comp (2022) 10:757–827 819  δ  = G r (z)G r (z  )γ (z − z  )dzdz  dr 0 R2d and  t  J (δ) := G t−r (x − z)G t−r (x − z  )γ (z − z  ) t−δ R2d   u(t, x)2 − u(r , z)u(r , z  ) dzdz  dr . Coming back to (5.3), we can write  t 1 1 Dr ,• u(t, x) 20 dr ≥ ψ0 (δ) − J (δ) − I (δ) on m. (5.4) 0 2m 2 2 We now give upper bounds for the first moments of J (δ) and I (δ). We will use the following facts, which were proved in [3]: Ct∗ := sup u(s, y) 2 <∞ (see also (3.16) in Remark 3.1) (s,y)∈[0,t]×Rd gt,x (δ) := sup sup u(t, x) − u(s, y) 2 → 0 as δ → 0. |t−s|<δ |x−y|<δ We first treat J (δ). By Cauchy-Schwarz inequality, for any r ∈ [0, t] and z, z  ∈ R2 ,   E |u(t, x)2 − u(r , z)u(r , z  )| ≤ u(t, x) 2 u(t, x) − u(r , z) 2 + u(r , z) 2 u(t, x) − u(r , z  ) 2   ≤ Ct∗ u(t, x) − u(r , z) 2 + u(t, x) − u(r , z  ) 2 . Since G t−r (x − z) contains the indicator of the set {|x − z| < t − r }, we obtain:  t  E(|J (δ)|) ≤ 2Ct∗ G t−r (x − z)G t−r (x − z  )γ (z − z  ) u(t, x) t−δ R2d − u(r , z) 2 dzdz  dr  t  ≤ 2Ct∗ G t−r (x − z)G t−r (x − z  )γ (z − z  ) t−δ R2d sup u(t, x) − u(s, y) 2 dzdz  dr . t−δ 2 0 dr < ∩ ψ0 (δ) m − 0 2 n 2m 2 n  −1   1 1 1 ≤ ψ0 (δ) − E[I (δ)] + E[|J (δ)|] 2m 2 n 2 (Ct + Ct )δφ(δ) + Ct∗ gt,x (δ)ψ0 (δ) ≤ . 1 ψ (δ) − n1 2m 2 0 Letting n → ∞, we obtain: 4 5    t φ(δ) P Dr ,• u(t, x) 2 0 dr =0 ∩ m ≤ 2m 2 (Ct + Ct )δ + Ct∗ gt,x (δ) . 0 ψ0 (δ) Note that using Fourier transform and the expression (2.29), we can rewrite (5.7) as   φ(δ) = G s  (ξ )γ0 (s − s  )μ(dξ )dsds  s (ξ )G [0,δ]2 Rd   1   s  (ξ )2 γ0 (s − s  )μ(dξ )dsds  ≤ G s (ξ )2 + G [0,δ]2 Rd 2   ≤ δ s (ξ )2 μ(dξ )ds, G [0,δ] Rd δ where δ = 2 0 γ0 (s)ds. That is, we have φ(δ) ≤ δ ψ0 (δ). Finally taking δ → 0 proves (5.1), since gt,x (δ) → 0 and δ ψφ(δ) 0 (δ) ≤ δδ → 0 as δ → 0.   123 822 Stoch PDE: Anal Comp (2022) 10:757–827 Acknowledgements The authors would like to thank Wangjun Yuan for carefully proofreading the manuscript and providing a list of typos. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 6 Appendix A.1 Auxiliary Results Let d = 2 and assume Hypothesis (H1). Suppose that S : R+ × R2 → R is a measurable function such that S ∈ L 2 (R+ ; L 2q (R2 )), where q is given in (2.20) in cases (a) and (b) and it is given in (2.23) in case (c). We assume also that S has support in [0, T ] × B M for some M > 0. We claim that S belongs to H and the following estimates hold true: S H≤  T S H0 ≤  T Dγ S L 2 (R+ ;L 2q (R2 )) . Indeed, the first inequality is due to (2.13) and the second one follows from (2.25). For d = 1, if S ∈ L 2 (R+ × R) has support in [0, T ] × B M for some M > 0, then S ∈ H and the following estimates hold true:  S H≤  T S H0 ≤ T γ 1 B2M L 1 (R) S L 2 (R+ ×R) . Indeed, the first inequality is due to (2.13) and the second one follows from  T  S 2H0 = S(t, y)S(t, y  )γ (y − y  )dydy  dt 0 R2   T S 2 (t, y) + S 2 (t, y  ) ≤ γ (y − y  )dydy  dt 0 R2 2 and   sup γ (y − y  )dy ≤ γ (y)dy. y  ∈B M BM B2M Let us recall the Hypothesis (H2): The measures μ0 and μ such that γ0 = Fμ0 and γ = Fμ are absolutely continuous with respect to the Lebesgue measures with strictly positive densities. 123 Stoch PDE: Anal Comp (2022) 10:757–827 823 Lemma A.1 Fix d ∈ {1, 2} and assume that the Hypothesis (H2) holds. Let the Hypoth- esis (H1) hold if in addition d = 2. Suppose that the function S : R+ × Rd → R has support in [0, T ] × B M for some M > 0 and S ∈ L 2 R+ ; L 2q (Rd ) , where  q is given by (2.20) in cases (a) and (b) and by (2.23) in case (c) if d = 2, q = 1 if d = 1. If  T   I := S(t, x)S(t, y)γ (x − y)d xd ydt > 0, (A.1) 0 Rd Rd then S H > 0. Proof Suppose that S H = 0. There exists a sequence of smooth functions (ψk )k≥1 in C ∞ (R+ × Rd ), with support in [0, T ] × B M , which converges to S in L 2 (R+ ; L 2q (Rd )). Then,  0 = S 2H = lim ψk 2H = lim |Fψk (τ, ξ )|2 μ0 (dτ )μ(dξ ), k→∞ k→∞ R+ ×Rd where γ0 = Fμ0 , γ = Fμ and Fψk stands for the Fourier transform of ψk in space-time variables in this proof. By choosing a subsequence (k j ) j≥1 we have that lim Fψk j (τ, ξ ) = 0 j→∞ for μ0 ⊗ μ-almost all (τ, ξ ). On the other hand, keeping in mind that the supports of S, ψk are contained in [0, T ] × B M , we have   1   ψk − S  1 ≤ (π M 2 T ) ψk − S  L 2q (R ×R2 ) 1− 2q  L (R + ×R 2) + 1  2 1− 2q 21   ≤ (π M )  T ψk − S L 2 (R ;L 2q (R2 )) , + from which we deduce that (ψk )k≥1 converge in L 1 ([0, T ]× B M ) to S. Thus Fψk (τ, ξ ) converges to F S(τ, ξ ) for all (τ, ξ ) and the convergence is uniform. As a consequence, F S(τ, ξ ) = 0 for μ0 ⊗ μ-almost all (τ, ξ ) ∈ R+ × Rd and by Hypothesis (H2), we obtain F S(τ, ξ ) = 0 for almost all (τ, ξ ) ∈ R+ × Rd with respect to the Lebesgue measure. Hence S(t, x) = 0 for almost all t > 0 and x ∈ Rd , i.e. there exists a Borel set N ⊂ R+ × Rd with λd+1 (N ) = 0 such that S(t, x) = 0 for all (t, x) ∈ / N . Here λk denotes the Lebesgue measure on Rk . Therefore,  ∞  I = 1 A (t, x, y)S(t, x)S(t, y)γ (x − y)d xd ydt, 0 Rd Rd where A := {(t, x, y) ∈ R+ × Rd × Rd ; (t, x) ∈ N , (t, y) ∈ N }. 123 824 Stoch PDE: Anal Comp (2022) 10:757–827 Let Nt = {x ∈ Rd ; (t, x) ∈ N } be the section of the set N at point t > 0. ∞ By Fubini’s theorem, λd+1 (N ) = 0 λd (Nt )dt. Since λd+1 (N ) = 0, we infer that λd (Nt ) = 0 for almost all t > 0. Note that the section of the set A at point t is At = {(x, y) ∈ Rd × Rd ; (t, x, y) ∈ A} = Nt × Nt , and its Lebesque measure is λ2d (At ) = λ2d(Nt ) = 0 for almost all t > 0. By applying Fubini again, we infer that ∞ λ2d+1 (A) = 0 λ2d (At )dt = 0. This shows I = 0, which contradicts (A.1).   A.2 Proof of Proposition 1.9 In this section, we only sketch the proof of Proposition 1.9 as the main body of the proof is almost identical to that in [42, Proposition 3.2]. Proof of (1.27) Using the duality relation (2.5) and the identity L = −δ D, we have   E D F, −DL −1 GH   = E F(−δ D)L −1 G = E[F L L −1 G] = E[F G] = Cov(F, G), which shows the equality in (1.27). Then, applying the Gaussian Poincaré inequality (2.12) and using Lemma 3.2 of [26], we can bound the variance appearing in the left-hand side of (1.27) by     E DD F, −DL −1 GH 2H ≤ 2E D 2 F, −DL −1 GH 2H   + 2E D F, −D 2 L −1 GH 2H . We will show that the first expectation-term is bounded by A1 and the other one can be estimated in the same way and bounded by A2 . Using the representation (see e.g. [25, Proposition 2.9.3])  ∞ −DL −1 G = dte−t Pt DG, 0 with {Pt , t ≥ 0} the Ornstein-Uhlenbeck semigroup, we can write  ∞ D 2 F, −DL −1 GH = dte−t D 2 F, Pt DGH . (6.2) 0 Note that if (M, M, ν) is a probability space on which s ∈ M −→ Vs ∈ |H|   2 is M-measurable such that M |Vs |H ν(ds) < ∞, then by Fubini’s theorem and Cauchy-Schwarz inequality,  2     Vs ν(ds) = Vs , Vs  H ν(ds)ν(ds  )   M H M2   Vs 2H + Vs  2H ≤ ν(ds)ν(ds  ) = Vs 2H ν(ds). M2 2 M 123 Stoch PDE: Anal Comp (2022) 10:757–827 825 Using the above inequality on (R+ , e−t dt), we deduce from (6.2) that   2  ∞  2 D F, −DL −1 GH 2 ≤ dte−t D 2 F, Pt DGH H . H 0 Observe that D 2 F, Pt DGH is nothing else but the one-contraction D 2 F ⊗1 Pt DG, so that  2  D F, Pt DGH 2 = D 2 F ⊗1 Pt DG, D 2 F ⊗1 Pt DGH H   = D 2 F ⊗1 D 2 F, (Pt DG) ⊗ (Pt DG) H⊗2 , where the last equality follows from the definition of contractions. Therefore, we have E[ D 2 F, −DL −1 GH 2H ]  ∞  −t ≤E dt e dr dr  dsds  dθ dθ  dzdz  0 R6+ ×R6d × dydy  dwdw  γ0 (θ − θ  )γ0 (s − s  )γ0 (r − r  )γ (z − z  )γ (w − w  )γ (y − y  )    × Dr ,z Dθ,w F Ds,y Dθ  ,w F Pt (Dr  ,z  G)Pt (Ds  ,y  G) and thus we end our estimation of E[ D 2 F, −DL −1 GH 2H ] by using Hölder inequality and the contraction property of Pt on L 4 ( ), that is, using Pt (Dr  ,z  G) 4 ≤ Dr  ,z  G 4 . To estimate the other expectation-term E[ D F, −D 2 L −1 GH 2H ], one can begin with  ∞ 2 −1 −D L G= dte−2t Pt D 2 G 0 and then follow the same arguments.   References 1. Balan, R.M.: The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach. Potential Anal. 36, 1–34 (2012) 2. Balan, R.M., Quer-Sardanyons, L., Song, J.: Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise. Electron. J. Probab. 24(106), 1–43 (2019) 3. 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Balan rbalan@uottawa.ca David Nualart nualart@ku.edu Lluís Quer-Sardanyons quer@mat.uab.cat 1 Department of Mathematics and Statistics, STEM Building, University of Ottawa, 150 Louis-Pasteur Private, Ottawa, ON K1N 6N5, Canada 2 Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045, USA 3 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Cerdanyola del Vallès Catalonia, Spain 4 School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK 123