nal r u o o f J P c r i o on ba Electr bility Electron. J. Probab. 24 (2019), no. 106, 1–43. ISSN: 1083-6489 https://doi.org/10.1214/19-EJP363 Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise Raluca M. Balan* Lluís Quer-Sardanyons† Jian Song‡ Abstract In this article, we consider the stochastic wave equation on R+ × R, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions γ (in time) and f (in space), which are the Fourier transforms of tempered measures ν on R, respectively µ on R. Our main result shows that the law of the solution u(t, x) of this equation is absolutely continuous with respect to the Lebesgue measure. Keywords: Gaussian noise; stochastic partial differential equations; Malliavin calculus. AMS MSC 2010: Primary 60H15; 60H07. Submitted to EJP on May 17, 2018, final version accepted on September 18, 2019. 1 Introduction In this article, we study the absolute continuity of the law of the solution to the stochastic wave equation with linear multiplicative noise, in spatial dimension d = 1:  2 ∂ u ∂2u (t, x) = (t, x) + u(t, x)Ẇ (t, x), t > 0, x ∈ R,   2 ∂x2   ∂t   u(0, x) = 1, x ∈ R, (1.1)    ∂u (0, x) = 0, x ∈ R,    ∂t where W is the spatially-homogeneous Gaussian noise considered in [3, 2], whose precise definition is given in Section 2 below. * University of Ottawa, Department of Mathematics and Statistics, STEM Building, 150 Louis-Pasteur Private, Ottawa, ON, K1N 6N5, Canada. E-mail: rbalan@uottawa.ca. Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada. † Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Catalo- nia, Spain. E-mail: quer@mat.uab.cat. Research supported by the grant MTM2015-67802P. ‡ School of Mathematics, Shandong University, Jinan, Shandong 250100, P. R. China. E-mail: txjsong@hotmail.com. Corresponding author. Existence of density for the stochastic wave equation The goal of this article is to show that the law of the solution to equation (1.1) has a density with respect to the Lebesgue measure. This problem has a long history which we outline below. Before this, we recall from [3] the definition of the solution, referring to Section 3 below for the precise definition of the Skorohod integral. We denote by G the fundamental solution of the wave equation on R+ × R: 1 G(t, x) = 1{|x| 0, and for any t > 0 and x ∈ R, the following equality holds in L2 (Ω): Z tZ u(t, x) = 1 + G(t − s, x − y)u(s, y)W (δs, δy), (1.2) 0 R where the stochastic integral is understood in the Skorohod sense, and the process v (t,x) = {v (t,x) (s, y) = 1[0,t] (s)G(t − s, x − y)u(s, y); s ≥ 0, y ∈ R} is Skorohod integrable. The existence of the solution to the wave equation with general initial condition and space-time homogeneous Gaussian noise has not been studied in the literature so far. Article [2] examined the heat equation with initial condition given by a measure, and the same noise W as in the present paper. In the case of constant initial condition, it was proved in [3] that the wave equation has a unique solution in any spatial dimension d, provided that the spatial spectral measure µ of the noise satisfies Dalang’s condition: Z 1 µ(dξ) < ∞. (1.3) Rd 1 + |ξ|2 Condition (1.3) was introduced simultaneously in articles [9] and [30], and plays a crucial role in the study of stochastic partial differential equations (SPDEs) with spatially homogeneous Gaussian noise (see for instance [9, 29, 27, 17] for a sample of relevant references). Owing to Remark 10(b) in [9], note that (1.3) holds automatically if d = 1. The problem of absolute continuity of the law and smoothness of the density for the solution of an SPDE goes back to article [7], in which the authors studied the equation: Lu(t, x) = σ(u(t, x))Ẇ (t, x) + b(u(t, x)) (1.4) with space-time white noise W , smooth functions σ and b, and L the wave operator on R+ × I , for an interval I ⊂ R which could be bounded, semi-bounded, or even R, with Dirichlet boundary conditions when I has a finite endpoint. These authors showed that the mild solution (defined similarly to (1.2) using the Green function of the wave operator) coincides with the weak solution (defined using integration against test functions) and proved that this solution has a smooth density. In [28], [5] and [24], it was shown that the same property holds for equation (1.4) in which L is replaced by the heat operator on R+ × [0, 1], with Dirichlet (respectively Neumann) boundary conditions. (In this case, the fact that the mild and weak solutions coincide was known from Walsh’ lecture notes [36].) In the recent years, several authors revisited the problem of existence and smoothness of density for the mild solution of an SPDE of form (1.4) on the entire space Rd , with Lipschitz functions σ and b, driven by the more general Gaussian noise introduced in [10, 9]. This noise is spatially homogeneous (with spatial spectral measure µ as above), EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 2/43 Existence of density for the stochastic wave equation but white in time, i.e. has temporal covariance structure given formally by the Dirac distribution at 0. The function σ is assumed to satisfy the condition: |σ(x)| ≥ c > 0 for all x ∈ R. (1.5) This non-degeneracy condition guarantees the genuine stochastic nature of the solution u, since the noise term will never vanish, and turns out to be crucial in order to prove conditions (1.7) and (1.8) below. Nevertheless, in the case of the stochastic heat equation, condition (1.5) can be substantially weaken (see [24, 28]). We refer the reader to [23] for the wave equation in spatial dimension d = 2, [27] for the heat equation in any dimension d and the wave equation in dimension d = 1, 2, 3 (see also [22, 32, 33]), and [34, 35] for the wave equation in dimension d ≥ 4. In the case of the space-time Gaussian noise which is colored in time (i.e. has temporal covariance structure given by a non-negative-definite locally integrable function), all references related to the problem of existence and smoothness of the density of the law of the solution focus on the stochastic heat equation with linear multiplicative noise (i.e. σ(x) = x and b = 0): ∂u 1 (t, x) = ∆u(t, x) + u(t, x)Ẇ (t, x), t > 0, x ∈ Rd , (1.6) ∂t 2 with initial condition u(0, x) = u0 (x), where u0 is a continuous bounded function. In this case, the noise is not a martingale in time, and the techniques of Itô stochastic integration cannot be used. This leads to several types of solution: the mild Skorohod solution defined similarly to (1.2) using the Skorohod integral involving the Green function Gh of the heat operator, the mild Stratonovich solution defined using a Stratonovich-type integral of the same term as in (1.2) involving Gh , and the weak solution defined using Stratonovich integration against test functions. According to some well-known criteria from Malliavin calculus, to show that u(t, x) has a density it is enough to prove that kDu(t, x)kH > 0 a.s. (1.7) and this density is smooth if u(t, x) is infinitely differentiable in the Malliavin sense and EkDu(t, x)k−2p H < ∞ for all p > 0. (1.8) The notation H stands for the Hilbert space associated to the space-time correlation of the noise (see Section 2 for the precise definition). In [16], it was proved that relation (1.8) holds for the weak solution of (1.6), if the covariance functions of the noise are given by γ(t) = ρH (t) := H(2H − 1)|t|2H−2 and Qd f (x) = i=1 ρHi (xi ) with parameters H, H1 , . . . , Hd ∈ (1/2, 1). This result was obtained using the Feynman-Kac (FK) representation of the weak solution, which holds only when Pd the parameters of the noise satisfy the condition 2H + i=1 Hi > d + 1. Under the same condition, it might be possible to prove that the mild Skorohod solution of equation (1.6) satisfies (1.8), using the FK formula of this solution given in [16]. In the case of general covariance functions γ and f , the authors of [14] established the FK formula for the mild Stratonovich solution to equation (1.6), assuming that 0 ≤ γ(t) ≤ Cβ |t|−(1−β) and Z  β 1 µ(dξ) < ∞, (1.9) Rd 1 + |ξ|2 for some β ∈ (0, 1) and Cβ > 0. Under this assumption, it may be possible to show that the mild Stratonovich solution satisfies relation (1.8). In the recent preprint [15], it was proved that the mild Skorohod solution u(t, x) to equation (1.6) satisfies (1.8) by EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 3/43 Existence of density for the stochastic wave equation deriving estimates for the small ball probability P(kDu(t, x)kH ≤ a) as a → 0+. This was proved using an FK formula for the regularization of the solution, and assuming that the noise is white in either space or time, or c1 tα0 ≤ γ(t) ≤ c2 t−α0 for all t ∈ R, for some c1 , c2 > 0 and α0 ∈ [0, 1), and f satisfies the scaling property f (cx) ≤ c−α f (x) for all c > 0 and x ∈ Rd , for some α ∈ (0, 2). In addition, these authors assume that γ = γ0 ∗ γ0 and f = f0 ∗ f0 for some functions γ0 and f0 . In the present article, we will prove the absolute continuity of the law of the solu- tion to the wave equation (1.1) with space-time homogeneous Gaussian noise, under Assumption A given below. For the wave equation, it is not known if there is a FK formula for the solution. Our method for proving (1.7) is similar to the one used in [27] for the white noise in time. Since σ(x) = x fails to satisfy condition (1.5), we will prove (1.7) by localizing on the event Ωm = {|u(t, x)| > m−1 } and let m → ∞. Note that, compared to parabolic type equations, the solution of the stochastic wave equation does not verify a comparison principle and therefore can hit zero with positive probability. A closer look at the inner product in H reveals that it is enough to prove that (see Corollary 2.4 below): Z tZ |Dr,z u(t, x)|2 dzdr > 0 a.s. on Ωm . (1.10) 0 R For this, we will show that the process {Dr,z u(t, x); r ∈ [0, t], z ∈ R} has a jointly mea- surable modification and satisfies a certain integral equation involving a Hilbert-space valued Skorohod integral. As shown in Sections 3, 4 and 5 below, the proofs of these facts contain some of the main technical difficulties encountered throughout the article, which are mainly due to the time-space correlation of the underlying noise. The recent preprints [18, 12] contain some estimates which are related to the present article. More precisely, Lemma 5.1 of [18] shows that if u is the solution of the stochastic heat equation with space-time white noise, then for any T > 0 and (t, x) ∈ [0, T ] × R, E|Dr,z u(t, x)|2 ≤ CT G2 (t − r, x − z), for all (r, z) ∈ [0, t] × R, (1.11) where G(t, x) = (2πt)−1/2 exp(−x2 /(2t)) is the heat kernel and CT > 0 is a constant depending on T . In the case of the wave equation in dimension d = 1 with Gaussian noise which is white in time and fractional in space (with Hurst index H ≥ 1/2), Lemma 2.2 in [12] gives an estimate similar to (1.11), except that it holds for almost all (r, z) ∈ [0, t]×R. In the present article, we could not obtain such a nice estimate for the solution of the wave equation in dimension d = 1, with general space-time homogeneous Gaussian noise. But we prove that EkDr,· u(t, x)k2L2 (R) is uniformly bounded in r ∈ [0, t] and (t, x) ∈ [0, T ] × R, which implies that Dr,· u(t, x) belongs to L2 (R) a.s. (see Theorem 3.11 below). We suppose that the following assumption holds, which is important for describing the space H and its inner product, as shown by Theorem 2.3 below. Assumption A. µ(dξ) = (2π)−d g(ξ)dξ , ν(dτ ) = (2π)−1 h(τ )dτ and 1/(hg)1{hg>0} is a slow growth (or tempered) function, i.e. there exists an integer k ≥ 1 such that Z  k 1 1 dτ dξ < ∞. {hg>0} 1 + τ 2 + |ξ|2 h(τ )g(ξ) Our basic example is γ(t) = H(2H − 1)|t|2H−2 with 1/2 < H < 1 and f is the Riesz kernel of order α, i.e. f (x) = |x|−1+α with 0 < α < 1. In this case, h(τ ) = |τ |2H−1 and g(ξ) = cα |ξ|−α for some constant cα > 0 depending on α. Assumption A is satisfied for this example. The following theorem is the main result of the present article. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 4/43 Existence of density for the stochastic wave equation Theorem 1.2. Let u be the solution of equation (1.1). If Assumption A holds, then the restriction of the law of the random variable u(t, x)1{u(t,x)6=0} to the set R\{0} is absolutely continuous with respect to the Lebesgue measure on R\{0}. This article is organized as follows. In Section 2, we describe our space-time homoge- neous Gaussian noise and we characterize the space H and its inner product. In Section 3, we review some basic elements of Malliavin calculus, we prove that the solution to equation (1.1) is infinitely differentiable in the Malliavin sense, and examine some properties of its Malliavin derivative. In Section 4 we examine the second order Malliavin derivative of the solution. In Section 5, we prove that the Malliavin derivative satisfies a certain integral equation. In Section 6, we give the proof of Theorem 1.2. In Appendix A, we discuss a Parseval-type identity, while in Appendix B we give a criterion for the existence of a measurable modification of a random field. Both these results are used in the present article. 2 Characterization of the space H In this section, we define the space-time homogeneous Gaussian noise W , we provide an alternative definition of the inner product in H in terms of the Fourier transform in the space and time variables, and we give a characterization of the space H (which is due essentially to [6]). This characterization plays an important role in the present article, because it allows us to focus on (1.10), instead of (1.7). The results in the present section are valid for any space dimension d ≥ 1. The noise W is given by a zero-mean Gaussian process {W (ϕ); ϕ ∈ D(Rd+1 )} defined on a complete probability space (Ω, F, P), with covariance Z E[W (ϕ1 )W (ϕ2 )] = γ(t − s)f (x − y)ϕ1 (t, x)ϕ2 (s, y)dxdydtds =: J(ϕ1 , ϕ2 ), R2 ×R2d where γ : R → [0, ∞] and f : Rd → [0, ∞] are continuous, symmetric, locally integrable functions, such that γ(t) < ∞ if and only if t 6= 0; f (x) < ∞ if and only if x 6= 0. Here D(Rd+1 ) is the space of C ∞ -functions on Rd+1 with compact support. We denote by H the completion of D(Rd+1 ) with respect to h·, ·iH defined by hϕ1 , ϕ2 iH = J(ϕ1 , ϕ2 ). We assume that f is non-negative-definite (in the sense of distributions), i.e. Z (ϕ ∗ ϕ)(x)f e (x)dx ≥ 0, for all ϕ ∈ S(Rd ), Rd where ϕ(x) e = ϕ(−x) and S(Rd ) is the space of rapidly decreasing C ∞ -functions on R . By the Bochner-Schwartz theorem, there exists a tempered measure µ on Rd such d 0 that f = Fµ, where Fµ denotes the Fourier transform of µ in the space SC (Rd ) of d C-valued R tempered distributions on R . We emphasize that this does not mean that −iξ·x f (x) = Rd e µ(dξ) for all x ∈ Rd , since µ may be an infinite measure. It means that Z Z f (x)ϕ(x)dx = Fϕ(ξ)µ(dξ) for all ϕ ∈ SC (Rd ), Rd R where SC (Rd ) is the space of C-valued rapidly decreasing C ∞ -functions on Rd , and Fϕ(x) = Rd e−iξ·x ϕ(x)dx is the Fourier transform of ϕ. Here ξ · x denotes the scalar R product in Rd . Similarly, we assume that γ is non-negative-definite (in the sense of EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 5/43 Existence of density for the stochastic wave equation distributions), and so there exists a tempered measure ν on Rd such that γ = Fν in SC0 (Rd ). Note that, by the definition of the measure µ, for any ϕ1 , ϕ2 ∈ SC (Rd ) Z Z Z f (x − y)ϕ1 (x)ϕ2 (y)dxdy = Fϕ1 (ξ)Fϕ2 (ξ)µ(dξ). (2.1) Rd Rd Rd Similarly, for any φ1 , φ2 ∈ SC (Rd ) Z Z Z γ(t − s)φ1 (t)φ2 (s)dtds = Fφ1 (τ )Fφ2 (τ )ν(dτ ). (2.2) R R R Building upon a remarkable result borrowed from [21], in Appendix A, we show that relation (2.1) holds for any functions ϕ1 , ϕ2 ∈ L1C (Rd ) whose absolute values have “finite energy” with respect to the kernel f . This fact is used in the proof of Lemma 3.1 below. By Lemma 2.1 of [3], for any ϕ1 , ϕ2 ∈ D(Rd+1 ), Z hϕ1 , ϕ2 iH = Fϕ1 (τ, ξ)Fϕ2 (τ, ξ)ν(dτ )µ(dξ), (2.3) Rd+1 where F denotes the Fourier transform in both variables t and x. Similarly to [20, 6], we consider the space: Z U = {S ∈ S 0 (Rd+1 ); FS is a (measurable) function and |FS(τ, ξ)|2 ν(dτ )µ(dξ) < ∞}. Rd+1 The space U is endowed with the inner product Z hS1 , S2 iU := FS1 (τ, ξ)FS2 (τ, ξ)ν(dτ )µ(dξ). Rd+1 By (2.3), hϕ1 , ϕ2 iH = hϕ1 , ϕ2 iU for any ϕ1 , ϕ2 ∈ D(Rd+1 ). We denote kSk2U = hS, SiU . Note that if S ∈ S 0 (Rd+1 ) is such that FS is a function, then FS(−τ, −ξ) = FS(τ, ξ) for almost all (τ, ξ) ∈ R × Rd . (2.4) The proof of this fact is very similar to that of Lemma 3.1 of [20]. This property and the symmetry of the measures ν and µ imply that h·, ·iU is a well-defined R-valued inner product, provided that we identify two elements S1 and S2 such that kS1 − S2 kU = 0. We let L2C (ν × µ) be the space of functions v : Rd+1 → C such that Z |v(τ, ξ)|2 ν(dτ )µ(dξ) < ∞, Rd+1 and Le 2 (ν × µ) be the subset of L2 (ν × µ) consisting of functions v such that v(−τ, −ξ) = C C v(τ, ξ) for almost all (τ, ξ) ∈ R × Rd . The next result generalizes Theorem 3.2 of [20] to higher dimensions. Assumption A is not needed for this result. Theorem 2.1. The space D(Rd+1 ) is dense in U , with respect to h·, ·iU , and hence U is included in H. Moreover, hS1 , S2 iU = hS1 , S2 iH for any S1 , S2 ∈ U . Proof. We only need to prove that D(Rd+1 ) is dense in U , with respect to h·, ·iU , i.e. for any S ∈ U , there exists a sequence (ϕn )n≥1 ⊂ D(Rd+1 ) such that kϕn − SkU → 0 as n → ∞. For this, it suffices to prove that F D(Rd+1 ) is dense in L e 2C (ν × µ),  (2.5) EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 6/43 Existence of density for the stochastic wave equation  where F D(Rd+1 ) is the image of D(Rd+1 ) under the Fourier transform, and L e 2 (ν ×µ) is C 2 endowed with the topology of LC (ν × µ). (To see this, let S ∈ U be arbitrary. Since FS ∈ e 2 (ν × µ), there exists a sequence (ϕn )n≥1 ⊂ D(Rd+1 ) such that kFϕn − FSkL2 (ν×µ) = L C C kϕn − SkU → 0 as n → ∞.) It remains prove (2.5). First, we claim that F DC (Rd+1 ) is dense in L2C (ν × µ).  (2.6) Indeed, this is an extension of Theorem 4.1 in [19], which is proved as follows. First, F DC (Rd+1 ) is dense in SC (Rd+1 ), because the Fourier transform defines an homeomor- phism from SC (Rd+1 ) onto itself. Secondly, using that ν and µ are tempered measures, one obtains that SC (Rd+1 ) ⊂ L2C (ν × µ) and that convergence in SC (Rd+1 ) implies conver-  gence in L2C (ν × µ). These two facts imply the third one, namely F DC (Rd+1 ) is dense in SC (Rd+1 ) with respect to the topology of L2C (ν × µ). Finally, the conclusion follows by observing that SC (Rd+1 ) is dense in L2C (ν × µ), since DC (Rd+1 ) is so. This proves (2.6). We now prove (2.5). Let v ∈ L e 2 (ν × µ) be arbitrary. By (2.6), there exists a sequence C (ϕn )n≥1 in DC (Rd+1 ) such that Fϕn → v in L2C (ν × µ). Note that this implies that the following limits hold in L2 (ν × µ): lim Re(Fϕn ) = Re(v) and lim Im(Fϕn ) = Im(v). n→∞ n→∞ Let us construct a sequence (ψn )n≥1 in D(Rd+1 ) such that Fψn → v in L2C (ν ×µ). For this, we will use the following notation. Namely, for any C-valued function κ, e(κ) := 12 (κ + κ e) and o(κ) := 21 (κ − κ e) denote the even and odd parts of κ, respectively. e 2 (ν ×µ), we have that Re(v) is a even function and Im(v) is an odd Since v belongs to L C function. Here, we say that a function g is even if g(−x) = g(x) for almost all x ∈ Rd+1 and is odd if g(−x) = −g(x) for almost all x ∈ Rd+1 . These properties, together with the symmetry of the measures ν and µ, imply that the following limits hold in L2 (ν × µ): lim e(Re(Fϕn )) = Re(v) and lim o(Im(Fϕn )) = Im(v). n→∞ n→∞ Define ψn := F −1 e(Re(Fϕn )) + io(Im(Fϕn )) . Then, ψn is a real function, belongs to  D(Rd+1 ) and, by construction, Fψn → v in L2C (ν × µ). The following result not only generalizes Theorem 3.4 of [20] to higher dimensions, but also specifies the necessary and sufficient condition for the completeness of U . This result is an immediate consequence of Theorem 3.5.(2) of [6], applied to the space Rd+1 and the measure F = ν × µ. Theorem 2.2. Assume that µ(ξ) = (2π)−d g(ξ)dξ and ν(dτ ) = (2π)−1 h(τ )dτ . Then U is complete if and only if for any function ϕ ∈ L2C (ν × µ) there exists an integer k ≥ 1 such that Z  k 1 |ϕ(τ, ξ)| dτ dξ < ∞. {h>0,g>0} 1 + τ 2 + |ξ|2 In particular, U is a complete if 1/(hg)1{hg>0} is a slow growth function. Combining Theorems 2.1 and 2.2, we obtain the following result, which can be viewed as a generalization of Theorem 3.5 of [20] to higher dimensions. Theorem 2.3. If Assumption A holds, then H coincides with the space U . Moreover, for any S1 , S2 ∈ H, Z hS1 , S2 iH = FS1 (τ, ξ)FS2 (τ, ξ)ν(dτ )µ(dξ). Rd+1 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 7/43 Existence of density for the stochastic wave equation Corollary 2.4. If Assumption A holds and S is a measurable function on R+ × R such that S ∈ H and Z ∞Z |S(t, x)|2 dxdt > 0, (2.7) 0 Rd then kSkH > 0. Proof. Suppose that kSkH = 0. By Theorem 2.3, FS(τ, ξ) = 0 for almost all (τ, ξ) ∈ R×Rd . Hence S(t, x) = 0 for almost all t > 0 and x ∈ Rd , which contradicts (2.7). Despite its complicated nature, the space H contains a nice space of functions, called |H|, which is defined as the set of all measurable functions ϕ : R+ × Rd → R such that Z 2 kϕk|H| := |ϕ(t, x)||ϕ(s, y)|γ(t − s)f (x − y)dtdxdsdy < ∞. (R+ ×Rd )2 Note that |H| is a Banach space equipped with the norm k · k|H| , and kϕkH ≤ kϕk|H| for any ϕ ∈ |H|. By Lemma A.1 (Appendix A), for any integrable functions ϕ, ψ ∈ |H|, Z hϕ, ψiH = ϕ(t, x)ψ(s, y)γ(t − s)f (x − y)dtdxdsdy. (R+ ×Rd )2 3 Malliavin derivative of the solution In this section, we examine some properties of the Malliavin derivative Du(t, x) of the solution u(t, x) to equation (1.1). We recall some basic elements of Malliavin calculus. We refer the reader to [26] for more details. Any random variable F ∈ L2 (Ω) which is measurable with respect to the P σ -field generated by {W (ϕ); ϕ ∈ H} admits the representation F = n≥0 Jn F where Jn F is the projection of F on the n-th Wiener chaos space Hn for n ≥ 1, and J0 F = E(F ). We denote by In the multiple integral of order n with respect to W , which is a linear continuous operator from H⊗n onto Hn . Let S be the class of “smooth” random variables, i.e. random variables of the form F = f (W (ϕ1 ), . . . , W (ϕn )), where n ≥ 1, ϕ1 , . . . , ϕn ∈ H and f is in the set Cb∞ (Rn ) of bounded infinitely differentiable functions on Rn whose partial derivatives are bounded. The Malliavin derivative of a random variable F of this form is the H-valued random variable given by: n X ∂f DF = (W (ϕ1 ), . . . , W (ϕn ))ϕi . i=1 ∂xi 1/2 We let D1,2 be the completion of S with respect to the norm kF k1,2 = E|F |2 +EkDF k2H . Similarly, the iterated derivative D k F can be defined as a H⊗k -valued random vari- able, for any natural number k ≥ 1. For any p > 1, let Dk,p be the completion of S with respect to the norm  k X 1/p kF kk,p = E|F |p + EkDj F kpH⊗j . j=1 It can be proved that: (see p. 28 of [26]) X F ∈ Dk,2 if and only if nk E|Jn F |2 < ∞. n≥1 For p > 1, it can be shown that X F ∈ Dk,p if nk/2 (p − 1)n/2 (E|Jn F |2 )1/2 < ∞ (3.1) n≥1 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 8/43 Existence of density for the stochastic wave equation (see p. 28 of [1]). The divergence operator δ is defined as the adjoint of the operator D . The domain of δ , denoted by Dom δ , is the set of u ∈ L2 (Ω; H) such that |EhDF, uiH | ≤ c(E|F |2 )1/2 , ∀F ∈ D1,2 , (3.2) where c is a constant depending on u. If u ∈ Dom δ , then δ(u) is the element of L2 (Ω) characterized by the following duality relation: E(F δ(u)) = EhDF, uiH , ∀F ∈ D1,2 . (3.3) If u ∈ Dom δ , we will use the notation Z ∞ Z δ(u) = u(t, x)W (δt, δx), 0 R even if u is not a function in (t, x), and we say that δ(u) is the Skorohod integral of u with respect to W . We return now to the solution u of equation (1.1). From [3], we know that u(t, x) has the Wiener chaos expansion X u(t, x) = 1 + In (fn (·, t, x)), n≥1 where In is the multiple Wiener integral of order n with respect to W , and fn (t1 , x1 , . . . , tn , xn , t, x) = G(t − tn , x − xn ) · · · G(t2 − t1 , x2 − x1 )1{0 1, u(t, x) ∈ Dk,p . We will now extend this result to general functions f and γ (in the case d = 1). We begin with a maximum principle, which is a refinement of Lemma 4.2 of [3], specific to the cases d = 1 and d = 2. Its proof is based on a Parseval-type identity given in Appendix A. Lemma 3.1. If G is the fundamental solution of the wave equation in spatial dimension d = 1 or d = 2, then Z Z 2 sup |FG(t, ·)(ξ + η)| µ(dξ) = |FG(t, ·)(ξ)|2 µ(dξ). η∈Rd Rd Rd Proof. We denote by Ef (ϕ) the energy of ϕ with respect to f given by (A.1) (Appendix A). R Note that G(t, ·) is a non-negative integrable function with Rd G(t, x)dx = t and Z Z |FG(t, ·)(ξ)|2 µ(dξ) < ∞.  Ef (G(t, ·)) = G(t, ·) ∗ G(t, ·) (x)f (x)dx = Rd Rd For any η ∈ Rd , we let Gη (t, x) = e−ix·η G(t, x). Then Gη (t, ·) ∈ L1C (Rd ) and Z  Ef (|Gη (t, ·)|) = |Gη (t, ·)| ∗ |Gη (t, ·)| (x)f (x)dx d ZR  ≤ G(t, ·) ∗ G(t, ·) (x)f (x)dx = Ef (G(t, ·)) < ∞. Rd EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 9/43 Existence of density for the stochastic wave equation Moreover, FGη (t, ·)(ξ) = FG(t, ·)(ξ + η) for any ξ ∈ Rd . It follows that Z Z 2 0≤ |FG(t, ·)(ξ + η)| µ(dξ) = |FGη (t, ·)(ξ)|2 µ(dξ) Rd Rd Z Z = (Gη (t, ·) ∗ Gη (t, ·))(x)f (x)dx = (Gη (t, ·) ∗ Gη (t, ·))(x)f (x)dx d Rd ZR  ≤ |Gη (t, ·)| ∗ |Gη (t, ·)| (x)f (x)dx ≤ Ef (G(t, ·)), Rd where for the second equality above, we used Lemma A.2 (Appendix A). The next result gives a stronger form of relation (4.10) of [3] (with a simplified proof). Its proof is based on Lemma 3.1. Lemma 3.2. For any k ≥ 0, X nk αn (t) < ∞. n! n≥0 Proof. We will borrow notations from [3, Theorem 4.4]. Recall that Z n Y αn (t) = γ(tj − sj )ψn (t, s)dtds, (3.4) [0,t]2n j=1 where t = (t1 , . . . , tn ) and s = (s1 , . . . , sn ), Z (n) (n) ψn (t, s) = Fgt (·, t, x)(ξ1 , . . . , ξn )Fgs (·, t, x)(ξ1 , . . . , ξn )µ(dξ1 ) . . . µ(dξn ) Rn and (n) gt (·, t, x) = n!fen (t1 , ·, . . . , tn , ·, t, x). (3.5) If the permutation ρ of {1, . . . , n} is chosen such that tρ(1) < . . . < tρ(n) , then Pn (n) Fgt (·, t, x)(ξ1 , . . . , ξn ) = e−i j=1 ξj ·x FG(tρ(2) − tρ(1) , ·)(ξρ(1) ) FG(tρ(3) − tρ(2) , ·)(ξρ(1) + ξρ(2) ) . . . FG(t − tρ(n) , ·)(ξρ(1) + . . . + ξρ(n) ). (3.6) By relation (4.15) of [3], Z αn (t) ≤ Γnt ψn (t, t)dt, [0,t]n Rt where Γt := 2 0 γ(s)ds. To estimate ψn (t, t) we use the first inequality in (4.16) of [3]. We denote uj = tρ(j+1) − tρ(j) for j = 1, . . . , n and tρ(n+1) = t. We have: n  Y Z  2 ψn (t, t) ≤ sup |FG(uj , ·)(ξj + η)| µ(dξj ) j=1 η∈R R n Z 2 Y sin (uj |ξj |) ≤ µ(dξj ), j=1 R |ξj |2 where for the last equality we used Lemma 3.1. Therefore, sin2 ((t2 − t1 )|ξ1 |) sin2 ((t − tn )|ξn |) Z Z αn (t) ≤ Γnt n! . . . µ(dξ1 ) . . . µ(dξn )dt. {0 0 and ξ ∈ R, ∞ sin2 (t|ξ|) Z 2 1 Iβw (ξ) := e−βt 2 dt = · 2 . 0 |ξ| β β + 4|ξ|2 Using the change of variables t2 − t1 = u1 , . . . t − tn = un , we obtain: sin2 ((t2 − t1 )|ξ1 |) sin2 ((t − tn )|ξn |) Z . . . dt1 . . . dtn {0 1. Proof. We apply (3.1) to the variable F = u(t, x). Let Jn (t, x) := Jn u(t, x) = In (fn (·, t, x)). 1 Then E|Jn (t, x)|2 = n!kfen (·, t, x)k2H⊗n = n! αn (t). By (3.7), we have:  1/2  n/2 X 1 X 2Γt nk/2 (p − 1)n/2 αn (t) ≤ eM t/2 enk/2 (p − 1)n/2 KM < ∞, n! M n≥1 n≥1 if we choose M large enough. Remark 3.4. Lemmas 3.2 and 3.3 remain valid in the case d = 2. We begin now to examine the Malliavin derivative of the variable u(t, x). Recall that the solution u has the chaos expansion: X X u(t, x) = 1 + In (fn (·, t, x)) = 1 + In (fen (·, t, x)), n≥1 n≥1 where fen (·, t, x) is the symmetrization of fn (·, t, x). By Proposition 2.4 of [1], the Malliavin derivative of u(t, x) has the chaos expansion: X Du(t, x) = nIn−1 (fen (·, ∗, t, x)), (3.8) n≥1 where ∗ denotes the missing (r, z)-variable and the convergence of the series is in L2 (Ω; H). We emphasize the presence of the symmetrized kernel fen (·, t, x) in this relation. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 11/43 Existence of density for the stochastic wave equation P We now examine the series n≥1 nIn−1 (fen (·, r, z, t, x)) for fixed (r, z) ∈ [0, t] × R. Note that fen (·, r, z, t, x) is the sum of n terms corresponding to the possible positions of (r, z) among the n variables of fen (·, t, x): n 1 X (n) fen (·, r, z, t, x) = h (·, r, z, t, x), (3.9) n j=1 j (n) where hj (·, r, z, t, x) ∈ H⊗(n−1) is a symmetric function given by: (n) hj (t1 , x1 , . . . , tn−1 , xn−1 , r, z, t, x) = (3.10) 1 X fn (tρ(1) , xρ(1) , . . . , tρ(j−1) , xρ(j−1) , r, z, tρ(j) , xρ(j) , . . . , tρ(n−1) , xρ(n−1) , t, x). (n − 1)! ρ∈Sn−1 (n) (n) Clearly, hj (·, r, z, t, x) is the symmetrization of the function fj (·, r, z, t, x) given by: (n) fj (t1 , x1 , . . . , tn−1 , xn−1 , r, z, t, x) (3.11) = fn (t1 , x1 , . . . , tj−1 , xj−1 , r, z, tj , xj , . . . , tn−1 , xn−1 , t, x) = G(t − tn−1 , x − xn−1 ) . . . G(tj − r, xj − z)G(r − tj−1 , z − xj−1 ) . . . G(t2 − t1 , x2 − x1 ) 1{0 0. Hence, for any j = 1, . . . , n,  n−1 (n) 1 n−1 2 n−1 k(n − 1)!hj (·, r, z, t, x))k2H⊗(n−1) ≤ Γ (n − 1)! eM t KM . 4 t M EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 13/43 Existence of density for the stochastic wave equation Coming back to (3.14), we obtain that for any (r, z) ∈ [0, t] ∈ R  n−1 1 2 E|An (r, z, t, x)| ≤ n2 Γn−1 2 t eM t n−1 KM . 4 M By choosing M large enough (M may depend on t), and using the orthogonality of the Wiener chaos spaces, we see that m 2 X sup E Ak (r, z, t, x) → 0 as n, m → ∞. (r,z)∈[0,t]×R k=n Remark 3.6. (The case d = 2) The proof of Lemma 3.5 does not work for the case (n) d = 2. Unfortunately, we could not find another argument to prove that fj (·, r, z, t, x) ∈ H⊗(n−1) for j < n, when d = 2. To see where the problem appears, consider n = 2 and j = 1: (2) f1 (t1 , x1 , r, z, t, x) = G(t − t1 , x − x1 )G(t1 − r, x1 − z)1{r 0. Regarding the other term, using the arguments in the proof of Lemma 3.2 it can be bounded, up to some constant, by Z tZ f (xj − x0j )|G(r + h − tj−1 , z − xj−1 ) − G(r − tj−1 , z − xj−1 )| 0 R2 × |G(r + h − tj−1 , z − x0j−1 ) − G(r − tj−1 , z − x0j−1 )|dxj−1 dx0j−1 dtj−1 Z tZ = f (xj − x0j )|G(r + h − tj−1 , xj−1 ) − G(r − tj−1 , xj−1 )| 0 R2 × |G(r + h − tj−1 , x0j−1 ) − G(r − tj−1 , x0j−1 )|dxj−1 dx0j−1 dtj−1 , (3.16) where we have applied a change of variables in order to get rid of z . Finally, in (3.16) we apply the bounded convergence theorem thanks to the continuity of the map t 7→ G(t, x) for fixed x and taking into account that, assuming 0 < h < 1, |G(r + h − tj−1 , xj−1 ) − G(r − tj−1 , xj−1 )| ≤ G(r + h − tj−1 , xj−1 ) ≤ G(T, xj−1 ), for some T > 0. We now deal with the norm in H⊗(n−1) of the term involving B . In this case, this norm can be directly written as a product of two terms, one of which is an integral on 2 i−1 j−1  {t1 < . . . < tj−1 < r} × R2(j−1) that is bounded by eM r M KM , for all M > 0. The other term is given by Z Z Γn−j t f (xj − x0j ) · · · f (xn−1 − x0n−1 )|G(tj − r − h, xj − z)−G(tj − r, xj − z)| [0,t]n−j (R2 )n−j × G(tj+1 − tj , xj+1 − xj ) · · · G(t − tn−1 , x − xn−1 )|G(tj −r − h, x0j − z) − G(tj − r, x0j − z)| × G(tj+1 − tj , x0j+1 − x0j ) · · · G(t − tn−1 , x − x0n−1 ) dxj · · · dxn−1 dx0j · · · dx0n−1 dtj · · · dtn−1 . EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 15/43 Existence of density for the stochastic wave equation In order to get rid of the dependence on z , we make a series of changes of variable and move z from the terms involving xj and x0j to those involving xn−1 and x0n−1 . Another change of variable allows us eventually remove the variable z . In the remaining term, one can easily apply the bounded convergence theorem, because the differences in absolute value can be bounded by a constant. This would let us conclude the proof of item (i). (n) Step 2. (The second term) Note that T 0 (r, z) = n≥1 An (r, z) = G(t − r, x − z)F (r, z), P where X X F (r, z) = nIn−1 (fn−1 (·, r, z)) = (n + 1)Jn (r, z), n≥1 n≥0 and Jn (r, z) = In (fn (·, r, z)). Since the map (r, z) 7→ G(t − r, x − z) is measurable on [0, t] × R, it suffices to show that F has a measurable modification. This will follow by Proposition B.1.a) (Appendix B), once we show that F is Lp (Ω)-continuous, for any p ≥ 2. For this, we use the same argument as in the proof of Theorem 7.1 of [3]. We denote by k · kp the Lp (Ω)-norm. Using the equivalence of norms k · kp for random variables in the same Wiener chaos space (see last line of p.62 of [26]) and (3.7), for any r ∈ [0, t] and z ∈ Rd , we have:  1/2 n/2 n/2 1 kJn (r, z)kp ≤ (p − 1) kJn (r, z)k2 = (p − 1) αn (r) n!  n/2 2Γr ≤ (p − 1)n/2 eM r/2 KM . M Using the fact that n + 1 ≤ en and choosing M sufficiently large, it follows that  n/2 X M t/2 X n n/2 2Γt (n + 1) sup kJn (r, z)kp ≤ e e (p − 1) KM < ∞. (r,z)∈[0,t]×Rd M n≥0 n≥0 Pn Hence, the sequence {Fn (r, z) = k=0 (k +1)Jk (r, z); n ≥ 1} converges to F (r, z) in Lp (Ω), uniformly in (r, z) ∈ [0, t] × Rd . Note that Fn is Lp (Ω)-continuous for any n, since Jn is Lp (Ω)-continuous for any n (by Lemma 7.1 of [3]). Therefore, F is Lp (Ω)-continuous. Remark 3.10. For the remaining part of the article, we fix t > 0 and x ∈ R, and we work with the measurable modification given by Theorem 3.9, which will be denoted also by {Dr,z u(t, x); r ∈ [0, t], z ∈ R}. We aim to prove that for any fixed r ∈ [0, t], the Malliavin derivative Dr,· u(t, x) satisfies an equation in L2 (Ω; L2 (R)). For this, we first need to prove that Dr,· u(t, x) belongs to L2 (R) a.s. This fact is established by the following result. Theorem 3.11. For any T > 0, Z sup sup E |Dr,z u(t, x)|2 dz ≤ CT , (t,x)∈[0,T ]×R r∈[0,t] R where CT > 0 is a constant depending on T . Proof. Using the orthogonality of Wiener chaos spaces, we have Z Z X 2 2 E |Dr,z u(t, x)| dz = E nIn−1 (fen (·, r, z, t, x)) dz R R n≥1 XZ 2 = E nIn−1 (fen (·, r, z, t, x)) dz. (3.17) n≥1 R EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 16/43 Existence of density for the stochastic wave equation (n) The proof of Lemma 3.5 shows that fj (·, r, z, t, x) ∈ H⊗(n−1) and 2 E nIn−1 (fen (·, r, z, t, x)) n X 1 (n) ≤n k(n − 1)!hj (·, r, z, t, x))k2H⊗(n−1) j=1 (n − 1)! n Z Z n−1 X 1 Y ≤n Γn−1 (n − 1)! f (xi − yi ) j=1 (n − 1)! t 0 0, t ∈ [0, T ], x ∈ R and r ∈ [0, t], Z X n−1 2 E |Dr,z u(t, x)|2 dz ≤ πT e2M T e2 e2 ΓT KM . R M n≥1 Choose M = MT > 2 large enough such that e2 ΓT KMT < 1/2. Then Z X  1 n−1 2 2MT T 2 E |Dr,· u(t, x)| dz ≤ πT e e . (3.24) R 2 n≥1 The conclusion follows with CT = 2πT e2MT T e2 . Remark 3.12. Fix 0 < r < t and x ∈ R. Theorem 3.11 shows that with probability 1, the functions z 7→ In−1 (fen (·, r, z, t, x)) and z 7→ Dr,z u(t, x) belong to L2 (R). We denote by Fz [In−1 (fen (·, r, ·, t, x))] and Fz [Dr,· u(t, x)] the Fourier transforms of these functions. Similarly to the proof of relation (4.18) below, it can be proved that for almost all ξ ∈ R,   X h i X  Fz Dr,· u(t, x) (ξ) = nFz In−1 (fen (·, r, ·, t, x)) (ξ) = nIn−1 Fz fen (·, r, ·, t, x)(ξ) , n≥1 n≥1 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 18/43 Existence of density for the stochastic wave equation which gives the chaos expansion of the Fourier transform of Dr,· u(t, x). Hence, Z EkDr,· u(t, x)k20 = E |Fz [Dr,· u(t, x)](ξ)|2 µ(dξ) R X Z 2 = n (n − 1)! kFz fen (·, r, ·, t, x)(ξ)k2H⊗(n−1) µ(dξ) n≥1 R X = n2 (n − 1)! kfen (·, r, ·, t, x)k2H⊗(n−1) ⊗P0 , n≥1 where for the last equality we expressed the norm in H⊗(n−1) using the Fourier transform with respect to the variables (t1 , x1 ), . . . , (tn−1 , xn−1 ). Similarly to the proof of Theorem 4.6 below, it can be shown that EkDr,· u(t, x)k20 < ∞, and hence Dr,· u(t, x) ∈ P0 a.s. Moreover, sup sup EkDr,· u(t, x)k20 < ∞. (t,x)∈[0,T ]×R r∈[0,t] These facts will not be used in the present article. 4 The second order Malliavin derivative In this section, we study the second order Malliavin derivative of the solution. Note that X  2  D(θ,w),(r,z) u(t, x) = Dθ,w Dr,z u(t, x) = Dθ,w nIn−1 (fen (·, r, z, t, x)) n≥1 X = n(n − 1)In−2 (fen (·, θ, w, r, z, t, x)). (4.1) n≥2 We will show below that fen (·, θ, w, r, z, t, x) ∈ H⊗(n−2) and the series above converges in L2 (Ω). First, note that for any j = 1, . . . , n n (n) 1 X (n) hj (·, θ, w, r, z, t, x) = hij (·, θ, w, r, z, t, x), (4.2) n−1 i=1,i6=j where (n) hij (t1 , x1 , . . . , tn−2 , xn−2 , θ, w, r, z, t, x) = 1 X fn (tρ(1) , xρ(1) , . . . , tρ(i−1) , xρ(i−1) , θ, w, tρ(i) , xρ(i) , . . . , tρ(j−1) , xρ(j−1) , r, z, (n − 2)! ρ∈Sn−2 tρ(j) , xρ(j) , . . . , tρ(n−2) , xρ(n−2) , t, x). (n) In the definition of hij (·, θ, w, r, z, t, x), (θ, w) is on position i and (r, z) is on position j , as arguments of the function fn (·, t, x). Since the function fn (·, t, x) contains the indicator (n) of the set {0 < t1 < . . . < tn < t}, this means that if θ < r , then hij (·, θ, w, r, z, t, x) = 0 (n) for all i > j ; on the other hand, if θ > r , then hij (·, θ, w, r, z, t, x) = 0 for all i < j . (n) For θ < r and i < j , hij (·, θ, w, r, z, t, x) is the symmetrization of the function (n) fij (·, θ, w, r, z, t, x) defined by: (n) fij (t1 , x1 , . . . , tn−2 , xn−2 , θ, w, r, z, t, x) = fn (t1 , x1 , . . . , ti−1 , xi−1 , θ, w, ti , xi , . . . , tj−1 , xj−1 , r, z, tj , xj , . . . , tn−2 , xn−2 , t, x) = G(t − tn−2 , x − xn−2 ) · · · G(ti − θ, xi − w)G(θ − ti−1 , w − xi−1 ) · · · G(tj − r, xj − z) G(r−xj−1 , z−xj−1 ) · · · G(t2 − t1 , x2 − x1 )1{0 j . So the previous sum has only 2 terms. Lemma 4.1. For any (r, z), (θ, w) ∈ [0, t] × R, fen (·, θ, w, r, z, t, x) ∈ H⊗(n−2) . Moreover, the series X n(n − 1)In−2 (fen (·, θ, w, r, z, t, x)) converges in L2 (Ω), n≥2 uniformly in (r, z), (θ, w) ∈ [0, t] × R. Proof. For the first statement, we fix (r, z), (θ, w) ∈ [0, t] × R. Say θ < r . To prove that (n) fen (·, θ, w, r, z, t, x) ∈ H⊗(n−2) , it suffices to show that fij (·, θ, w, r, z, t, x) ∈ H⊗(n−2) for any 1 ≤ i < j ≤ n. We proceed as in the proof of Lemma 3.5. In the computation of the (n) squared H⊗(n−2) -norm of fij (·, θ, w, r, z, t, x), we bound by 1/2 the terms G(tj − r, xj − z), G(ti − θ, xi − w), G(sj − r, yj − z), G(si − θ, yi − w). We obtain that: (n) kfij (·, θ, w, r, z, t, x)k2H⊗(n−2) ≤ 1 kfi−1 (·, θ, w)k2H⊗(i−1) kfj−i (·, r − θ, z)k2H⊗(j−i) kfn−j−1 (·, t − r, x)k2H⊗(n−j−1) < ∞. 16 To prove the convergence of the series, we let n (n) X Bn (θ, w, r, z, t, x) := n(n − 1)In−2 (fen (·, θ, w, r, z, t, x)) = In−2 (hij (·, θ, w, r, z, t, x)). i,j=1,i6=j PN PN Using the inequality ( i=1 ai )2 ≤ N i=1 a2i , we see that n (n) X E|Bn (θ, w, r, z, t, x)|2 ≤ n(n − 1) E|In−2 (hij (·, θ, w, r, z, t, x))|2 i,j=1,i6=j n X 1 (n) = n(n − 1) k(n − 2)!hij (·, θ, w, r, z, t, x))k2H⊗(n−2) . (n − 2)! i,j=1,i6=j (4.4) Proceeding as in the proof of Lemma 3.5, we see that for any i < j and θ < r , (n) k(n − 2)!hij (·, θ, w, r, z, t, x))k2H⊗(n−2) ≤ (4.5) Z Z n−2 Y Γn−2 t (n − 2)! f (xk − yk ) t1 <... 0. A similar bound holds for j < i and r < θ . Hence, for any i, j = 1, . . . , n with i 6= j ,  n−2 (n) 1 n−2 2 n−2 k(n − 2)!hij (·, θ, w, r, z, t, x))k2H⊗(n−2) ≤ Γ (n − 2)! eM t KM . 16 t M Coming back to (4.4), we obtain that for any (θ, w), (r, z) ∈ [0, t] ∈ R  n−2 1 2 E|Bn (θ, w, r, z, t, x)| ≤ [n(n − 1)]2 Γn−2 2 t eM t n−2 KM . 16 M By choosing M = Mt large enough, and using the orthogonality of the Wiener chaos spaces, we see that m 2 X sup E Bk (θ, w, r, z, t, x) → 0 as n, m → ∞. (θ,w),(r,z)∈[0,t]×R k=n Theorem 4.2. For any T > 0, Z Z sup sup E 2 |D(θ,w),(r,z) u(t, x)|2 dwdz ≤ CT0 , (t,x)∈[0,T ]×R θ,r∈[0,t] R R where CT0 > 0 is a constant depending on T . Proof. Fix θ < r < t. Using (4.1), Fubini’s theorem and orthogonality, we have: Z Z XZ Z 2 2 E |D(θ,w),(r,z) u(t, x)| dwdz = E n(n − 1)In−2 (fen (·, θ, w, r, z, t, x)) dwdz. R R n≥2 R R (4.7) To estimate the second moment of n(n − 1)In−2 (fen (·, θ, w, r, z, t, x)), we will use (4.4). Let 1 ≤ i < j ≤ n. Using (4.5) and Fubini’s theorem, Z Z (n) k(n − 2)!hij (·, θ, w, r, z, t, x))k2H⊗(n−2) dwdz R R Z ≤ Γn−2 t (n − 2)! t1 <... 0. Using (4.4), it follows that Z Z E |n(n − 1)In−2 (fen (·, θ, w, r, z, t, x))|2 dwdz R R  n−2 2 ≤ [n(n − 1)]2 Γtn−2 (πT )2 e3M t n−2 KM . (4.8) M Coming back to (4.7), it follows that Z Z  n−2 X 2 E 2 |D(θ,w),(r,z) u(t, x)|2 dwdz ≤ (πT )2 e3M t [n(n − 1)]2 Γtn−2 n−2 KM . R R M n≥2 The conclusion follows as in the last part of the proof of Theorem 3.11. Remark 4.3. Fix 0 < r < t and x ∈ R. Taking the integral dθ on [0, t] in (4.8), we infer that with probability 1, for almost all z ∈ R, the map (θ, w) 7→ In−2 ((fen (·, θ, w, r, z, t, x))) belongs to L2 (R2 ). We denote by Fθ,w [In−2 ((fen (·, ∗, r, z, t, x)))] its Fourier transform with respect to the missing (θ, w)-variable, denoted by ∗. Similarly, by Theorem 4.2, with probability 1, for almost all z ∈ R, the map (θ, w) 7→ D(θ,w),(r,z) u(t, x) belongs to L2 (R2 ). We denote by Fθ,w [D∗,(r,z) 2 2 u(t, x)] the Fourier trans- form of this function. We claim that: X 2   Fθ,w [D∗,(r,z) u(t, x)] = n(n − 1)Fθ,w In−2 (fen (·, ∗, r, z, t, x)) . (4.9) n≥2 Indeed, using the proof of Theorem 4.2 one verifies that the series on the right-hand side of (4.9) converges in L2 (Ω; L2 (R)); for this, it suffices to prove that X   n(n − 1)kFθ,w In−2 (fen (·, ∗, r, z, t, x)) kL2 (Ω;L2 (R)) < ∞. n≥2 Next, fix N > 2 and observe that N hX i XN   Fθ,w n(n − 1)In−2 (fen (·, ∗, r, z, t, x)) = n(n − 1)Fθ,w In−2 (fen (·, ∗, r, z, t, x)) . n=2 n=2 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 22/43 Existence of density for the stochastic wave equation In order to conclude that (4.9) holds, it suffices to prove that the left-hand side above 2 converges to Fθ,w [D∗,(r,z) u(t, x)], as N → ∞. This follows by applying Plancherel’s theorem and using the arguments in the proof of Lemma 4.1. Lemma 4.4. For any t > 0, x ∈ R and (t1 , x1 ), . . . , (tn−2 , xn−2 ), (r, z) ∈ [0, t] × R, the function (θ, w) 7→ fen (t1 , x1 , . . . , tn−2 , xn−2 , θ, w, r, z, t, x) is in L2 (R2 ). If we denote by Fθ,w fen (t1 , x1 , . . . , tn−2 , xn−2 , ∗, , r, z, t, x) the Fourier transform of this function, then Fθ,w fen (·, ∗, , r, z, t, x)(τ, ξ) ∈ H⊗(n−2) (4.10) for almost all z ∈ R and for almost all (τ, ξ) ∈ R2 . (n) Proof. By (4.3), it is enough to prove that (θ, w) 7→ fij (t1 , x1 , . . . , tn−2 , xn−2 , θ, w, r, z, t, x) is in L2 (R2 ) for any i, j = 1, . . . , n with i 6= j . This is clear since, if for instance i < j , Z tZ G(tj − θ, xj − w)G(θ − tj−1 , w − xj−1 )dwdθ < ∞. 0 R RtR We now prove (4.10). By (4.8), 0 R E|In−2 (fen (·, θ, w, r, z, t, x))|2 dwdθ < ∞, for almost all z ∈ R. This integral is equal to (n − 2)! multiplied by the following quantity: Z Z kfen (·, θ, w, r, z, t, x)k2H⊗(n−2) dwdθ R R Z Z = |Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, θ, w, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 )|2 R2 R2(n−2) n−2 Y n−2 Y × ν(dτi ) µi (dξi )dwdθ i=1 i=1 where Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, θ, w, r, z, t, x) denotes the Fourier transform with respect to the variables t1 , x1 , . . . , tn−2 , xn−2 . We apply Fubini’s theorem, followed by Plancherel’s theorem: Z |Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, θ, w, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 )|2 dwdθ R2 Z 1 = |Fθ,w [Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, ∗, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 )](τ, ξ)|2 dτ dξ, (2π)2 R2 where ∗ denotes missing (θ, w)-variable and Fθ,w is the Fourier transform with respect to this variable. Since fen (·, ∗, r, z, t, x) ∈ L1 (Rn−1 ), we can switch the order of the variables (t1 , x1 ), . . . , (tn−2 , xn−2 ) and (θ, w) when calculating the Fourier transform, i.e. Fθ,w [Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, ∗, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 )](τ, ξ) = Ft1 ,x1 ,...,tn−2 ,xn−2 [Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ)](τ1 , ξ1 , . . . , τn−2 , ξn−2 ) (4.11) = F fen (·, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 , τ, ξ). (4.12) Applying Fubini’s theorem again, we conclude that for almost all z ∈ R, Z Z |Ft1 ,x1 ,...,tn−2 ,xn−2 [Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ)](τ1 , ξ1 , . . . , τn−2 , ξn−2 )|2 R2 R2(n−2) n−2 Y n−2 Y × ν(dτi ) µi (dξi )dτ dξ < ∞. i=1 i=1 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 23/43 Existence of density for the stochastic wave equation Hence, for almost all z ∈ R and for almost all (τ, ξ) ∈ R2 , the integral kFθ,w f˜n (·, ∗, r, z, t, x) (τ, ξ) k2H⊗(n−2)    (4.13) Z = |Ft1 ,x1 ,...,tn−2 ,xn−2 [Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ)](τ1 , ξ1 , . . . , τn−2 , ξn−2 )|2 R2(n−2) n−2 Y n−2 Y × ν(dτi ) µi (dξi ) (4.14) i=1 i=1 is finite. Relation (4.10) follows using a characterization of the space H⊗(n−2) (in terms of the integral of the squared Fourier transform) similar to the one given by Theorem 2.3 for space H. The next result shows that in our context, we can interchange the order of the Fourier transform and the multiple integral, a fact which is highly non-trivial  in the general case. Lemma 4.4 shows that the integral In−2 Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ) is well-defined. Lemma 4.5. With probability 1, for all r ∈ [0, t] and for almost all z ∈ R and (τ, ξ) ∈ R2 ,  Fθ,w [In−2 (fen (·, ∗, r, z, t, x))](τ, ξ) = In−2 Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ) . Proof. Let h1 (τ, ξ) = Fθ,w [In−2 (fen (·, ∗, r, z, t, x))](τ, ξ) and h2 (τ, ξ) = In−2 (Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ)). For an arbitrary function g ∈ L2 (R2 ), we denote Z Z Xg = h1 (τ, ξ)Fg(τ, ξ)dτ dξ and Yg = h2 (τ, ξ)Fg(τ, ξ)dτ dξ. R2 R2 We will show that for any g ∈ L2 (R2 ) fixed, E(GXg ) = E(GYg ) for all G ∈ L2 (Ω), (4.15) which implies that Xg = Yg a.s., the negligible set depending on g . Since L2 (R2 ) is separable, there exists a countable dense set S in L2 (R2 ). Hence, there exists an event Ω0 of probability 1 on which Xg = Yg for any g ∈ S . Since {Fg; g ∈ S} is dense in L2 (R2 ), it follows that on Ω0 , hh1 , φiL2 (R2 ) = hh2 , φiL2 (R2 ) for all φ ∈ L2 (R2 ), and hence h1 = h2 a.e. First, we prove that Xg ∈ Hn−2 . (A similar argument shows that Yg ∈ H.) For this, it suffices to prove that E[Im (hm )Xg ] = 0, for all m 6= n − 2 and all symmetric hm ∈ H⊗m . By Plancherel’s theorem, Z Xg = (2π) In−2 (fen (·, θ, w, r, z, t, x))g(θ, w)dwdθ, (4.16) R2 and hence, Z  E[Im (hm )Xg ] = (2π)E Im (hm )In−2 (fn (·, θ, w, r, z, t, x))g(θ, w)dwdθ . e (4.17) R2 Now, we would like to apply Fubini theorem, for which we need to check that Z E |Im (hm )In−2 (fen (·, θ, w, r, z, t, x))g(θ, w)|dwdθ < ∞. R2 But we have, by Fubini theorem (because the integrand is now non-negative) and EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 24/43 Existence of density for the stochastic wave equation Cauchy-Schwarz inequality (applied twice), Z E |Im (hm )In−2 (fen (·, θ, w, r, z, t, x))g(θ, w)|dwdθ R2 Z = E[|Im (hm )In−2 (fen (·, θ, w, r, z, t, x))|] |g(θ, w)|dwdθ R2 Z 1 1 ≤ E[Im (hm )2 ] 2 E[In−2 (fen (·, θ, w, r, z, t, x))2 ] 2 |g(θ, w)|dwdθ R2 Z 1/2 1 = (m!) khm kH⊗m E[In−2 (fen (·, θ, w, r, z, t, x))2 ] 2 |g(θ, w)|dwdθ R2 Z  12 ≤ (m!)1/2 khm kH⊗m E[In−2 (fen (·, θ, w, r, z, t, x))2 ]dwdθ kgkL2 (R2 ) < ∞, R2 where we have applied (4.8). By Fubini’s theorem, we obtain that for any m 6= n − 2, Z E[Im (hm )Xg ] = E[Im (hm )In−2 (fen (·, θ, w, r, z, t, x))]g(θ, w)dwdθ = 0. R2 We now prove (4.15). Since both Xg and Yg are in Hn−2 , it is enough to assume that G = In−2 (gn−2 ) for some symmetric gn−2 ∈ H⊗(n−2) . Using Fubini’s theorem and (4.16), Z h i E(GXg ) = (2π) E In−2 (gn−2 )In−2 (fen (·, θ, w, r, z, t, x)) g(θ, w)dθdw 2 ZR = (2π) (n − 2)!hgn−2 , fen (·, θ, w, r, z, t, x)iH⊗(n−2) g(θ, w)dθdw R2 and E(GYg ) Z = E[In−2 (gn−2 )In−2 (Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ))]Fg(τ, ξ)dτ dξ R2 Z = (n − 2)!hgn−2 , Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ))iH⊗(n−2) Fg(τ, ξ)dτ dξ R2 Z Z = (n − 2)! Ft1 ,x1 ,...,tn−2 ,xn−2 (Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ))(τ1 , ξ1 , . . . , τn−2 , ξn−2 ) R2 R2(n−2) n−2 Y n−2 Y Fgn−2 (τ1 , ξ1 , . . . , τn−2 , ξn−2 ) Fg(τ, ξ) ν(dτi ) µ(dξi )dτ dξ. i=1 i=1 We use (4.11), followed by Fubini’s theorem. For the dτ dξ integral, we use Plancherel’s theorem to eliminate the Fourier transform in the (θ, w)-variable. We obtain: Z  Z E(GYg ) = (n − 2)! 2π Ft1 ,x1 ,...,tn−2 ,xn−2 fen (·, θ, w, r, z, t, x)(τ1 , ξ1 , . . . , τn−2 , ξn−2 ) R2(n−2) R2  n−2 Y n−2 Y g(θ, w)dθdw Fgn−2 (τ1 , ξ1 , . . . , τn−2 , ξn−2 ) ν(dτi ) µ(dξi ) i=1 i=1 Z = (n − 2)!(2π) hgn−2 , fen (·, θ, w, r, z, t, x)iH⊗(n−2) g(θ, w)dθdw. R2 This proves (4.15). In summary, relation (4.9) and Lemma 4.5 show that for all r ∈ [0, t] and almost all EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 25/43 Existence of density for the stochastic wave equation z ∈ R and for almost all (τ, ξ) ∈ R2 ,  2  X h i Fθ,w D∗,(r,z) u(t, x) (τ, ξ) = n(n − 1)Fθ,w In−2 (fen (·, ∗, r, z, t, x)) (τ, ξ) n≥2 X  = n(n − 1)In−2 Fθ,w fen (·, ∗, r, z, t, x)(τ, ξ) , (4.18) n≥2 2 which gives the chaos expansion of the Fourier transform of D∗,(r,z) u(t, x). Theorem 4.6. Let Dr2 u(t, x) denote D∗,(r,·) 2 , where ∗ is the missing (θ, w)-variable and · is the missing z -variable. For any T > 0, sup sup EkDr2 u(t, x)k2H⊗L2 (R) ≤ CT00 , (t,x)∈[0,T ]×R r∈[0,t] where CT00 > 0 is a constant depending on T . Proof. By (4.18), we have: Z EkDr2 u(t, x)k2H⊗L2 (R) = E 2 kD∗,(r,z) u(t, x)k2H dz R Z Z  2 u(t, x) (τ, ξ)|2 ν(dτ )µ(dξ)dz  =E |Fθ,w D∗,(r,z) R R 2 X Z Z n2 (n − 1)2 (n − 2)! kFθ,w fen (·, ∗, r, z, t, x)(τ, ξ) k2 ⊗(n−2) ν(dτ )µ(dξ)dz.  = H n≥2 R R2  Using relation (4.13) for expressing kFθ,w fen (·, ∗, r, z, t, x)(τ, ξ) k2H⊗(n−2) , followed by (4.12), we see that Z kFθ,w fen (·, ∗, r, z, t, x)(τ, ξ) k2H⊗(n−2) ν(dτ )µ(dξ) = kfen (·, r, z, t, x)k2H⊗(n−1) .  R2 Hence, X Z EkDr2 u(t, x)k2H⊗L2 (R) ≤ 2 2 n (n − 1) (n − 2)! kfen (·, r, z, t, x)k2H⊗(n−1) dz. n≥2 R By (3.23), for any t ∈ [0, T ], x ∈ R, r ∈ [0, t] and M > 0, Z  n−1 2 2 n (n − 1)! kfen (·, r, z, t, x))k2H⊗(n−1) dz ≤ πtn2 Γn−1 t e2M t n−1 KM , R M and hence,  n−1 X 2 EkDr2 u(t, x)k2H⊗L2 (R) ≤ πte2M t n2 (n − 1)Γn−1 t n−1 KM . M n≥2 We use the fact that n ≤ en for all n ≥ 1. We conclude that for any M > 0, t ∈ [0, T ], x ∈ R and r ∈ [0, t], X n−1 2 EkDr2 u(t, x)k2H⊗L2 (R) ≤ πT e 2M T 3 e 3 e ΓT KM . M n≥2 Choose M = MT0 > 2 large enough such that e3 ΓT KMT0 < 1/2. Then 0 X  1 n−1 EkDr2 u(t, x)k2H⊗L2 (R) ≤ πT e2MT T e3 . (4.19) 2 n≥1 0 The conclusion follows with CT00 = 2πT e2MT T e3 . EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 26/43 Existence of density for the stochastic wave equation 5 Equation satisfied by Dr,· u(t, x) In this section, we show that Dr,· u(t, x) satisfies an integral equation. By Theorem 6.7 of [1], we know that Malliavin derivative Du of the solution satisfies the following equation in L2 (Ω; H): Z tZ D· u(t, x) = G(t − ·, x − ·)u(·) + U (s, y)W (δ ∗ s, δ ∗ y) (5.1) 0 R where U = U (t,x) ∈ L2 (Ω; H ⊗ H) belongs to the domain of the δ ∗ and is given by: U ((r0 , z 0 ), (s, y)) = G(t − s, x − y)Dr0 ,z0 u(s, y). Here δ ∗ is the Skorohod integral with values in H. In what follows we will show that Dr,· u satisfies an equation similar to (5.1), but in L2 (Ω; L2 (R)) for fixed r ∈ [0, t]. Let δ be the L2 (R)-valued Skorohod integral defined in Section 6 of [1] with A = L (R). By Proposition 6.2 of [1], D1,2 (H ⊗ L2 (R)) ⊂ Dom δ . Let (un )n≥0 be the sequence 2 of Picard iterations given by: u0 (t, x) = 1 and n X un (t, x) = 1 + Ik (fk (·, t, x)). k=1 We fix t > 0, x ∈ Rd and r ∈ [0, t]. For any s ∈ [0, t], y ∈ Rd and z ∈ Rd , we let K (r) (s, y), z = 1[0,t] (s)G(t − s, x − y)Dr,z u(s, y),  Kn(r) (s, y), z = 1[0,t] (s)G(t − s, x − y)Dr,z un (s, y).  We denote Z Z φ(t) := G(s, y)G(s0 , y 0 )γ(s − s0 )f (y − y 0 )dydy 0 dsds0 = α1 (t), (5.2) [0,t]2 R2 where we recall that α1 (t) has been defined in (3.4). The following result is similar to Lemma 6.5 of [1]. (r) Lemma 5.1. For any r ∈ [0, t], K (r) ∈ D1,2 (H ⊗ L2 (R)) and Kn ∈ D1,2 (H ⊗ L2 (R)). (r) Hence, K (r) and Kn belong to Dom δ , for any r ∈ [0, t]. Proof. To prove that K (r) ∈ D1,2 (H⊗L2 (R)) it suffices to show that kK (r) k2D1,2 (H⊗L2 (R)) < ∞, where kK (r) k2D1,2 (H⊗L2 (R)) = EkK (r) k2H⊗L2 (R) + EkDK (r) k2H⊗H⊗L2 (R) . It suffices to show that both terms above are finite. Note that kK (r) kH⊗L2 (R) ≤ kK (r) k|H|⊗L2 (R) . Applying Cauchy-Schwarz inequality twice and Theorem 3.11, we have EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 27/43 Existence of density for the stochastic wave equation EkK (r) k2|H|⊗L2 (R) Z Z =E G(t − s, x − y)G(t − s0 , x − y 0 ) hDr,· u(s, y), Dr,· u(s0 , y 0 )iL2 (R) [0,t]2 R2 × γ(s − s0 )f (y − y 0 )dydy 0 dsds0 Z Z ≤E G(t − s, x − y)G(t − s0 , x − y 0 )kDr,· u(s, y)kL2 (R) kDr,· u(s0 , y 0 )kL2 (R) [0,t]2 R2 × γ(s − s0 )f (y − y 0 )dydy 0 dsds0 Z Z  1/2 ≤ G(t − s, x − y)G(t − s0 , x − y 0 ) EkDr,· u(s, y)k2L2 (R) [0,t]2 R2  1/2 × EkDr,· u(s0 , y 0 )k2L2 (R) γ(s − s0 )f (y − y 0 )dydy 0 dsds0 ≤ Ct φ(t) < ∞. (5.3) (r) A similar calculation shows that EkKn k2|H|⊗L2 (R) < Ct φ(t). For the second term, we proceed similarly. First, note that kDK (r) kH⊗H⊗L2 (R) ≤ kDK (r) k|H|⊗H⊗L2 (R) . Applying Cauchy-Schwarz inequality twice and Theorem 4.6, we have EkDK (r) k2|H|⊗H⊗L2 (R) Z Z =E G(t − s, x − y)G(t − s0 , x − y 0 ) hDr2 u(s, y), Dr2 u(s0 , y 0 )iH⊗L2 (R) [0,t]2 R2 × γ(s − s0 )f (y − y 0 )dydy 0 dsds0 Z Z ≤E G(t − s, x − y)G(t − s0 , x − y 0 )kDr2 u(s, y)kH⊗L2 (R) kDr2 u(s0 , y 0 )kH⊗L2 (R) [0,t]2 R2 × γ(s − s0 )f (y − y 0 )dydy 0 dsds0 Z Z  1/2 ≤ G(t − s, x − y)G(t − s0 , x − y 0 ) EkDr2 u(s, y)k2H⊗L2 (R) [0,t]2 R2  1/2 × EkDr2 u(s0 , y 0 )k2H⊗L2 (R) γ(s − s0 )f (y − y 0 )dydy 0 dsds0 ≤ Ct00 φ(t) < ∞. (5.4) (r) A similar calculation shows that EkDKn k2|H|⊗H⊗L2 (R) < ∞. We use the following more suggestive notation: Z tZ δ(K (r) ) = G(t − s, x − y)Dr,· u(s, y)W (δs, δy) 0 R Z tZ δ(Kn(r) ) = G(t − s, x − y)Dr,· un (s, y)W (δs, δy). 0 R By Lemma 5.1, these stochastic integrals are well-defined. The following result establishes a recursive relation involving the Malliavin derivative of the Picard iteration scheme (see Proposition 6.6 of [1] for a similar result). Proposition 5.2. For any r ∈ [0, t], the following equality holds in L2 (Ω; L2 (R)): Z tZ Dr,· un (t, x) = G(t − r, x − ·)un−1 (r, ·) + G(t − s, x − y)Dr,· un−1 (s, y)W (δs, δy). (5.5) 0 R Proof. For clarity, we split the proof into several steps. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 28/43 Existence of density for the stochastic wave equation Step 1. To prove (5.5), we will show (in Step 2 below) that for all ϕ ∈ L2 (R) fixed, the following equality holds in L2 (Ω): hϕ, Dr,· un (t, x)iL2 (R) = hϕ, G(t − r, x − ·)un−1 (r, ·)iL2 (R) Z tZ + G(t − s, x − y)hϕ, Dr,· un−1 (s, y)iL2 (R) W (δs, δy). (5.6) 0 R The integral on the right-hand side of (5.6) is a real-valued Skorohod integral. In this step of the proof, we show that this integral is well-defined. For this, let vn (s, y) = G(t − s, x − y)hϕ, Dr,· un−1 (s, y)iL2 (R) . Since D1,2 (H) ⊂ Dom δ , it is enough to prove that vn ∈ D1,2 (H), i.e. Ekvn k2H < ∞ and EkDvn k2H⊗H < ∞. By Cauchy-Schwarz inequality and Theorem 3.11, it follows that Ekvn k2H ≤ Ekvn k2|H| Z =E γ(s − s0 )f (y − y 0 )G(t − s, x − y)G(t − s0 , x − y 0 ) ([0,t]×R)2 × |hϕ, Dr,· un−1 (s, y)iL2 (R) | · |hϕ, Dr,· un−1 (s0 , y 0 )iL2 (R) |dydy 0 dsds0 ≤ kϕk2L2 (R) Ct φ(t) < ∞. We now prove that: EkDvn k2H⊗H < ∞. (5.7) Note that Z 2 Dθ,w vn (s, y) = G(t − s, x − y)ϕ(z)D(θ,w),(r,z) un−1 (s, y)dz. R Then, we have EkDvn k2H⊗H ≤ EkDvn k2|H|⊗H Z =E γ(s − s0 )f (y − y 0 )G(t − s, x − y)G(t − s0 , x − y 0 ) ([0,t]×R)2 Z Z  × 2 ϕ(z)D∗,(r,z) un−1 (s, y)dz, 2 ϕ(z)D∗,(r,z) un−1 (s0 , y 0 )dz dsds0 dydy 0 . R R H By Cauchy-Schwarz inequality, in order to have (5.7) it suffices to check that Z 2 2 sup E ϕ(z)D∗,(r,z) un−1 (s, y)dz < ∞. (s,y)∈[0,t]×R R H Observe that Z 2 2 E ϕ(z)D∗,(r,z) un−1 (s, y)dz R H Z Z  2 2 =E Fθ,w ϕ(z)D∗,(r,z) un−1 (s, y)dz (τ, ξ) ν(dτ )µ(dξ). R2 R One verifies that, with probability 1, for all r ∈ [0, t] and (s, y) ∈ [0, t] × R, the function 2 un−1 (s, y)dz belongs to L2 (R2 ) and R (θ, w) → R ϕ(z)D(θ,w),(r,z) Z  Z 2  2  Fθ,w ϕ(z)D∗,(r,z) un−1 (s, y)dz = ϕ(z)Fθ,w D∗,(r,z) un−1 (s, y) dz. R R EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 29/43 Existence of density for the stochastic wave equation Indeed, for the latter equality to be fulfilled, it suffices to show the equality of the respective inner products against an arbitrary element of L2 (R2 ). Thus, by Cauchy- Schwarz inequality and Fubini theorem, Z 2 2 E ϕ(z)D∗,(r,z) un−1 (s, y)dz R H Z Z 2  2  =E ϕ(z)Fθ,w D∗,(r,z) un−1 (s, y) (τ, ξ)dz ν(dτ )µ(dξ) R2 R Z Z 2  2  2 ≤ kϕkL2 (R) E Fθ,w D∗,(r,z) un−1 (s, y) (τ, ξ) dzν(dτ )µ(dξ) R2 R Z 2 2 = kϕkL2 (R) E kD∗,(r,z) un−1 (s, y)k2H dz R ≤ kϕk2L2 (R) sup sup EkDr2 un−1 (t, x)k2H⊗L2 (R) , (t,x)∈[0,t]×R r∈[0,t] and this is finite by Theorem 4.6. This concludes the proof of (5.7). Step 2. We now give the proof of (5.6). By (3.8) and (3.13), Dr,z un (t, x) = Pn Pn (n) k=1 Ak (r, z, t, x) with An (r, z, t, x) = j=1 Aj (r, z, t, x) and Z (n) Aj (r, z, t, x) = G(t − tn−1 , x − xn−1 ) . . . G(tj − r, xj − z) 0 1, then the measure (kDF k2H · P) ◦ F −1 is absolutely continuous with respect to the Lebesgue measure on R. In particular, for any Borel set B ⊂ R with Lebesgue measure zero, P(F ∈ B, kDF kH > 0) = 0. (6.1) Proof. Let ϕ ∈ Cb∞ (R) be arbitrary. By the chain rule for the Malliavin derivative (see Proposition 1.2.3 of [26]), ϕ(F ) ∈ D1,2 and D(ϕ(F )) = ϕ0 (F )DF . Taking the scalar product with DF , we obtain that hD(ϕ(F )), DF iH = ϕ0 (F )kDF k2H . EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 33/43 Existence of density for the stochastic wave equation Note that F ∈ D2,p implies that DF ∈ D1,p (H). (Recall that D1,p (H) is the completion of the space S(H) of “smooth” H-valued random variables F with respect to the norm 1/p kF kD1,p (H) = EkF kpH + EkDF kpH⊗H .) Since D1,2 (H) ⊂ Dom δ , it follows that DF ∈ Dom δ . By definition (3.2) of Dom δ , we have 1 E kDF k2H ϕ0 (F ) = E hDϕ(F ), DF iH ≤ c E|ϕ(F )|2 2 ≤ ckϕk∞ .   We have thus proved that, for all ϕ ∈ Cb∞ (R), Z ϕ0 (x) (kDF k2H · P) ◦ F −1 (dx) ≤ ckϕk∞ .  R By Lemma 2.1.1 of [26], the measure µ = (kDF k2H · P ) ◦ F −1 is absolutely continuous with respect to the Lebesgue measure. To prove the last statement, let B ⊂ R be a Borel set with Lebesgue measure zero. Then F −1 (B) kDF k2H dP = µ(B) = 0, which implies (6.1). R We continue with a result about existence of density of a truncated random variable. First note that if Γ is a Borel set in R such that 0 6∈ Γ, then the law of the truncated variable G = F 1{F ∈Γ} is a probability measure PG on Γ ∪ {0} given by PG (A) = P(F ∈ A) for any Borel set A ⊂ Γ, and PG ({0}) = P(G = 0) = P(F 6∈ Γ). Lemma 6.2. Let Γ be a Borel set in R such that 0 6∈ Γ. Let F ∈ D2,p for some p > 1 be such that kDF kH > 0 a.s. on {F ∈ Γ}. (6.2) Then, the restriction of the law of the variable G = F 1{F ∈Γ} to the set Γ is absolutely continuous with respect to the Lebesgue measure on Γ. Proof. Let B ⊂ R be a Borel set with zero Lebesgue measure. By (6.2), we have P(F ∈ Γ, kDF kH = 0) = 0. (6.3) By (6.1) and (6.3), we obtain: P(F ∈ B, F ∈ Γ) = P(F ∈ B, F ∈ Γ, kDF kH > 0) + P(F ∈ B, F ∈ Γ, kDF kH = 0) = 0. Lemma 6.3. Let (Γm )m≥1 be a sequence of open sets in R such that 0 6∈ Γm and Γm ⊂ Γm+1 , for all m ≥ 1. Let Γ = ∪m≥1 Γm . Let F ∈ D2,p for some p > 1 be such that, for all m ≥ 1, kDF kH > 0 a.s. on {F ∈ Γm }. Then, the restriction of the law of the variable F 1{F ∈Γ} to the set Γ is absolutely continuous with respect to the Lebesgue measure on Γ. Proof. Let B ⊂ R be a Borel set with zero Lebesgue measure. By Lemma 6.2, P(F ∈ B, F ∈ Γm ) = 0 for all m ≥ 1. By taking m → ∞, we infer that P(F ∈ B, F ∈ Γ) = 0. In the proof of Theorem 1.2, we will use the following notation: Z tZ Z tZ 2 1 1 2 ψ(t) := G(r, z) dzdr = 1{|z| 0, φ(t) ≤ Γt ψ0 (t). 1 2 Proof. By Cauchy-Schwarz inequality and the inequality ab ≤ 2 (a + b2 ), 1  hG(s, ·), G(s0 , ·)i0 ≤ kG(s, ·)k0 kG(s0 , ·)k0 ≤ kG(s, ·)k20 + kG(s0 , ·)k20 . 2 By symmetry, it follows that Z tZ t Z tZ t φ(t) = γ(s − s0 )hG(s, ·), G(s0 , ·)i0 dsds0 ≤ γ(s − s0 )kG(s, ·)k20 dsds0 . 0 0 0 0 The last integral is bounded by Γt ψ0 (t) by Lemma 4.3 of [3] (with n = 1). Proof of Theorem 1.2. We will apply Lemma 6.3 in the case F = u(t, x) and Γm = {v ∈ 1 1 R; |v| > m }. For any m ≥ 1, let Ωm := {|u(t, x)| > m }∩Ω e , where Ω e is an event of 2,p probability 1 which will be defined below. By Lemma 3.3 u(t, x) ∈ D for any p > 1. We will prove that kDu(t, x)kH > 0 a.s. on Ωm . In view of Corollary 2.4 (applied to S = Du(t, x)(ω) for fixed ω ), it is enough to prove that Z tZ |Dr,z u(t, x)|2 dzdr > 0 a.s. on Ωm . (6.6) 0 R By Remark 3.7 the map (r, z) → 7 Dr,z u(t, x)(ω) is measurable on RR+ × R, for any ω ∈ Ω. We use Remark 5.4. By Fubini’s theorem, (P × λ)(N ) = Ω λ(Nω )P(dω), where Nω := {r ∈ [0, t]; (ω, r) ∈ N } is the section of N at point ω ∈ Ω. Since (P × λ)(N ) = 0, λ(Nω ) = 0 for P-almost  all ω ∈ Ω. Say this happens on the event Ω of probability 1. e Let A = Ω × [0, t] \N . Note that the section of A at ω ∈ Ω is the set Aω := {r ∈ [0, t]; (ω, r) ∈ A} = {r ∈ [0, t]; (ω, r) 6∈ N } = [0, t] − Nω . Fix ω ∈ Ω e . For the remaining part of the proof, we use an argument similar to the proof of Theorem 5.2 of [27] (for the white noise in time). For any r ∈ Aω , equality (5.12) holds. Using the inequality ka + bk2L2 (R) ≥ 12 kak2L2 (R) − kbk2L2 (R) , we obtain: Z tZ 2 kDr· u(t, x)k2L2 (R) = G(t − r, x − ·)u(r, ·) + G(t − s, x − y)Dr· u(s, y)W (δs, δy) 0 R L2 (R) Z tZ 2 1 ≥ kG(t − r, x − ·)u(r, ·)k2L2 (R) − G(t − s, x − y)Dr· u(s, y)W (δs, δy) . 2 0 R L2 (R) Let δ ∈ (0, 1) be arbitrary. Taking the integral with respect to r on Aω (and using the fact that λ(Nω ) = 0), we obtain that on Ωe, Z t Z t kDr· u(t, x)k2L2 (R) dr ≥ kDr· u(t, x)k2L2 (R) dr 0 t−δ Z t 1 ≥ kG(t − r, x − ·)u(r, ·)k2L2 (R) dr − I(δ), (6.7) 2 t−δ EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 35/43 Existence of density for the stochastic wave equation where Z t Z tZ 2 I(δ) = G(t − s, x − y)Dr· u(s, y)W (δs, δy) dr t−δ 0 R L2 (R) Z t Z t Z 2 = G(t − s, x − y)Dr· u(s, y)W (δs, δy) dr. t−δ t−δ R L2 (R) The second equality is due to the fact that Dr· u(s, y) = 0 if r > s, and so we must have s ≥ r ≥ t − δ. On the event Ωm , Z t Z Z t Z 1 |G(t − r, x − z)u(r, z)|2 dzdr ≥ G(t − r, x − z)2 dzdr − J(δ) t−δ R m2 t−δ R 1 1 2 = 2 ψ(δ) − J(δ) = δ − J(δ), m 4m2 where Z t Z G2 (t − r, x − z) u(t, x)2 − u(r, z)2 dzdr.  J(δ) = t−δ R So, by (6.7), we have that Z tZ 1 2 1 |Dr,z u(t, x)|2 dzdr ≥ δ − J(δ) − I(δ). (6.8) 0 R 8m2 2 Let us now estimate the first moment of J(δ) and I(δ), respectively. To start with, we have Z t Z G2 (t − r, x − z)E |u(t, x)2 − u(r, z)2 | dzdr,   E[|J(δ)|] ≤ t−δ R and we have E |u(t, x)2 − u(r, z)2 | = E |u(t, x) + u(r, z)| × |u(t, x) − u(r, z)|      1  1 ≤ E |u(t, x) + u(r, z)|2 2 E |u(t, x) − u(r, z)|2 2    1  1 E |u(s, y)|2 2 E |u(t, x) − u(r, z)|2 2   ≤2 sup (s,y)∈[0,t]×R  1 = 2Ct∗ E |u(t, x) − u(r, z)|2 2 ,  using the fact that Ct∗ := sup(s,y)∈[0,t]×R E[|u(s, y)|2 ] < ∞, by Theorem 7.1 of [3]. Thus, Z t Z  1 E[|J(δ)|] ≤ 2Ct∗ G2 (t − r, x − z) E |u(t, x) − u(r, z)|2 2 dzdr  t−δ R Z t Z  1 ≤ 2Ct∗ G2 (t − r, x − z) E |u(t, x) − u(s, y)|2 2 dzdr  sup sup t−δ R t−δ δ − 0 R n 2 8m2 n  −1  1 2 1 1  ≤ 2 δ − E[I(δ)] + E[|J(δ)|] 8m n 2  −1  1 2 1 1  ≤ 2 δ − δ 2 c0 (Ct + Ct00 )Γδ + Ct∗ gt,x (δ) . 8m n 4 Taking n → ∞, one gets Z t Z    −1  1 2 1  P |Dr,z u(t, x)|2 dzdr = 0 ∩ Ωm ≤ 2 δ δ 2 c0 (Ct +Ct00 )Γδ + Ct∗ gt,x (δ) . 0 R 8m 4 Next, we take δ → 0. Since limδ→0 Γδ = 0 and limδ→0 gt,x (δ) = 0, we obtain: Z t Z   2 P |Dr,z u(t, x)| dzdr = 0 ∩ Ωm = 0. 0 R This concludes the proof of (6.6) and the proof of Theorem 1.2. A A Parseval-type identity In this section, we give the Parseval-type identity which is used in the proof of Lemma 3.1. We begin by recalling a remarkable result of [21], and comment on a small correction of a related result from the same paper. Let f : Rd → [0, ∞] be a kernel of positive type, i.e. f is locally integrable and its Fourier transform in S 0 (Rd ) is a function g which is non-negative almost everywhere. In addition, we suppose that f is continuous, symmetric and f (x) < ∞ if and only if x 6= 0. By Lemma 5.6 of [21], for any Borel probability measure µ on Rd , we have: Z Z Z 1 f (x − y)µ(dx)µ(dy) = |Fµ(ξ)|2 g(ξ)dξ =: Ef (µ). Rd Rd (2π)d Rd In particular, if µ(dx) = ϕ(x)dx where ϕ is a density function on Rd , Z Z Z 1 f (x − y)ϕ(x)ϕ(y)dxdy = |Fϕ(ξ)|2 g(ξ)dξ =: Ef (ϕ). (A.1) Rd Rd (2π)d Rd It follows that relation (A.1) holds for any non-negative function ϕ ∈ L1 (Rd ). Relation (5.37) of [21] says that Z Z Z 1 f (x − y)µ(dx)ν(dy) = Fµ(ξ)Fν(ξ)g(ξ)dξ (A.2) Rd Rd (2π)d Rd for any Borel probability measures µ and ν on Rd . When ν = δ0 and µ = δx for x ∈ R arbitrary, this relation becomes Z 1 f (x) = e−iξ·x g(ξ)dξ, x ∈ R, (2π)d Rd which is not true if g is not integrable. The problem is caused by the fact that in the proof of (5.37) in [21], on the right hand side of (5.39) therein, we may have ∞ − ∞. But (A.2) does hold for any Borel probability measures µ and ν on Rd such that Ef (µ) < ∞ and Ef (ν) < ∞. In particular, Z Z Z 1 f (x − y)ϕ(x)ψ(y)dxdy = Fϕ(ξ)Fψ(ξ)g(ξ)dξ =: Ef (ϕ, ψ) (A.3) Rd Rd (2π)d Rd EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 38/43 Existence of density for the stochastic wave equation for any density functions ϕ and ψ on Rd with Ef (ϕ) < ∞ and Ef (ψ) < ∞. It follows that relation (A.3) holds for any non-negative functions ϕ, ψ ∈ L1 (Rd ) with Ef (ϕ) < ∞ and Ef (ψ) < ∞. (To see this, we write (A.3) for the density functions ϕ/kϕk1 and ψ/kψk1 and then we multiply by kϕk1 kψk1 .) Moreover, in this case |Ef (ϕ, ψ)| < ∞ since by the Cauchy-Schwarz inequality Z Z 1/2 Z 1/2 Fϕ(ξ)Fψ(ξ)g(ξ)dξ ≤ |Fϕ(ξ)|2 g(ξ)dξ |Fψ(ξ)|2 g(ξ)dξ Rd Rd Rd In fact, we have the following more general result. Lemma A.1. Relation (A.3) holds for any functions ϕ, ψ ∈ L1 (Rd ) with Ef (|ϕ|) < ∞ and Ef (|ψ|) < ∞, and in this case |Ef (ϕ, ψ)| < ∞. Proof. We write ϕ = ϕ+ − ϕ− , where ϕ+ = max(ϕ, 0) and ϕ− = max(−ϕ, 0). Similarly, ψ = ψ + − ψ − , where ψ + = max(ψ, 0) and ψ − = max(−ψ, 0). Then |ϕ| = ϕ+ + ϕ− . Since ϕ+ (x) ≤ |ϕ(x)| for any x ∈ R, we have Z Z Ef (ϕ+ ) = f (x − y)ϕ+ (x)ϕ+ (y)dxdy Rd Rd Z Z ≤ f (x − y)|ϕ(x)||ϕ(y)|dxdy = Ef (|ϕ|) < ∞. Rd Rd Similarly, we obtain that Ef (ϕ− ) < ∞, Ef (ψ + ) < ∞ and Ef (ψ − ) < ∞. Hence relation (A.3) holds for the pairs (ϕ+ , ψ + ), (ϕ− , ψ − ), (ϕ+ , ψ − ) and (ϕ− , ψ + ): Z Z Z 1 f (x − y)ϕ+ (x)ψ + (y)dxdy = Fϕ+ (ξ)Fψ + (ξ)g(ξ)dξ d d (2π)d Rd ZR ZR Z 1 f (x − y)ϕ− (x)ψ − (y)dxdy = Fϕ− (ξ)Fψ − (ξ)g(ξ)dξ Rd Rd (2π)d Rd Z Z Z 1 f (x − y)ϕ+ (x)ψ − (y)dxdy = Fϕ+ (ξ)Fψ − (ξ)g(ξ)dξ Rd Rd (2π)d Rd Z Z Z 1 f (x − y)ϕ− (x)ψ + (y)dxdy = Fϕ− (ξ)Fψ + (ξ)g(ξ)dξ Rd Rd (2π)d Rd and all the integrals appearing above are finite. We take the sum of the first two relations above and from this, we subtract the sum of the last two. We obtain Z Z f (x − y)(ϕ+ (x) − ϕ− (x))(ψ + (y) − ψ − (y))dxdy = Rd Rd Z 1 (Fϕ+ (ξ) − Fϕ− (ξ))(Fψ + (ξ) − Fψ − (ξ))g(ξ)dξ, (2π)d Rd which is exactly relation (A.3). Finally, we note that Z Z |Ef (ϕ, ψ)| ≤ f (x − y)|ϕ(x)||ψ(y)|dxdy = Ef (|ϕ|, |ψ|) ≤ Ef (|ϕ|)1/2 Ef (|ψ|)1/2 < ∞. Rd Rd For complex-valued functions, we have the following result. Lemma A.2. For any functions ϕ, ψ ∈ L1C (Rd ) with Ef (|ϕ|) < ∞ and Ef (|ψ|) < ∞, we have: Z Z Z 1 f (x − y)ϕ(x)ψ(y)dxdy = d Fϕ(ξ)Fψ(ξ)g(ξ)dξ =: Ef (ϕ, ψ), (A.4) Rd R d (2π) Rd and in this case, |Ef (ϕ, ψ)| < ∞. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 39/43 Existence of density for the stochastic wave equation Proof. We write ϕ = ϕ1 + iϕ2 and ψ = ψ1 + iψ2 , where ϕ1 , ϕ2 , ψ1 , ψ2 ∈ L1 (Rd ). Note that |ϕ|2 = |ϕ1 |2 + |ϕ2 |2 . It follows that |ϕ1 (x)| ≤ |ϕ(x)| for all x ∈ R, and hence Z Z Ef (|ϕ1 |) = f (x − y)|ϕ1 (x)||ϕ1 (y)|dxdy d d ZR ZR ≤ f (x − y)|ϕ(x)||ϕ(y)|dxdy = Ef (|ϕ|) < ∞. Rd Rd Similarly, Ef (|ϕ2 |) < ∞, Ef (|ψ1 |) < ∞ and Ef (|ψ2 |) < ∞. Note that ϕ(x)ψ(y) = [ϕ1 (x) + iϕ2 (x)][ψ1 (y) − iψ2 (y)] = [ϕ1 (x)ψ1 (y) + ϕ2 (y)ψ2 (y)] + i[ϕ2 (x)ψ1 (y) − ϕ1 (y)ψ2 (y)] and Fϕ(ξ)Fψ(ξ) = [Fϕ1 (ξ) + iFϕ2 (ξ)][Fψ1 (ξ) − iFψ2 (ξ)] = Fϕ1 (ξ)Fψ1 (ξ) + Fϕ2 (ξ)Fψ2 (ξ) + i[Fϕ2 (ξ)Fψ1 (ξ) − Fϕ1 (ξ)Fψ2 (ξ)]. We apply Lemma A.1 to the pairs of functions (ϕ1 , ψ1 ), (ϕ2 , ψ2 ), (ϕ2 , ψ1 ), (ϕ1 , ψ2 ): Z Z Z 1 f (x − y)ϕ1 (x)ψ1 (y)dxdy = Fϕ1 (ξ)Fψ1 (ξ)g(ξ)dξ d d (2π)d Rd ZR ZR Z 1 f (x − y)ϕ2 (x)ψ2 (y)dxdy = Fϕ2 (ξ)Fψ2 (ξ)g(ξ)dξ d d (2π)d Rd ZR ZR Z 1 f (x − y)ϕ2 (x)ψ1 (y)dxdy = Fϕ2 (ξ)Fψ1 (ξ)g(ξ)dξ d d (2π)d Rd ZR ZR Z 1 f (x − y)ϕ1 (x)ψ2 (y)dxdy = Fϕ1 (ξ)Fψ2 (ξ)g(ξ)dξ, Rd Rd (2π)d Rd where all the integrals above are finite. We take the sum of the first two relations and then we add the difference between the third and fourth relations multiplied by i. We obtain: Z Z f (x − y){ϕ1 (x)ψ1 (y) + ϕ2 (x)ψ2 (y) − i[ϕ2 (x)ψ1 (y) + ϕ1 (x)ψ2 (y)]}dxdy = Rd Rd Z 1 {Fϕ1 (ξ)Fψ1 (ξ) + Fϕ2 (ξ)Fψ2 (ξ) + i[Fϕ2 (ξ)Fψ1 (ξ) − Fϕ1 (ξ)Fψ2 (ξ)]}g(ξ)dξ, (2π)d Rd which is exactly relation (A.4). The fact that |Ef (ϕ, ψ)| < ∞ follows as in Lemma A.1. B Existence of measurable modifications In this section, we prove a result about existence of measurable modifications of random fields, which is used frequently in the literature on SPDEs using Walsh’ approach [36]. Parts a) and b) of this result are extensions to random fields of Theorem 30 in Chapter IV of [13], respectively Proposition 3.21 of [31]. Part a) of this result is used in the proof of Theorem 3.9. Part b) of this result is not needed in the present article, but we include it here since its proof requires only a minor modification of the proof of part a). We include the proof for the sake of completeness. A random field is a collection X = {X(t, x); t ≥ 0, x ∈ Rd } of random variables defined on the same probability space (Ω, F, P ). We say that X is stochastically continuous P if it is continuous in probability, i.e. X(tn , xn ) → X(t, x) if tn → t and xn → x. X is predictable with respect to a filtration (Ft )t≥0 if the map (ω, t, x) 7→ X(ω, t, x) is EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 40/43 Existence of density for the stochastic wave equation measurable with respect to the predictable σ -field on Ω × R+ × Rd , which is the σ -field generated by elementary processes. (An elementary process is a process of the form Y (ω, t, x) = Y0 (ω)1(a,b] (t)1A (x), where Y0 is Fa -measurable, 0 < a < b and A ∈ B(Rd ) is a bounded set.) X is adapted with respect to a filtration (Ft )t≥0 if X(t, x) is Ft -measurable, for any t ≥ 0 and x ∈ Rd . Proposition B.1. a) Any stochastically continuous random field has a measurable modi- fication. b) Any stochastically continuous random field which is adapted with respect to a filtration (Ft )t≥0 has a predictable modification, with respect to the same filtration. Proof. a) Let X = {X(t, x); t ≥ 0, x ∈ Rd } be a stochastically continuous random field defined on a probability space (Ω, F, P ). Let (Em )m≥1 be an increasing sequence of compact sets in Rd such that ∪m Em = Rd . Fix m ≥ 1 and let I = [0, m] × Em . Since X is stochastically continuous, it is uniformly stochastically continuous on I , i.e. for any ε > 0 there exists δε > 0 such that for any (t, x), (s, y) ∈ I with |t − s|2 + |x − y|2 ≤ δm 2 , P(|X(t, x) − X(s, y)| > 2−m ) ≤ 2−m . (m) (m) (m) Let 0 = t0 < t1 < . . . < tnm = m be a partition of [0, m] into subintervals of length (m) smaller than δm and (Ul )l=1,...,Km be a partition of Em into Borel sets of diameter smaller than δm . (The diameter of a set S is defined as sup{|x − y|; x, y ∈ S}.) Let (m) (m) (m) (m) (m) xl ∈ U l be arbitrary. For any t ∈ (tk , tk+1 ] and x ∈ Ul , (m) (m) P(|X(tk , xl ) − X(t, x)| > 2−m ) ≤ 2−m . (B.1) Define m −1 X nX Km (m) (m) Xm (ω, t, x) = X(ω, tk , xl )1(t(m) ,t(m) ] (t)1U (m) (x). k k+1 l k=0 l=1 (m) (m) Note that Xm is a measurable process, since X(tk , xl ) is F -measurable. Let A = {(ω, t, x) ∈ Ω × R+ × Rd ; (Xm (ω, t, x))m≥0 converges}. Then A ∈ F × B(R+ ) × B(Rd ) and the process X e defined by X(ω, e t, x) = 1A (ω, t, x) lim Xm (ω, t, x) m→∞ is measurable. e is a modification of X . Let (t, x) ∈ R+ × Rd be arbitrary. Then We now show that X (m) (m) (m) (t, x) ∈ Im for m large enough, and t ∈ (tk , tk+1 ] for some k = km and x ∈ Ul for some l = lm . Let (m) (m) Am = {|X(tk , xl ) − X(t, x)| > 2−m }. P By (B.1), m P(Am ) < ∞. By the Borel-Cantelli Lemma, P(Ωt,x ) = 1 where Ωt,x = lim inf m Acm = ∪m0 ∩m≥m0 Acm . Let ω ∈ Ωt,x . Then there exists m0 = m0 (ω) such that (m) (m) |X(ω, tk , xl ) − X(ω, t, x)| ≤ 2−m ∀m ≥ m0 (ω). By the definition of Xm , it follows that |Xm (ω, t, x) − X(ω, t, x)| ≤ 2−m ∀m ≥ m0 (ω). From this, we infer that limm→∞ Xm (ω, t, x) = X(ω, t, x), i.e. (ω, t, x) ∈ A. This shows that X(ω, e t, x) = X(ω, t, x) for any ω ∈ Ωt,x . Since P(Ωt,x ) = 1, we infer that P(X(t, e x) = X(t, x)) = 1, as required. b) In this part, we assume, in addition, that X is adapted. Then Xm is predictable, being a linear combination of elementary processes. Hence A and X e are predictable. Remark B.2. Proposition B.1.a) follows from a general result of Cohn: see Theorem 3 of [8], and the (first) Remark on page 164 of [8]. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 41/43 Existence of density for the stochastic wave equation References [1] Balan, R. M. (2012). The stochastic wave equation with multiplicative fractional noise: a Malliavin calculus approach. Potential Anal. 36, 1–34. MR-2886452 [2] Balan, R. M. and Chen, L. (2018). Parabolic Anderson model with space-time homogeneous Gaussian noise and rough initial conditions. J. Theor. Probab. 31, 2216–2265. MR-3866613 [3] Balan, R. M. and Song, J. (2017). Hyperbolic Anderson Model with space-time homogeneous Gaussian noise. Latin Amer. J. Probab. Math. Stat. 14, 799–849. MR-3731795 [4] Balan, R. M. and Song, J. (2017). Second order Lyapunov exponents for parabolic and hyperbolic Anderson models. Preprint available on arXiv:1704.02411. MR-4003574 [5] Bally, V. and Pardoux, E. (1998). Malliavin calculus for white noise driven parabolic SPDEs. Potential Anal. 9, 27–64. MR-1644120 [6] Basse-O’Connor, A., Graversen, S.-E. and Pedersen, J. (2012). Multiparameter processes with stationary increments. Spectral representation and intregration. Electr. J. Probab. 17, paper no. 74, 21 pages. MR-2968681 [7] Carmona, R. and Nualart, D. (1988). Random non-linear wave equations: smoothness of the solutions. Probab. Th. Rel. Fields 79, 469–508. MR-0966173 [8] Cohn, D. L. (1972). Measurable choice of limit points and the existence of separable and measurable processes. Z. Wahr. verw. Geb. 22, 161–165. MR-0305444 [9] Dalang R.C. (1999) Extending martingale measure stochastic integral with applications to spatially homogeneous spde’s. Electr. J. Probab. 4. MR-1684157 [10] Dalang, R. C. and Frangos, N. (1998). The stochastic wave equation in two spatial dimensions. Ann. Probab. 26, 187–212. MR-1617046 [11] Dalang, R. C. and Sanz-Solé, M. (2005). Regularity of the sample paths of a class of second- order spde’s. J. Funct. Anal. 227, 304–337. MR-2168077 [12] Delgado-Vences, F., Nualart, D. and Zheng, G. (2018). A central limit theorem for the stochas- tic wave equation with fractional noise. Preprint available on arXiv:1812.05019. [13] Dellacherie, C. and Meyer, P.A. (1975). Probabilités et Potentiel. Hermann, Paris. MR-0488194 [14] Hu, Y., Huang, J., Nualart, D. and Tindel, S. (2015). Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electr. J. Probab. 20, no. 55, 1–50. MR-3354615 [15] Hu, Y. and Lê, K. (2018). Asumptotics of the density of parabolic Anderson random fields. Preprint available on arXiv:1801.03386. [16] Hu, Y., Nualart, D. and Song, J. (2011). Feynman-Kac formula for heat equation driven by fractional white noise. Ann. Probab. 39, 291–326. MR-2778803 [17] Huang, J., Lê, K. and Nualart, D. (2017). Large time asymptotics for the Parabolic Anderson model driven by spatially correlated noise. Ann. Inst. Henri Poincaré 53, 1305–1340. MR- 3689969 [18] Huang, J., Nualart, D. and Viitasaari, L. (2018). A central limit theorem for the stochastic heat equation. Preprint available on arXiv:1810.09492. [19] Itô, K. (1954) Stationary random distributions. Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 28, 209–223. MR-0065060 [20] Jolis, M. (2010). The Wiener integral with respect to second-order processes with stationary increments. J. Math. Anal. Appl. 366, 607–620. MR-2600506 [21] Khoshnevisan, D. and Xiao, Y. (2009). Harmonic analysis of additive Lévy processes. Probab. Th. Rel. Fields 145, 459–515. MR-2529437 [22] Márquez-Carreras, D., Mellouk, M., Sarrà, M. (2001). On stochastic partial differential equations with spatially correlated noise: smoothness of the law. Stoch. Proc. Appl. 93, 269–284. MR-1828775 [23] Millet, A. and Sanz-Solé, M. (1999). A stochastic wave equation in two space dimension: smoothness of the law. Ann. Probab. 27, 803–844. MR-1698971 [24] Mueller, C. and Nualart, D. (2008). Regularity of the density for the stochastic heat equation. Electr. J. Probab. 13, Paper no. 74, 2248–2258. MR-2469610 EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 42/43 Existence of density for the stochastic wave equation [25] Nualart, D. (1998). Analysis on Wiener space and anticipative stochastic calculus. Lecture Notes Math. 1690. Ecole d’ete de probabilités de Saint Flour XXV-1995, P. Bernard Ed. 123–227. MR-1668111 [26] Nualart, D. (2006). Malliavin Calculus and Related Topics. Second Edition. Springer. MR- 2200233 [27] Nualart, D. and Quer-Sardanyons, L. (2007). Existence and smoothness of the density for spatially homogeneous SPDEs. Potential Anal. 27, 281–299. MR-2336301 [28] Pardoux, E. and Zhang, T. (1993). Absolute continuity for the law of the solution of a parabolic SPDE. J. of Funct. Anal. 112, 447–458. MR-1213146 [29] Peszat, S. (2002). The Cauchy problem for a nonlinear stochastic wave equation in any dimension. J. Evol. Equ. 2, 383–394. MR-1930613 [30] Peszat, S. and Zabczyk, J. (2000). Nonlinear stochastic wave and heat equations. Probab. Th. Rel. Fields 116, 421–443. MR-1749283 [31] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press. MR-2356959 [32] Quer-Sardanyons, L., Sanz-Solé, M. (2004). Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation. J. Funct. Anal. 206, no. 1, 1–32. MR-2024344 [33] Quer-Sardanyons, L., Sanz-Solé, M. (2004) A stochastic wave equation in dimension 3: Smoothness of the law. Bernoulli 10, no. 1, 165–186. MR-2044597 [34] Sanz-Solé, M. and Süß, A. (2013). The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity. Electr. J. Probab. 18, no. 64, 1–28 MR-3078023 [35] Sanz-Solé, M. and Süß, A. (2015). Absolute continuity for SPDEs with irregular fundamental solution. Electr. Comm. Probab. 20, no. 14, 1–11. MR-3314649 [36] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. Ecole d’Eté de Probabilités de Saint-Flour XIV. Lecture Notes in Math. 1180, 265–439, Springer-Verlag. MR-0876085 Acknowledgments. The first author is grateful to Mohammud Foondun who asked her (in a personal communication) about the existence of a predictable modification of a random field. The answer to his question is given by Proposition B.1.b). The authors would like to thank David Nualart for useful discussions, and for sending them the preliminary version of the preprint [18]. The authors would like to thank an anonymous referee for his/her valuable comments which greatly improved this paper. EJP 24 (2019), paper 106. http://www.imstat.org/ejp/ Page 43/43 Electronic Journal of Probability Electronic Communications in Probability Advantages of publishing in EJP-ECP • Very high standards • Free for authors, free for readers • Quick publication (no backlog) • Secure publication (LOCKSS1 ) • Easy interface (EJMS2 ) Economical model of EJP-ECP • Non profit, sponsored by IMS3 , BS4 , ProjectEuclid5 • Purely electronic Help keep the journal free and vigorous • Donate to the IMS open access fund6 (click here to donate!) • Submit your best articles to EJP-ECP • Choose EJP-ECP over for-profit journals 1 LOCKSS: Lots of Copies Keep Stuff Safe http://www.lockss.org/ 2 EJMS: Electronic Journal Management System http://www.vtex.lt/en/ejms.html 3 IMS: Institute of Mathematical Statistics http://www.imstat.org/ 4 BS: Bernoulli Society http://www.bernoulli-society.org/ 5 Project Euclid: https://projecteuclid.org/ 6 IMS Open Access Fund: http://www.imstat.org/publications/open.htm