Absolute continuity for SPDEs with irregular fundamental solution

For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Sol\'e and S\"u\ss, 2013].


Introduction
The seminal article [8] begins with a criterion for the absolute continuity with respect to the Lebesgue measure for nonnegative finite measures on R m . Using tools of harmonic analysis, it is proved that if κ is such a measure and there exist a constant c such that for every Φ with compact support, we have ∂ k Φdκ ≤ c Φ ∞ , for 1 ≤ k ≤ m, then κ(dx) = k(x)dx and k ∈ L 1 . By iteration, it is possible to strengthen this criterion and obtain the existence of an infinitely differentiable density. In the same article, Malliavin sets up the grounds of a stochastic calculus of variation with the purpose to be able to apply this criterion to the probability law of Gaussian functionals. It was coined as Malliavin Calculus. The book [9] contains an extensive list of references on applications of this calculus which is still continuing to grow. Among them, the study of the law of random field solutions to stochastic partial differential equations, in the sequel referred to as SPDEs (see [13] for an introduction).
The use of Malliavin calculus for the analysis of densities requires some regularity properties that are not met by all SPDEs and neither by stochastic differential equations with non-smooth coefficients. This problem has motivated the search for alternatives to Malliavin's criterion, quite similar in spirit, but giving weaker conclusions, applicable to cases that exhibit irregularities. For the sake of brevity, we only mention the references [7,5,6] where the approach used in this paper is developed, and the recent related article [1].
Throughout the paper we consider the setting of [2,15]. More explicitly, we deal with an SPDE Lu(t, x) = b(u(t, x)) + σ(u(t, x))Ḟ (t, x), with suitable initial conditions, that we express in its mild formulation as Here Λ denotes the fundamental solution to Lu = 0 and M is the martingale measure derived from a random noise F white in time and with a stationary covariance measure in space. The spectral measure of the covariance (its inverse Fourier transform) will be denoted by µ.
Let {u(t, x), (t, x) ∈ [0, T ] × R d } be the random field solution to (1.1). Assume that the function σ in (1.1) is constant, and fix (t, x) ∈ (0, T ] × R d . In [15], using Malliavin Calculus, it is proved that the probability law of u(t, x) has a density. The purpose of this article is to extend this result allowing σ to be a nonlinear Lipschitz continuous function. Moreover, we prove that the density belongs to some Besov space.
In [15], the restriction on σ is forced by the method of the proof. Indeed, in the examples where the fundamental solution Λ is a nonnegative distribution, σ and b are differentiable, and σ is bounded away from zero, we can prove that the Malliavin matrix is invertible. However, for more general Λ, for example the fundamental solution to the wave equation in dimension d ≥ 4, this does not seem to be feasible, except for constant σ. In contrast, the method of [6], based on Lemma 2.2 of Section 2, can be successfully applied, and also the regularity of the coefficients σ and b can be relaxed.
The paper is structured in the following way. In Section 2 we prove the main result on existence of density, and find the Besov space that contains this density. In Section 3, we study the example of stochastic wave equations in any spatial dimension d ≥ 1, the interesting and novel case being d ≥ 4 (see [11], [12] for related results). We consider two cases of spectral measures µ: with densities given by a Riesz kernel, and finite measures. We also provide a comment about the stochastic heat equation. The existence of a density for this equation is well-known, see [10]; however, with this different approach we can allow for less smooth coefficients. We end this section by fixing some notation. Throughout the article we write C for any positive constant, which may change from line to line. The set of Schwartz functions on R d is denoted by S(R d ), S ′ r (R d ) is the set of tempered distributions with rapid decrease, and F the Fourier transform operator on R d . We denote by {F t , t ∈ [0, T ]} the filtration generated by the martingale measure {M t , t ∈ [0, T ]}.

Statement and proof of the main result
The objective of this section is to prove Theorem 2.1. We begin by introducing a first set of relevant assumptions: Under the assumptions (A1) and either (A2) or (A3), [2, Theorem 3.1] assures that the integrals in (1.1) are well-defined and the equation has a unique random field solution. Moreover, for all t ∈ [0, T ], u(t, x) has the same distribution as u(t, 0), for all L 2 -continuous and has uniformly bounded second moments. We also recall the following estimates: (see [2] for the details). The proof of the existence of density for the law of the solution requires the following second set of assumptions.
The assumption (A5) (strong ellipticity) appears frequently when studying the absolute continuity of probability measures induced by solutions to SDEs and SPDEs, while (A6) (or similar ones) has been usually required to prove regularity properties of the density. This is the main result of the paper.
We assume that the coefficients σ and b are Lipschitz continuous functions. Moreover, suppose that (A1), either (A2) or (A3), (A4), (A5) and (A6) hold, and that Then, the probability law of u(t, x) is absolutely continuous and its density belongs to all Besov spaces B s 1,∞ , with The spaces B s 1,∞ , s > 0, can be defined as follows. Let f : For any 0 < s < n, we define the norm It can be proved that for two distinct n, n ′ > s the norms obtained using n or n ′ are equivalent. Then we define B s 1,∞ to be the set of We refer the reader to [16] for more details. The proof of Theorem 2.1 is based on the following lemma from [5] (based on [6]). In the following, we denote by C α b the set of bounded Hölder continuous functions of degree α.
Lemma 2.2. Let κ be a finite nonnegative measure. Assume that there exist 0 < α ≤ a < 1, n ∈ N and a constant C n such that for all φ ∈ C α b , and all h ∈ R with |h| ≤ 1, Then κ has a density with respect to the Lebesgue measure, and this density belongs to the Besov space B a−α 1,∞ (R).
We will apply this lemma to κ = P • u(t, x) −1 , where u is the solution to the SPDE (1.1) at some fixed point (t, x) ∈ (0, T ] × R d . We prove the claim in Theorem 2.1 for x = 0. Since the probability distribution of u(t, x) does not depend on x ∈ R d , this yields the result.
Note that in this application the constant C n in (2.8) may depend on n, on the elements defining the SPDE (1.1) and on the assumptions. In particular, on σ, σ 0 , b, β, T, d, γ, γ 1 , γ 2 or the total mass |µ| if the measure µ is finite, and also on t ∈ (0, T ].
In order to apply Lemma 2.2, we rely on the three next lemmas.
Lemma 2.3. The density ϕ of a one-dimensional normal distribution N (0, σ 2 ) satisfies ϕ (n) L 1 = C n σ 2 −n/2 , for all n ∈ N, where ϕ (n) (y) = d n ϕ dy n (y), y ∈ R. Proof. Let H n denote the n-th Hermite polynomial. It is well-known that With a change of variables we obtain Since the last integral is finite, we have the result.
The following lemma gives a first bound for the expected values of the iterated differences of functions of the solution to the SPDE (1.1). Proof. The left-hand side of (2.10) satisfies
For the first term, the property ∆ n h φ C α b ≤ C n φ C α b , the spatial stationarity of the solution and Hölder's inequality yield For the study of the term I 2 (h, n, φ, ε, t), we consider the decomposition where Notice that U ε t is F t−ε -measurable, and conditionally to F t−ε , is a Gaussian random variable with zero mean and independent of U ε t . The conditional variance of V ε t is computed as follows: where in the last step we have used (A5). Therefore the conditional law of V ε t with respect to F t−ε has a C ∞ b -density, which we denote by ϕ t,Λ,ε . For any f ∈ C m , we have (2.14) Indeed, this inequality holds since ∆ n h f (x) = n 0 l n (v)f (n) (x+hv)h n dv, for some bounded function l n which is independent of f . Therefore, by conditioning with respect to F t−ε , and applying a discrete integration by parts, (2.14), Lemma 2.3 and (2.13), we obtain = C n φ ∞ |h| n (σ 2 Λ (ε)) −n/2 ≤ C n,σ0 φ ∞ |h| n g(ε) −n/2 , which together with (2.11) yields (2.10), since φ ∞ ≤ φ C α b . Lemma 2.5. Under the assumptions in Theorem 2.1 we have for all t ∈ [0, T ] and all ε ∈ (0, t), with δ as in (A4), and g 1 , g 2 are defined in (2.4), (2.5), respectively.
We are now in a position to show Theorem 2.1.

Examples
In this section we consider mainly the stochastic wave equation in any spatial dimension d ≥ 1, which is studied in [2,Section 4]. In the last part, we will give some remarks on the heat equation (see Remark 3.4) which complement known results.
Let us consider (1.1) where Λ is the fundamental solution to the wave equation, whose Fourier transform is We will assume that the spatial covariance of the noise F is a Riesz kernel of parameter β ∈ (0, 2 ∧ d) and therefore, that the spectral measure µ is This kernel is fairly common in the literature on SPDEs with spatially homogeneous covariance. For example, it is a particular case of those considered in [4]. To prove this theorem it suffices to check that the assumptions of Theorem 2.1 hold with δ = 2 − β, γ = γ 1 = 3 − β and γ 2 = 3. The conditions (A1) and (A2) have already been proved in [2]. The remaining conditions are established in the next result. Proof. We basically follow the same method as in [4,Proposition 4.1]. Using (2.1) and (2.2), we have Using the identity sin x − sin y = 2 sin x−y 2 cos x+y 2 , and the changes of variable, ζ → t−s 2 (ξ + η) and ξ → ζ − η, we get I 1 (s, t) The last step hold because the integral is finite. Indeed, as in [13, Lemma 6.1] we can show that sin 2 (|ξ+η|) and the integral on the right-hand side is finite (uniformly in η) if and only if β ∈ (0, 2 ∧ d).
Remark 3.3. Assume that the spectral measure µ is finite. With the same hypotheses as in Theorem 3.1 we can prove the existence of density for the law of u(t, x), for (t, x) ∈ (0, T ] × R d , and that this density belongs to the spaces B s 1,∞ , with s ∈ (0, 2/5). Indeed, referring to the notations in Lemma 3.2, for a finite measure µ we have I 1 (s, t) ≤ CI 3 (s, t) and I 2 (s, t) ≤ CI 4 (s, t), 0 ≤ s < t ≤ T . Hence, (A4) holds with δ = 2.
Also g 1 (t) ≤ Cg 2 (t), which yields (2.4), (2.5) with γ 1 = γ 2 = 3. Moreover, since  We assume that the spectral measure of the noise F is either given by (3.2), with β ∈ [0, 2 ∧ d) or finite. Suppose that (A5) is satisfied and that the coefficients σ and b are Lipschitz continuous. Fix (t, x) ∈ (0, T ] × R d . Then, the probability law of u(t, x) is absolutely continuous with respect to the Lebesgue measure on R and its density belongs to all Besov spaces B s 1,∞ with s ∈ 0, 1 2 . Let µ be given by (3.2). The case µ finite is left to the reader. Under the standing assumptions, there exists a random field solution to (1.1) (see [3]). In [14] it is proved that (A4) holds with δ = 1 − β/2. Hence, going through the proof of Theorem 2.1 we see that we only need to check hypotheses (A6).
The Fourier transform of the fundamental solution to the heat equation is given by F Λ(t)(ξ) = exp(−4π 2 t|ξ| 2 ). Using this expression, we immediately see that γ 2 = 1. Moreover γ = γ 1 = 1 − β/2. Indeed, with the changes of variables ξ → ξ − η and ζ = √ sξ we get because the integral can be shown to be finite. Therefore,γ in Theorem 2.1 is equal to 2, which implies the claim. In contrast with [10], with the method of this article, the density for the solution to the heat equation in any spatial dimension d ≥ 1 is proved under weaker conditions on σ and b (no differentiability is required).