Properties of the density for a three-dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let pt,x(y) be the density of the law of the solution u(t, x) of such an equation at points (t, x) ∈ ]0,T ]×R3. We prove that the mapping (t, x) →pt,x(y) owns the same regularity as the sample paths of the process {u(t,x), (t,x) ∈ ]0,T ] × R3} established in [R.C. Dalang, M. Sanz-Solé, Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three, Mem. Amer. Math. Soc., in press]. The proof relies on Malliavin calculus and more explicitly, the integration by parts formula of [S. Watanabe, Lectures on Stochastic Differential Equations and Malliavin Calculus, Tata Inst. Fund. Res./Springer-Verlag, Bombay, 1984] and estimates derived from it. © 2008 Elsevier Inc. All rights reserved.