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

AbstractWe 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