Hlder-Sobolev regularity of the solution to the stochastic wave equation in dimension 3

Abstract

We study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. We prove that at any fixed time, a.s. the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $x\in\IR^3$, the sample paths in time are Hölder continuous functions. Hence, we obtain joint Hölder continuity in the time and space variables. Our results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller (2003). Sharp results on one and two dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, we show that the Hölder exponents that we obtain are optimal.