Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three

摘要

We study the sample path regularity of the solution of a stochastic wave equa-tion in spatial dimension d = 3. The driving noise is white in time and witha spatially homogeneous covariance defined as a product of a Riesz kernel and asmooth function. We prove that at any fixed time, a.s., the sample paths in thespatial variable belong to certain fractional Sobolev spaces. In addition, for anyfixed x ∈ R~3, the sample paths in time are Holder continuous functions. Further,we obtain joint Wilder continuity in the time and space variables. Our results relyon a detailed analysis of properties of the stochastic integral used in the rigourousformulation of the s.p.d.e., as introduced by Dalang and Mueller (2003). Sharpresults on one- and two-dimensional space and time increments of generalized Rieszpotentials are a crucial ingredient in the analysis of the problem. For spatial co-variances given by Riesz kernels, we show that the Holder exponents that we obtainare optimal.