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

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

The authors of this title 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. The authors prove that at any fixed time, a.s., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigorous formulation of the s.p.d.e., 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, the authors show that the Holder exponents that they obtain are optimal.