On Hölder continuity of the solution of stochastic wave equations in dimension three

Abstract

In this paper, we study the stochastic wave equations in the three spatial dimensions driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path Holder continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean Holder continuity of the covariance function or in terms of its spectral measure. Some examples of the covariance functions are proved to satisfy our conditions, which include the case of the work Dalang and Sanz-Sole (Holder–Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three. Memoirs of the American Mathematical Society Number 931 2009). In particular, we obtain the Holder continuity results for the solution of the stochastic wave equations driven by (space inhomogeneous) fractional Brownian noises. For this particular noise, the optimality of the obtained Holder exponents is also discussed.