On H\"older continuity of the solution of stochastic wave equations

Abstract

In this paper, we study the stochastic wave equations in the spatial dimension 3 driven by a Gaussian noise which is white in time and correlated in space. Our main concern is the sample path H\"older continuity of the solution both in time variable and in space variables. The conditions are given either in terms of the mean H\"older 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 \cite{dalang4}. In particular, we obtain the H\"older 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 H\"older exponents is also discussed.