The Stochastic Wave Equation with Fractional Noise: a random field approach

Abstract

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in \cite{dalang99}, when the noise is white in time. Under this condition, we show that the solution is $L^2(\Omega)$-continuous. Similar results are obtained for the heat equation. Unlike the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is {\em different} (and more general) than the one obtained for the wave equation.