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

TL;DR

This paper investigates the existence and regularity of solutions to the stochastic wave and heat equations driven by fractional Gaussian noise in time, providing new conditions that generalize previous white noise results.

Contribution

It establishes necessary and sufficient conditions for solution existence under fractional noise, extending classical results to more general noise types and equations.

Findings

Solution exists under relaxed conditions compared to white noise case

Solution is $L^2(\Omega)$-continuous in time

Different conditions apply for heat and wave equations

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.