Hyperbolic Anderson Model with space-time homogeneous Gaussian noise

TL;DR

This paper investigates the Hyperbolic Anderson Model driven by space-time homogeneous Gaussian noise, establishing existence, uniqueness, and sample path regularity of solutions across arbitrary spatial dimensions.

Contribution

It proves the existence and uniqueness of solutions for the stochastic wave equation with multiplicative noise in any spatial dimension, extending previous results to more general Gaussian noises.

Findings

Existence and uniqueness of solutions in the Skorohod sense

Hölder continuity of sample paths

Applicable to general Gaussian noise with homogeneous space-time structure

Abstract

In this article, we study the stochastic wave equation in arbitrary spatial dimension $d$, with a multiplicative term of the form $\sigma(u)=u$, also known in the literature as the Hyperbolic Anderson Model. This equation is perturbed by a general Gaussian noise, which is homogeneous in both space and time. We prove the existence and uniqueness of a solution of this equation (in the Skorohod sense) and the H\"older continuity of its sample paths, under the same respective conditions on the spatial spectral measure of the noise as in the case of the white noise in time, regardless of the temporal covariance function of the noise.