Solving the hyperbolic Anderson model 1: Skorohod setting

TL;DR

This paper investigates the existence and uniqueness of solutions to a wave equation in low dimensions driven by fractional Gaussian noise, establishing precise conditions on the noise's covariance.

Contribution

It provides necessary and sufficient conditions for the existence and uniqueness of a mild Skorohod solution to the hyperbolic Anderson model with fractional Gaussian noise.

Findings

Established covariance conditions for solution existence

Proved uniqueness of the mild Skorohod solution

Focused on low-dimensional wave equations with fractional noise

Abstract

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.