Fractional stochastic wave equation driven by a Gaussian noise rough in space

TL;DR

This paper studies a fractional stochastic wave equation driven by Gaussian noise with rough spatial properties, proving existence, uniqueness, moment bounds, and regularity of solutions.

Contribution

It introduces a new analysis for wave equations driven by Gaussian noise with fractional Brownian motion covariance in space, establishing well-posedness and regularity results.

Findings

Existence and uniqueness of the mild Skorohod solution.

Bounds for the p-th moment of the solution.

Hölder continuity in time and space variables.

Abstract

In this article, we consider fractional stochastic wave equations on $\mathbb R$ driven by a multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in(\frac14, \frac12)$ in space. We prove the existence and uniqueness of the mild Skorohod solution, establish lower and upper bounds for the $p$-th moment of the solution for all $p\ge2$, and obtain the H\"older continuity in time and space variables for the solution.