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

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 $H\in(\frac{1}{4},\frac{1}{2})$ 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 Hölder continuity in time and space variables for the solution.