Existence of density for the stochastic wave equation with space-time homogeneous Gaussian noise

Abstract

In this article, we consider the stochastic wave equation on $\mathbb{R}_{+} \times \mathbb{R}$, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions $γ$ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $ν$ on $\mathbb{R}$, respectively $μ$ on $\mathbb{R}$. Our main result shows that the law of the solution $u(t,x)$ of this equation is absolutely continuous with respect to the Lebesgue measure.

This is a major revision of the previous version of the paper, where we have corrected an important error