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 $\gamma$ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $\nu$ on $\mathbb{R}$, respectively $\mu$ 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.