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.