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

Raluca M. Balan, Lluís Quer-Sardanyons, Jian Song

Published 2018-05-17Version 1

In this article, we consider the stochastic wave equation on $\mathbb{R}_{+} \times \mathbb{R}^d$, in spatial dimension $d=1$ or $d=2$, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structure 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 $\mathbb{R}^d$. 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, provided that the spatial spectral measure $\mu$ satisfies an integrability condition which ensures that the sample paths of the solution are H\"older continuous.