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

TL;DR

This paper proves that the solution to a stochastic wave equation driven by space-time homogeneous Gaussian noise has a probability distribution that is absolutely continuous, ensuring the existence of a density function under certain covariance conditions.

Contribution

It establishes the existence of a density for the solution of the stochastic wave equation with general space-time homogeneous Gaussian noise, extending previous results to broader covariance structures.

Findings

Law of the solution is absolutely continuous w.r.t. Lebesgue measure.

Density exists under conditions on covariance functions.

Generalizes previous results to broader noise structures.

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.