Properties of the density for a three dimensional stochastic wave equation

TL;DR

This paper investigates the regularity properties of the density function of solutions to a three-dimensional stochastic wave equation driven by spatially correlated noise, using Malliavin calculus techniques.

Contribution

It establishes that the density's regularity matches the sample path regularity of the solution, extending previous results to a more complex noise structure.

Findings

Density function inherits the regularity of the solution paths

Uses Malliavin calculus and Watanabe's integration by parts formula

Provides explicit estimates for the density's regularity

Abstract

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y)$ be the density of the law of the solution $u(t,x)$ of such an equation at points $(t,x)\in]0,T]\times \IR^3$. We prove that the mapping $(t,x)\mapsto p_{t,x}(y)$ owns the same regularity as the sample paths of the process $\{u(t,x), (t,x)\in]0,T]\times \mathbbR^3\}$ established Dalang and Sanz-Sol\'e [Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and more explicitely, Watanabe's integration by parts formula and estimates derived form it.