Absolute continuity for SPDEs with irregular fundamental solution

TL;DR

This paper establishes the existence and regularity of probability densities for solutions to certain SPDEs with irregular fundamental solutions, extending previous results to wave and heat equations with multiplicative noise.

Contribution

It proves the density existence and Besov regularity for solutions of SPDEs with irregular fundamental solutions, including stochastic wave and heat equations, using a novel approach.

Findings

Density exists for solutions at fixed points

Density belongs to a Besov space

Results extend previous work to new SPDE classes

Abstract

For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Sol\'e and S\"u\ss, 2013].