Gaussian upper density estimates for spatially homogeneous SPDEs

TL;DR

This paper establishes Gaussian upper bounds for the density of solutions to a broad class of spatially homogeneous SPDEs driven by Gaussian noise, using Malliavin calculus techniques.

Contribution

It provides new sufficient conditions on coefficients and spectral measures ensuring Gaussian upper density estimates for solutions of SPDEs, including heat and wave equations.

Findings

Gaussian upper density estimates for SPDE solutions

Applicable to stochastic heat and wave equations in specific dimensions

Conditions on spectral measure are proven to be optimal in certain cases

Abstract

We consider a general class of SPDEs in $\mathbb{R}^d$ driven by a Gaussian spatially homogeneous noise which is white in time. We provide sufficient conditions on the coefficients and the spectral measure associated to the noise ensuring that the density of the corresponding mild solution admits an upper estimate of Gaussian type. The proof is based on the formula for the density arising from the integration-by-parts formula of the Malliavin calculus. Our result applies to the stochastic heat equation with any space dimension and the stochastic wave equation with $d\in \{1,2,3\}$. In these particular cases, the condition on the spectral measure turns out to be optimal.