Optimal Gaussian density estimates for a class of stochastic equations with additive noise

TL;DR

This paper derives optimal Gaussian bounds for the density of solutions to certain stochastic integral equations driven by Gaussian noise, with applications to stochastic heat and wave equations.

Contribution

It establishes the first optimal Gaussian bounds for densities of solutions to a broad class of SPDEs with additive Gaussian noise, using Malliavin calculus techniques.

Findings

Optimal Gaussian bounds for solution densities are obtained.

Results apply to stochastic heat and wave equations in various dimensions.

Bounds are expressed in terms of the variance of the stochastic integral.

Abstract

In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in $\mathbb{R}^d$ with $d\geq 1$ and $d\leq 3$, respectively. The upper and lower Gaussian bounds have the same form and are given in terms of the variance of the stochastic integral term in the mild form of the equation.