Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations

TL;DR

This paper derives Gaussian bounds for solutions to stochastic heat and wave equations driven by Gaussian noise, utilizing Malliavin calculus to provide precise density estimates for these solutions.

Contribution

It introduces new Gaussian bounds for solutions to quasi-linear stochastic PDEs with additive Gaussian noise, extending existing density estimates to higher dimensions and different equations.

Findings

Established Gaussian bounds for 1D stochastic heat equation with white noise.

Extended density estimates to multi-dimensional heat and wave equations.

Provided explicit bounds applicable to equations with spatially homogeneous noise.

Abstract

In this paper we establish lower and upper Gaussian bounds for the solutions to the heat and wave equations driven by an additive Gaussian noise, using the techniques of Malliavin calculus and recent density estimates obtained by Nourdin and Viens. In particular, we deal with the one-dimensional stochastic heat equation in $[0,1]$ driven by the space-time white noise, and the stochastic heat and wave equations in $\mathbb{R}^d$ ($d\geq 1$ and $d\leq 3$, respectively) driven by a Gaussian noise which is white in time and has a general spatially homogeneous correlation.