Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

TL;DR

This paper derives Gaussian bounds for the probability density of solutions to the stochastic heat equation with multiplicative noise across any spatial dimension, using Malliavin calculus techniques.

Contribution

It provides the first comprehensive Gaussian density estimates for the nonlinear stochastic heat equation in arbitrary space dimensions.

Findings

Established Gaussian upper bounds for the density.

Derived Gaussian lower bounds using adapted methods.

Results apply to equations with spatially homogeneous Gaussian noise.

Abstract

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.