Gaussian estimates of the density for systems of non-linear stochastic heat equations

TL;DR

This paper investigates Gaussian bounds for the density of solutions to non-linear stochastic heat equations driven by Gaussian noise, extending previous smoothness results to include explicit lower and upper density estimates.

Contribution

It provides new lower and upper bounds for the density of solutions, using Malliavin calculus and advanced probabilistic techniques, building on prior smoothness analyses.

Findings

Established explicit Gaussian bounds for the density

Extended previous smoothness results to density estimates

Applied Malliavin calculus to non-linear stochastic heat equations

Abstract

In this paper we consider a system of non-linear stochastic heat equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and has a homogeneous spatial covariance. Under some suitable regularity and non degeneracy conditions, the smoothness of the joint density of the solution for this system has been studied by E. Nualart in [11]. The purpose of this paper is further to study the lower and upper bounds of the density. The main tools are the Malliavin calculus and the method developed by Kohatasu-Higa in [6] or E. Nualart and Quer-Sardanyons in [12].