Convergence of densities of spatial averages of the parabolic Anderson model driven by colored noise

TL;DR

This paper establishes a rate of convergence in the uniform norm for the densities of spatial averages of solutions to a d-dimensional parabolic Anderson model driven by Gaussian noise with Riesz kernel covariance, using Malliavin calculus and Stein's method.

Contribution

It provides the first explicit convergence rate for densities of spatial averages in the parabolic Anderson model with colored noise.

Findings

Derived a quantitative convergence rate in the uniform norm

Applied Malliavin calculus and Stein's method to the problem

Enhanced understanding of the distributional behavior of the model's solutions

Abstract

In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.