Convergence of Densities of Spatial Averages of Stochastic Heat Equation

TL;DR

This paper studies the convergence of spatial averages of the stochastic heat equation to a normal distribution, providing explicit rates using Stein's method and Malliavin calculus in two different initial condition scenarios.

Contribution

It introduces new estimates on the Malliavin derivatives and establishes convergence rates for the densities of spatial averages in the stochastic heat equation.

Findings

Established convergence rates to normal density for spatial averages.

Developed new estimates on the second Malliavin derivative.

Applied Stein's method combined with Malliavin calculus techniques.

Abstract

In this paper, we consider the one-dimensional stochastic heat equation driven by a space time white noise. In two different scenarios: {\it (i)} initial condition $u_0=1$ and general nonlinear coefficient $\sigma$ and {\it (ii)}: initial condition $u_0=\delta_0$ and $\sigma(x)=x$ (Parabolic Anderson Model), we establish rates of convergence for the uniform distance between the density of (renormalized) spatial averages and the standard normal density. These results are based on the combination of Stein method for normal approximations and Malliavin calculus techniques. A key ingredient in Case (i) is a new estimate on the $L^p$-norm of the second Malliavin derivative.