Gaussian fluctuations of spatial averages of a system of stochastic heat equations

TL;DR

This paper investigates the asymptotic Gaussian fluctuations of spatial averages in a system of nonlinear stochastic heat equations driven by space-time white noise, establishing convergence rates and a functional CLT.

Contribution

It provides the first detailed analysis of the Gaussian fluctuations and convergence rates for spatial averages of nonlinear stochastic heat equations driven by space-time white noise.

Findings

Established a rate of convergence to multivariate normal distribution in Wasserstein distance.

Proved a functional central limit theorem for spatial averages.

Analyzed the asymptotic behavior of the system over large spatial intervals.

Abstract

We consider a system of $d$ non-linear stochastic heat equations driven by an $m$-dimensional space-time white noise on $\mathbb{R}_+\times \mathbb{R}$. In this paper we study the asymptotic behavior of spatial averages over large intervals $[-R,R]$. We establish a rate of convergence to a multivariate normal distribution in the Wasserstein distance and a functional central limit theorem.