Quantitative normal approximations for the stochastic fractional heat equation

TL;DR

This paper establishes quantitative central limit theorems for the stochastic fractional heat equation driven by various Gaussian noises, demonstrating Gaussian convergence of spatial averages as the domain size grows, extending previous results to fractional Laplacians.

Contribution

It provides the first quantitative CLT for the stochastic fractional heat equation with general Gaussian noise, including space-time white noise and Riesz kernel-based noise.

Findings

Spatial averages converge to Gaussian limits as domain size increases.

Quantitative bounds in total variation distance are established.

Results extend previous CLT findings to fractional Laplacian cases.

Abstract

In this article we present a {\it quantitative} central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space-time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius $R$ converges, as $R$ tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.