Quantitative normal approximations for the stochastic fractional heat equation

Abstract

In this article we present a 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.

Central limit theorems for nonlinear stochastic wave equations in dimension three

“…For other similar results for stochastic heat equations, see e.g. [1,9,22,23,29,33] and the reference therein. The same approach as in [17] also led to similar results for one-and two-dimensional stochastic wave equations as shown in [7,15,31].…”

Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation

“…More precisely, they established the first Gaussian fluctuation result for the spatial integral of the solution to a stochastic nonlinear heat equation driven by space-time Gaussian white noise. Since then, we have witnessed a rapidly growing literature on similar CLT results for heat equations with various Gaussian homogeneous noises; see, for example, [26,42,10,11,12,1,41,44,51,36]. Meanwhile, such a program was carried out by Nualart, the second author, and their collaborators to investigate the stochastic nonlinear wave equation driven by Gaussian noises; see [20,8,44,43,6].…”