Central limit theorems for nonlinear stochastic wave equations in dimension three

TL;DR

This paper proves Gaussian fluctuation and functional central limit theorems for spatial averages of solutions to three-dimensional nonlinear stochastic wave equations driven by Gaussian noise with spatial correlations, using Malliavin-Stein's method.

Contribution

It introduces new central limit theorems for nonlinear stochastic wave equations with spatially correlated noise in three dimensions, employing Malliavin calculus techniques.

Findings

Gaussian fluctuation for spatial averages established

Functional central limit theorems proven

Results apply to noise with integrable and Riesz kernel correlations

Abstract

In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin-Stein's method, we prove the Gaussian fluctuation for the spatial average of the solution under the Wasserstein distance in the cases where the spatial correlation is given by an integrable function and by the Riesz kernel. In both cases we also establish functional central limit theorems.