Central limit theorems for stochastic wave equations in high dimensions

TL;DR

This paper proves central limit theorems for solutions of stochastic wave equations in high dimensions with Gaussian noise, showing convergence to normal distribution under certain conditions using Malliavin-Stein's method.

Contribution

It establishes the first CLTs for high-dimensional stochastic wave equations with spatially correlated Gaussian noise, including the Riesz kernel case.

Findings

Normalized spatial integrals converge to normal distribution

Constructed approximation sequences for solutions

Applied Malliavin-Stein's method for proofs

Abstract

We consider stochastic wave equations in spatial dimensions $d \geq 4$. We assume that the driving noise is given by a Gaussian noise that is white in time and has some spatial correlation. When the spatial correlation is given by the Riesz kernel, we also establish that the spatial integral of the solution with proper normalization converges to the standard normal distribution under the Wasserstein distance. The convergence is obtained by first constructing the approximation sequence to the solution and then applying Malliavin-Stein's method to the normalized spatial integral of the sequence. The corresponding functional central limit theorem is presented as well.