Central limit theorems for stochastic wave equations in dimensions one and two

TL;DR

This paper establishes quantitative and functional central limit theorems for the spatial average of solutions to stochastic wave equations in dimensions one and two, driven by Gaussian noise with spatial correlation.

Contribution

It provides the first quantitative CLTs for spatial averages of stochastic wave equations in low dimensions, using Malliavin calculus techniques.

Findings

Quantitative CLTs for spatial averages as the domain grows

Functional CLTs for the solutions

Pointwise $L^p$-estimates for Malliavin derivatives

Abstract

Fix $d\in\{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition. In this setting, we provide quantitative central limit theorems for the spatial average of the solution over a Euclidean ball, as the radius of the ball diverges to infinity. We also establish functional central limit theorems. A fundamental ingredient in our analysis is the pointwise $L^p$-estimate for the Malliavin derivative of the solution, which is of independent interest. This paper is another addendum to the recent research line of averaging stochastic partial differential equations.