Density convergence of spatial average of solution to a one dimensional stochastic wave equation

TL;DR

This paper investigates how the spatial averages of solutions to a one-dimensional stochastic wave equation converge in distribution to a normal distribution, providing explicit convergence rates using advanced probabilistic techniques.

Contribution

It introduces a novel analysis of the density convergence rate for spatial averages of stochastic wave equations driven by Gaussian noise, combining Malliavin calculus and Stein's method.

Findings

Established the rate of convergence to normality for spatial averages

Derived bounds on the density difference using Malliavin calculus

Addressed technical challenges in estimating Malliavin derivatives

Abstract

In this paper we study the spatial averages of the solution of a one-dimensional stochastic wave equation driven by a Gaussian multiplicative noise, which is white in time and has a homogeneous spatial covariance described by the Riesz kernel. We establish the rate of convergence for the uniform distance between the density of spatial averages and the standard normal density. The proof combines Malliavin calculus with Stein's method for normal approximations. The key technical challenges lie in estimating the Lp-norm of the second Malliavin derivative and the existence of negative moments of Malliavin covariance matrix.