Spatial integral of the solution to hyperbolic Anderson model with time-independent noise

TL;DR

This paper investigates the asymptotic distribution of the spatial integral of solutions to a hyperbolic Anderson model driven by time-independent Gaussian noise, demonstrating convergence to a normal distribution using Malliavin calculus and Stein's method.

Contribution

It introduces a novel approach combining Malliavin calculus and Stein's method to analyze the asymptotic behavior of the spatial integral in a hyperbolic Anderson model with time-independent noise.

Findings

Spatial integral converges to a normal distribution with proper normalization.

Established a functional limit theorem for the spatial integral process.

Provided convergence rates in total variation distance.

Abstract

In this article, we study the asymptotic behavior of the spatial integral of the solution to the hyperbolic Anderson model in dimension $d\leq 2$, as the domain of the integral gets large (for fixed time $t$). This equation is driven by a spatially homogeneous Gaussian noise, whose covariance function is either integrable, or is given by the Riesz kernel. The novelty is that the noise does not depend on time, which means that It\^o's martingale theory for stochastic integration cannot be used. Using a combination of Malliavin calculus with Stein's method, we show that with proper normalization and centering, the spatial integral of the solution converges to a standard normal distribution, by estimating the speed of this convergence in the total variation distance. We also prove the corresponding functional limit theorem for the spatial integral process.