Gaussian fluctuations for hyperbolic Anderson model with L\'evy colored noise

TL;DR

This paper investigates the asymptotic Gaussian fluctuations of the spatial integral of solutions to the hyperbolic Anderson model driven by Lévy colored noise, establishing normal convergence and rates under specific conditions.

Contribution

It extends the understanding of Gaussian fluctuations in hyperbolic Anderson models with Lévy colored noise, providing new limit theorems and convergence rates.

Findings

Normalized spatial integral converges to normal distribution as domain size grows.

Provides explicit convergence rate estimates in various probability metrics.

Establishes functional limit theorems for the model.

Abstract

In this article, we study the asymptotic behaviour of the spatial integral $F_R(t)$ of the solution to the hyperbolic Anderson model in dimension $d=1$, driven by the L\'evy colored noise introduced in Balan and Jim\'enez (2026). We assume that the spatial coloration kernel of the noise is either integrable on $\mathbb{R}$, or is the Riesz kernel of order $\alpha \in (0,1)$, and the L\'evy measure of the noise has finite moments of order $p$ and $2p$ for some $p \in (1,2]$. By applying a recent result of Trauthwein (2025), we prove that $F_R(t)/\sqrt{{\rm Var}\big(F_R(t)\big)}$ converges to the standard normal distribution as $R \to \infty$, and we give an estimate for the rate of this convergence in the Fortet-Mourier distance, the 1-Wasserstein distance, or the Kolmogorov distance. We also provide the corresponding functional limit result.