Gaussian fluctuations for hyperbolic Anderson model with Lévy colored noise

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 Lévy colored noise introduced in Balan and Jiménez (2026). We assume that the spatial coloration kernel of the noise is either integrable on $\mathbb{R}$, or is the Riesz kernel of order $α\in (0,1)$, and the Lévy 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.