Gaussian fluctuations for hyperbolic Anderson model with Lévy colored noise
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917296971186176 |
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| author | Balan, Raluca M. Stephenson, William D. |
| author_facet | Balan, Raluca M. Stephenson, William D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23137 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gaussian fluctuations for hyperbolic Anderson model with Lévy colored noise Balan, Raluca M. Stephenson, William D. Probability 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. |
| title | Gaussian fluctuations for hyperbolic Anderson model with Lévy colored noise |
| topic | Probability |
| url | https://arxiv.org/abs/2602.23137 |