Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise

Balan, Raluca; Xia, Panqiu; Zheng, Guangqu

doi:10.1090/proc/17204

Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
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by Raluca M. Balan, Panqiu Xia and Guangqu Zheng;

Proc. Amer. Math. Soc.

DOI: https://doi.org/10.1090/proc/17204

Published electronically: April 14, 2025

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Abstract:

In this paper, we present an almost sure central limit theorem (ASCLT) for the hyperbolic Anderson model (HAM) with a Lévy white noise in a finite-variance setting, complementing a recent work by Balan and Zheng [Trans. Amer. Math. Soc. 377 (2024), pp. 4171–4221] on the (quantitative) central limit theorems for the solution to the HAM. We provide two different proofs: one uses the Clark-Ocone formula and takes advantage of the martingale structure of the white-in-time noise, while the other is obtained by combining the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits’ method of characteristic functions. Both approaches are different from the one developed in the PhD thesis of C. Zheng [Multi-dimensional Malliavin-Stein method on the Poisson space and its applications to limit theorems (PhD dissertation), Université Pierre et Marie Curie, Paris VI, 2011], allowing us to establish the ASCLT without lengthy computations of star contractions. Moreover, the second approach is expected to be useful for similar studies on SPDEs with colored-in-time noises, when the former approach, based on Itô calculus, is not applicable.

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Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
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Abstract: