Almost sure central limit theorem for the hyperbolic Anderson model with L\'evy white noise

TL;DR

This paper establishes an almost sure central limit theorem for the hyperbolic Anderson model driven by Lévy white noise, providing two novel proofs that avoid complex contractions and extend applicability to colored-in-time noises.

Contribution

The paper introduces two distinct proofs of the ASCLT for the HAM with Lévy white noise, expanding theoretical understanding and methodological tools for SPDEs.

Findings

Proves ASCLT for hyperbolic Anderson model with Lévy white noise.

Develops two different proof techniques avoiding lengthy contractions.

Extends potential applications to SPDEs with colored-in-time noises.

Abstract

In this paper, we present an almost sure central limit theorem (ASCLT) for the hyperbolic Anderson model (HAM) with a L\'evy white noise in a finite-variance setting, complementing a recent work by Balan and Zheng (\emph{Trans.~Amer.~Math.~Soc.}, 2024) 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 Poincar\'e inequality with Ibragimov and Lifshits' method of characteristic functions. Both approaches are different from the one developed in the PhD thesis of C. Zheng (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, whereas the former, based on It\^o calculus, is not applicable.