Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
Raluca M. Balan, Panqiu Xia, Guangqu Zheng
TL;DR
This work establishes an almost-sure central limit theorem for the spatial integral fluctuations of the 1D hyperbolic Anderson model driven by a finite-variance Lévy white noise. It develops two independent proofs: a Clark-Ocone/Malliavin calculus approach that uses Itô representations of the centered spatial function and covariance bounds, and a second approach that combines the Ibragimov-Lifshits criterion with second-order Gaussian Poincaré inequalities to manage contractions. The ASCLT is proved for the normalized functional $\widetilde{F}_\theta=F_\theta/\sigma_\theta$ under the moment condition $m_{1+\alpha}+m_{2+2\alpha}<\infty$, with potential applicability to SPDEs driven by colored-in-time noise. Overall, the results extend ASCLT phenomena to SPDEs with Lévy noise in the finite-variance regime and provide robust methodological tools for ergodic and fluctuation analyses in such systems.
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 (\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 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 (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 Itô calculus, is not applicable.