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

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.

arXiv commentV2:15pages, minor revision; V1: 15pages

References in corpus (11)

• A Central Limit Theorem for the stochastic wave equation with fractional noise
• Gaussian fluctuations for the stochastic heat equation with colored noise
• Central limit theorems for stochastic wave equations in dimensions one and two
• Averaging Gaussian functionals
• Averaging 2d stochastic wave equation
• Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals
• The hyperbolic Anderson model: Moment estimates of the Malliavin derivatives and applications
• Spatial averages for the Parabolic Anderson model driven by rough noise
• Spatial ergodicity of stochastic wave equations in dimensions 1,2 and 3
• Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition
• Central limit theorems for nonlinear stochastic wave equations in dimension three