Central limit theorem for stochastic nonlinear wave equation with pure-jump Lévy white noise
Raluca M. Balan, Guangqu Zheng
TL;DR
This work addresses the Gaussian fluctuations of spatial averages for the stochastic nonlinear wave equation driven by pure-jump Lévy noise with Lipschitz nonlinearity. By developing Malliavin calculus on Poisson space and a discrete Malliavin-Stein framework, the authors prove Malliavin differentiability of the solution, establish spatial ergodicity, and derive quantitative central and functional limit theorems for the spatial averages with explicit Wasserstein-rate bounds. The results extend previous linear or Gaussian-noise analyses to nonlinearities and Lévy noise, yielding almost sure limits and asymptotic independence results for the spatial statistics. The methods offer a robust probabilistic description of fluctuations in SPDEs with jump noise, with potential applications to intermittency and spatial statistics in random media.
Abstract
In this paper, we study the random field solution to the stochastic nonlinear wave equation (SNLW) with constant initial conditions and multiplicative noise $σ(u)\dot{L}$, where the nonlinearity is encoded in a Lipschitz function $σ: \mathbb{R}\to\mathbb{R}$ and $\dot{L}$ denotes a pure-jump Lévy white noise on $\mathbb{R}_+\times\mathbb{R}$ with finite variance. Combining tools from Itô calculus and Malliavin calculus, we are able to establish the Malliavin differentiability of the solution with sharp moment bounds for the Malliavin derivatives. As an easy consequence, we obtain the spatial ergodicity of the solution to SNLW that leads to a law of large number result for the spatial integrals of the solution over $[-R, R]$ as $R\to\infty$. One of the main results of this paper is the obtention of the corresponding Gaussian fluctuation with rate of convergence in Wasserstein distance. To achieve this goal, we adapt the discrete Malliavin-Stein bound from Peccati, Solé, Taqqu, and Utzet ({\it Ann. Probab.}, 2010), and further combine it with the aforementioned moment bounds of Malliavin derivatives and Itô tools. Our work substantially improves our previous results (\textit{Trans.~Amer.~Math.~Soc.}, 2024) on the linear equation that heavily relied on the explicit chaos expansion of the solution. In current work, we also establish a functional version, an almost sure version of the central limit theorems, and the (quantitative) asymptotic independence of spatial integrals from the solution. The asymptotic independence result is established based on an observation of L. Pimentel (\textit{Ann.~Probab.}, 2022) and a further adaptation of Tudor's generalization (\textit{Trans.~Amer.~Math.~Soc.}, 2025) to the Poisson setting.