Extremes of the stochastic heat equation with additive Lévy noise

Chong, Carsten; Kevei, Péter

doi:10.1214/22-ejp855

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…At the end of this introduction, let us also mention that the stochastic heat equation with multiplicative Lévy noises σ(u) L, with σ Lipschitz, has been studied in a series of recent papers. The existence of the solution was proved in [13], weak intermittency property was established in [15], some path properties were obtained in [14], and the exact tail behavior was described in [16] in the case of additive noise (i.e. when u L is replaced by L).…”

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confidence: 99%

Hyperbolic Anderson model with Lévy white noise: spatial ergodicity and fluctuation

Balan¹,

Zheng²

2023

Preprint

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In this paper, we study one-dimensional hyperbolic Anderson models (HAM) driven by space-time Lévy white noise in a finite-variance setting. Motivated by recent active research on limit theorems for stochastic partial differential equations driven by Gaussian noises, we present the first study in this Lévy setting. In particular, we first establish the spatial ergodicity of the solution and then a quantitative central limit theorem (CLT) for the spatial averages of the solution to HAM in both Wasserstein distance and Kolmogorov distance, with the same rate of convergence. To achieve the first goal (i.e. spatial ergodicity), we exploit some basic properties of the solution and apply a Poincaré inequality in the Poisson setting, which requires delicate moment estimates on the Malliavin derivatives of the solution. Such moment estimates are obtained in a soft manner by observing a natural connection between the Malliavin derivatives of HAM and a HAM with Dirac delta velocity. To achieve the second goal (i.e. CLT), we need two key ingredients: (i) a univariate second-order Poincaré inequality in the Poisson setting by Last, Peccati, and Schulte (Probab. Theory Related Fields, 2016); (ii) aforementioned moment estimates of Malliavin derivatives up to second order. We also establish a corresponding functional central limit theorem by (a) showing the convergence in finite-dimensional distributions and (b) verifying Kolmogorov's tightness criterion. Part (a) is made possible by a multivariate version of second-order Poincaré inequality by Schulte and Yukich (Electron. J. Probab., 2019), while part (b) follows from a standard moment estimate with an application of Rosenthal's inequality.

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