Stochastic wave equation with heavy-tailed noise: Uniqueness of solutions and past light-cone property

TL;DR

This paper investigates the stochastic wave equation in low dimensions with heavy-tailed Lévy noise, establishing existence and uniqueness of solutions based on the noise's integrability properties, and leveraging the wave equation's past light-cone feature.

Contribution

It demonstrates the existence and uniqueness of solutions for the stochastic wave equation with heavy-tailed noise, under minimal integrability conditions, especially in the critical case of spatial dimension two.

Findings

Existence and uniqueness of solutions for $d \,\leq 2$

No restrictions on Lévy measure in 1D

Conditions on Lévy measure for 2D case

Abstract

In this article, we study the stochastic wave equation in spatial dimensions $d \le 2$ with multiplicative L\'evy noise that can have infinite $p$-th moments. Using the past light-cone property of the wave equation, we prove the existence and uniqueness of a solution, considering only the $p$-integrability of the L\'evy measure $\nu$ for the region corresponding to the small jumps of the noise. For $d=1$, there are no restrictions on $\nu$. For $d=2$, we assume that there exists a value $p \in (0,2)$ for which $\int_{ \{|z|\le 1 \} } |z|^p \nu (dz) < + \infty$.