SPDEs with Infinite-Variance Lévy Noise

Abstract

This thesis is devoted to the study of the existence and uniqueness of solutions for stochastic partial differential equations (SPDEs) driven by Lévy noise. The main contributions of this work are contained in the recent publications [32] and [5]. Article [32] focuses on a stochastic wave equation with multiplicative Lévy noise. We establish the existence and uniqueness of a random field solution, relying only on the integrability of the Lévy measure on the region |z| ≤ 1. Furthermore, we show that this solution has finite moments up to a certain stopping time, which depends on a bounded region of space. Article [5] studies a broader class of SPDEs driven by heavy-tailed Lévy noise, which includes the Parabolic Anderson Model (PAM) and the Hyperbolic Anderson Model (HAM). Specifically, we demonstrate the existence of solutions for SPDEs driven by symmetric α-stable Lévy noise. Using the Lepage representation of the noise and techniques borrowed from the theory of multiple stable integrals, we construct a solution that has a series representation which depends only on the points of the jump measure associated with the noise.