Multiparameter Lévy white noise theory and applications

TL;DR

This work constructs a white-noise theory and calculus for the multiparameter Lévy sheet and its compensated Poisson random measures, enabling analysis of SPDEs driven by Lévy noise with space-time structure. It builds a Lévy white-noise probability space on $\mathcal{S}'(\mathbb{R}^n)$, develops a chaos expansion, and defines Lévy white noises $\overset{\bullet}{L}$ and $\overset{\bullet}{\widetilde N}$, integrating these into a unified stochastic-distribution framework. As an application, it solves the fractional stochastic heat equation $\partial^{\alpha}_t Y = \lambda \Delta Y + \sigma W + \gamma V$ with Caputo time derivative, obtaining a mild solution $Y(t,x)=I_1+I_2+I_3$ with kernels involving the Mittag–Leffler function $E_{\alpha}$ and stochastic convolutions against $W$ and $V$. The framework is illustrated through a tumor invasion model where the Lévy noise captures localized space-time jumps, highlighting how jump noise influences anomalous diffusion and invasion dynamics.

Abstract

We construct a white noise theory and white noise calculus for the (multi-parameter) L\' evy sheet and its compensated Poisson random measures. The theory applies to stochastic partial differential equations subject to L\' evy noise.

Theorems & Definitions (16)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Lemma 2.5
• Theorem 2.6
• Example 2.7
• Example 2.8
• Theorem 2.9
• Example 2.10