Intermittency for the stochastic heat equation with L\'evy noise

TL;DR

This paper studies the growth of moments in solutions to the stochastic heat equation driven by Lévy noise, revealing stronger intermittency effects compared to Gaussian noise and analyzing how parameters influence this behavior.

Contribution

It introduces a new moment lower bound for stochastic integrals with Lévy noise and characterizes intermittency in the heat equation driven by jump processes across dimensions.

Findings

Intermittency occurs for all p in (1,3) in 1D and some p in (1,1+2/d) in higher dimensions.

Solutions exhibit stronger intermittency than in Gaussian noise cases.

Parameters like diffusion constant and noise intensity significantly affect intermittency.

Abstract

We investigate the moment asymptotics of the solution to the stochastic heat equation driven by a $(d+1)$-dimensional L\'evy space--time white noise. Unlike the case of Gaussian noise, the solution typically has no finite moments of order $1+2/d$ or higher. Intermittency of order $p$, that is, the exponential growth of the $p$th moment as time tends to infinity, is established in dimension $d=1$ for all values $p\in(1,3)$, and in higher dimensions for some $p\in(1,1+2/d)$. The proof relies on a new moment lower bound for stochastic integrals against compensated Poisson measures. The behavior of the intermittency exponents when $p\to 1+2/d$ further indicates that intermittency in the presence of jumps is much stronger than in equations with Gaussian noise. The effect of other parameters like the diffusion constant or the noise intensity on intermittency will also be analyzed in detail.