The stochastic heat equation with multiplicative L\'evy noise: Existence, moments, and intermittency

TL;DR

This paper investigates the stochastic heat equation driven by multiplicative Lévy noise, establishing existence, uniqueness, moment bounds, and demonstrating strong intermittency across all dimensions and noise intensities.

Contribution

It provides the first comprehensive analysis of the SHE with Lévy noise, including optimal existence conditions, moment estimates, and the proof of strong intermittency for all non-trivial Lévy measures.

Findings

Existence of solutions under optimal conditions in dimensions 1 and 2.

Finite moments of solutions for all positive orders when noise has finite moments.

Solution exhibits strong intermittency for any non-trivial Lévy measure at all noise levels.

Abstract

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 \Delta u + \beta u \xi$ driven by a multiplicative L\'evy noise $\xi$ with positive jumps and amplitude $\beta>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple L\'evy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $\beta\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial L\'evy measure, at any disorder intensity $\beta>0$, in any dimension $d\geq1$.