Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

TL;DR

This paper investigates the behavior of solutions to a stochastic heat equation on LCA groups, revealing a dichotomy in energy growth depending on whether the group is discrete or connected, especially under large noise excitation.

Contribution

It establishes a near-dichotomy in the energy growth of solutions to stochastic PDEs on LCA groups, linking the topology of the group to the solution's intermittency behavior.

Findings

Energy grows as exp(constant * λ^2) on discrete groups.

Energy grows as exp(constant * λ^4) on connected groups.

Provides a topological classification of intermittency in stochastic PDEs.

Abstract

Consider the stochastic heat equation $\partial_tu=\mathscr{L}u+\lambda\sigma(u)\xi$, where $\mathscr{L}$ denotes the generator of a L\'{e}vy process on a locally compact Hausdorff Abelian group $G$, $\sigma:\mathbf{R}\to\mathbf{R}$ is Lipschitz continuous, $\lambda\gg1$ is a large parameter, and $\xi$ denotes space-time white noise on $\mathbf{R}_+\times G$. The main result of this paper contains a near-dichotomy for the (expected squared) energy $\mathrm{E}(\|u_t\|_{L^2(G)}^2)$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $u$ is intermittent, the energy of the solution behaves generically as $\exp\{\operatorname {const}\cdot\,\lambda^2\}$ when $G$ is discrete and $\ge\exp\{\operatorname {const}\cdot\,\lambda^4\}$ when $G$ is connected.