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

Abstract

Consider the stochastic heat equation $\partial_{t}u=\mathscr{L}u+\lambda\sigma(u)\xi$, where $\mathscr{L}$ denotes the generator of a Lévy 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.