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

Consider the stochastic heat equation $partial_tu=mathscr{L}u+lambdasigma(u)xi$, where $mathscr{L}$ denotes the generator of a L{e}vy process on a locally compact Hausdorff Abelian group $G$, $sigma:mathbf{R}tomathbf{R}$ is Lipschitz continuous, $lambdagg1$ 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 $geexp{operatorname {const}cdot,lambda^4}$ when $G$ is connected.