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

Consider the stochastic heat equation ${\mathrm{\partial }}_{t}u=\mathcal{L}u+\lambda \sigma (u)\xi$, where $\mathcal{L}$ 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, $\lambda \gg 1$ 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 $\mathrm{E}(\mathrm{\parallel }{u}_t{\mathrm{\parallel }}_{L^2(G)}^2)$ 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 $\exp \{\mathrm{const}\cdot {\lambda }^2\}$ when $G$ is discrete and $\ge \exp \{\mathrm{const}\cdot {\lambda }^4\}$ when $G$ is connected.