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.
Categories: math.PR