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

Consider the stochastic heat equation .tu=Lu+..(u)., where L denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group G, .:R.R is Lipschitz continuous, ..1 is a large parameter, and . denotes space.time white noise on R+.G . The main result of this paper contains a near-dichotomy for the (expected squared) energy E(.ut.2L2(G)) 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{const..2} when G is discrete and .exp{const..4} when G is connected.