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

Abstract

Consider the stochastic heat equation partial derivative(t)u = Lu + lambda sigma (u)xi, where L denotes the generator of a Levy process on a locally compact Hausdorff Abelian group G, sigma : R -> R is Lipschitz continuous, lambda >> 1 is a large parameter, and xi denotes space time white noise on R+ x G.

The main result of this paper contains a near-dichotomy for the (expected squared) energy E(parallel to u(t)parallel to(2)(L2(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.lambda(2)} when G is discrete and > exp(const.lambda(4)) when G is connected.