NONLINEAR NOISE EXCITATION OF INTERMITTENT STOCHASTIC PDES AND THE TOPOLOGY OF LCA GROUPS

Consider the stochastic heat equation ∂tu = ℒu +λσ(u)ξ, where ℒ denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group G, σ: R → R is Lipschitz continuous, λ ≫ 1 is a large parameter, and ξ denotes space–time white noise on R+ × G. The main result of this paper contains a near-dichotomy for the (expected squared) energy $E(\left \| u_t \right \|^2_L{_{2(G)}})$ 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·λ2} when G is discrete and ≥ exp{const·λ4} when G is connected.