Intermittence and nonlinear parabolic stochastic partial differential equations

We consider nonlinear parabolic SPDEs of the form ∂ _tu = ℒu+σ(u)w, where ?w denotes spacetime white noise, σ : R → R is [globally] Lipschitz continuous, and ℒ is the L ²-generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is "weakly intermittent," provided that the symmetrization of ℒ is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for ℒ in dimension (1+1). When ℒ = Κ∂ _xx for κ > 0, these formulas agree with the earlier results of statistical physics [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u ₀ = δ ₀ or u ₀ = 1; in those cases the moments of the solution to the SPDE can be computed [1].