Intermittence and nonlinear parabolic stochastic partial differential equations

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$,where$\dot w$denotesspace-timewhitenoise,$\sigma:\R\to\R$is[globally]Lipschitzcontinuous,and$\sL$isthe$L^2$-generator of a L'evy 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 $\sigma$ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,‘’ provided that the symmetrization of $\text{sL}$ 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 $\text{sL}$ in dimension $(1+1)$. When $\text{sL}=\kappa {\mathrm{\partial }}_{xx}$ for $\kappa >0$, these formulas agree with the earlier results of statistical physics \cite{Kardar,KrugSpohn,LL63}, and also probability theory \cite{BC,CM94} in the two exactly-solvable cases where $u_0={\delta }_0$ and $u_0\equiv 1$.