Intermittence and nonlinear parabolic stochastic partial differential equations

TL;DR

This paper analyzes nonlinear parabolic stochastic PDEs driven by space-time white noise, providing existence, uniqueness, growth rate estimates, and conditions for weak intermittency, with applications to the parabolic Anderson model.

Contribution

It offers new criteria for solution existence and uniqueness, growth bounds, and characterizes intermittency for a broad class of nonlinear SPDEs with Lévy generators.

Findings

Solutions grow at most exponentially in time.

Weak intermittency occurs under specific conditions.

Formulas for the second-moment Lyapunov exponent are derived.

Abstract

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-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 $\sL$ 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 $\sL$ in dimension $(1+1)$. When $\sL=\kappa\partial_{xx}$ 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 $u_0\equiv 1$.