标题： 间歇性和非线性抛物型随机偏微分方程 显示英文标题

标题： Intermittence and nonlinear parabolic stochastic partial differential equations

摘要： 我们考虑形式为$\partial_t u=\sL u + \sigma(u)\dot w$的非线性抛物型随机偏微分方程，其中$\dot w$表示时空白噪声，$\sigma:\R\to\R$是[全局]利普希茨连续的，$\sL$是一个 Lévy 过程的$L^2$-生成元。我们给出了存在性和唯一性的精确条件。更重要的是，我们证明这些解随时间增长的速率至多为精确的指数速率。我们还证明，当$\sigma$是全局利普希茨且渐近次线性时，如果$\sL$的对称化是常返的，并且初始数据足够大，则非线性热方程的解是“弱间歇性”的。 在其他方面，我们的结果导致了抛物线安德森模型在维度$(1+1)$中的上二阶矩李雅普诺夫指数的一般公式，对于$\sL$。 当$\sL=\kappa\partial_{xx}$对于$\kappa>0$时，这些公式与统计物理的早期结果\cite{Kardar,KrugSpohn,LL63}以及概率理论\cite{BC,CM94}在两个精确可解的情况下一致，其中$u_0=\delta_0$和$u_0\equiv 1$。 显示英文摘要

摘要： 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 $u_0\equiv 1$.