An asymptotic theory for randomly forced discrete nonlinear heat equations

Abstract

We study discrete nonlinear parabolic stochastic heat equations of the form, $u_{n+1}(x)-u_n(x)=(\mathcal{L}{u}_n)(x)+\sigma (u_n(x)){\xi }_n(x)$, for $n\in {\mathbf{Z}}_{+}$ and $x\in {\mathbf{Z}}^d$, where $\mathbf{\xi }:=\{{\xi }_n(x){\}}_{n\ge 0,x\in {\mathbf{Z}}^d}$ denotes random forcing and $\mathcal{L}$ the generator of a random walk on ${\mathbf{Z}}^d$. Under mild conditions, we prove that the preceding stochastic PDE has a unique solution that grows at most exponentially in time. And that, under natural conditions, it is "weakly intermittent." Along the way, we establish a comparison principle as well as a finite support property.

Published in at http://dx.doi.org/10.3150/11-BEJ357 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• The universality classes in the parabolic Anderson model
• A two cities theorem for the parabolic Anderson model
• Intermittency in a catalytic random medium