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 ${\boldsymbol\xi}:=\{\xi_n(x)\}_{n\ge 0,x\in{\mathbf Z}^d}$ 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.