An asymptotic theory for randomly-forced discrete nonlinear heat equations

We study discrete nonlinear parabolic stochastic heat equations of the form, $u_{n+1}(x) - u_n(x) = (\sL u_n)(x) + \sigma(u_n(x))\xi_n(x)$, for $n\in \Z_+$ and $x\in \Z^d$, where $\bm\xi:=\{\xi_n(x)\}_{n\ge 0,x\in\Z^d}$ denotes random forcing and $\sL$ the generator of a random walk on $\Z^d$. 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.