An asymptotic theory for randomly forced discrete nonlinear heat equations

We study discrete nonlinear parabolic stochastic heat equations of the form, $u_{n+1}(x)-u_n(x)=(\mathcal{L}{u}_n)(x)+\sigma (u_n(x)){\xi }_n(x)$, for $n\in {$\mathbf{Z}$}+$and$x\in {$\mathbf{Z}$}^d$,where$\boldsymbol \xi:={\xi_n(x)}{n\ge 0,x\in {$\mathbf{Z}$}^d}$denotesrandomforcingand$\mathcal {L}$thegeneratorofarandomwalkon${$\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.