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) = (Lu_n)(x)+ σ(u _n(x))ζ_n(x), for n ∈ Z₊ and x ∈ Z^d, where ζ := {ζ_n(x)} _n≥0,x∈Zd denotes random forcing and L 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.