An asymptotic theory for randomly forced discrete nonlinear heat equations

TL;DR

This paper develops an asymptotic theory for discrete nonlinear stochastic heat equations with random forcing, establishing existence, growth bounds, weak intermittency, and key properties like comparison and finite support.

Contribution

It introduces a comprehensive asymptotic framework for analyzing solutions to nonlinear stochastic heat equations on discrete spaces, including new growth and intermittency results.

Findings

Unique solutions grow at most exponentially in time

Solutions exhibit weak intermittency under natural conditions

Established comparison principle and finite support property

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.