The heat equation with time-correlated random potential in d=2: Edwards-Wilkinson fluctuations

TL;DR

This paper studies a stochastic heat equation in two dimensions with a mollified white noise potential, demonstrating that its fluctuations converge to the Edwards-Wilkinson universality class after renormalization, with explicit parameters.

Contribution

It establishes the convergence of fluctuations to the Edwards-Wilkinson limit for a 2D stochastic PDE with mollified noise, extending the Kallianpur-Robbins law to a regenerative process.

Findings

Fluctuations converge to Edwards-Wilkinson limit with explicit variance.

Explicit effective diffusivity is derived.

Renormalization is necessary for convergence.

Abstract

We consider the stochastic PDE: $\partial_tu(t,x)=\frac{1}{2}\Delta u(t,x)+{\beta}{}u(t,x)V(t,x),$ in dimension $d=2$, where the potential V is the space and time mollification of the two-dimensional space-time white noise. We show that after renormalizing, the fluctuations of the solution converge to the Edwards-Wilkinson limit with an explicit effective variance and constant effective diffusivity. Our main tool is a Markov chain on the space of paths which we use to establish an extension of the Kallianpur-Robbins law to a specific regenerative process.