The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher

TL;DR

This paper studies the behavior of the heat equation with random potential in three or more dimensions, showing convergence to a deterministic diffusion and Edwards-Wilkinson fluctuations.

Contribution

It establishes the Edwards-Wilkinson limit for the renormalized solution of the stochastic heat equation in higher dimensions, linking it to a well-known fluctuation model.

Findings

Renormalized solutions converge to a deterministic diffusion equation.

Large-scale fluctuations are described by the Edwards-Wilkinson model.

Effective diffusivity and variance are explicitly characterized.

Abstract

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\geq 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.