Homogenization with large spatial random potential

TL;DR

This paper studies the homogenization of parabolic equations with large spatial Gaussian random potentials, deriving effective equations and analyzing the behavior of fluctuations in different spatial dimensions and potential types.

Contribution

It provides new homogenization results for equations with large spatially-dependent Gaussian potentials, including cases with long-range correlations and different spatial dimensions.

Findings

Fluctuations converge to Gaussian random variables in law.

Homogenized equations derived for vanishing correlation length.

Results extend to long-range random potentials.

Abstract

We consider the homogenization of parabolic equations with large spatially-dependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We characterize the leading effect in the random fluctuations and show that their spatial moments converge in law to Gaussian random variables. Both results hold for sufficiently small times and in sufficiently large spatial dimensions $d\geq\m$, where $\m$ is the order of the spatial pseudo-differential operator in the parabolic equation. In dimension $d<\m$, the solution to the parabolic equation is shown to converge to the (non-deterministic) solution of a stochastic equation in the companion paper [2]. The results are then extended to cover the case of long range random potentials, which generate larger, but still asymptotically Gaussian, random fluctuations.