Central limits and homogenization in random media

TL;DR

This paper studies how solutions to elliptic operators with rapidly varying random potentials behave, showing they converge to Gaussian processes as the scale of variation shrinks, providing central limit corrections to homogenization.

Contribution

It introduces a rigorous analysis of the asymptotic Gaussian fluctuations in solutions to perturbed elliptic operators with mixing random potentials, extending homogenization theory.

Findings

Perturbed solutions can be expressed as Gaussian processes in the limit.

The results apply to general elliptic equations and one-dimensional cases.

A new integral formulation facilitates the analysis of the asymptotic behavior.

Abstract

We consider the perturbation of elliptic operators of the form $P(\bx,\bD)$ by random, rapidly varying, sufficiently mixing, potentials of the form $q(\frac{\bx}\eps,\omega)$. We analyze the source and spectral problems associated to such operators and show that the properly renormalized difference between the perturbed and unperturbed solutions may be written asymptotically as $\eps\to0$ as explicit Gaussian processes. Such results may be seen as central limit corrections to the homogenization (law of large numbers) process. Similar results are derived for more general elliptic equations in one dimension of space. The results are based on the availability of a rapidly converging integral formulation for the perturbed solutions and on the use of classical central limit results for random processes with appropriate mixing conditions.