A central limit theorem for fluctuations in one dimensional stochastic homogenization

TL;DR

This paper establishes a central limit theorem for the fluctuations in one-dimensional stochastic homogenization, showing that the first order fluctuations are Gaussian and described by an SPDE with white noise.

Contribution

It provides a probabilistic proof of the Gaussian fluctuation limit and decomposes the limiting process, linking it to the corrector from two-scale expansion.

Findings

First order fluctuations are Gaussian.

The limiting process solves an SPDE with additive white noise.

Error decomposition up to first order is achieved.

Abstract

In this paper, we analyze the random fluctuations in a one dimensional stochastic homogenization problem and prove a central limit result, i.e., the first order fluctuations can be described by a Gaussian process that solves an SPDE with additive spatial white noise. Using a probabilistic approach, we obtain a precise error decomposition up to the first order, which helps to decompose the limiting Gaussian process, with one of the components corresponding to the corrector obtained by a formal two scale expansion.