# Higher-order pathwise theory of fluctuations in stochastic homogenization

## TL;DR

This paper develops a higher-order, pathwise theory for Gaussian fluctuations in stochastic homogenization of elliptic equations, revealing the algebraic structure of correctors and providing new estimates for random coefficients.

## Contribution

It extends the homogenization theory to higher-order fluctuations, offering a detailed algebraic framework and technical estimates for Gaussian random environments.

## Key findings

- Fluctuations are Gaussian at order d/2 in dimension d.
- Higher-order correctors and expansions clarify the structure of fluctuations.
- Introduction of annealed Calderón-Zygmund estimates enhances technical analysis.

## Abstract

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $\frac{d}{2}$, where $d$ is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the homogenization commutator, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calder\'on-Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.