# Quantitative stochastic homogenization and large-scale regularity

## TL;DR

This work provides a comprehensive and simplified presentation of quantitative stochastic homogenization and large-scale regularity theory for elliptic equations, including new results and optimal estimates for various related problems.

## Contribution

It introduces new simplified proofs and several novel results, such as optimal estimates for the Dirichlet problem and Green functions, advancing the understanding of stochastic homogenization.

## Key findings

- Algebraic convergence rate for variational subadditive quantities
- Large-scale Lipschitz and higher regularity estimates
- Optimal quantitative estimates for homogenization error and two-scale expansion

## Abstract

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. There are several chapters containing new results, such as: quantitative estimates for the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion; optimal estimates for the homogenization of the parabolic and elliptic Green functions; and $W^{1,p}$-type estimates for two-scale expansions.