The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations

TL;DR

This paper establishes optimal decay rates and stochastic integrability bounds for the gradient and flux of the corrector in stochastic homogenization, revealing their sub-Gaussian behavior and fluctuations.

Contribution

It introduces new optimal decay rates and stochastic integrability bounds for the corrector's gradient and flux in stochastic homogenization, utilizing the self-averaging property of the semi-group.

Findings

Gradient and flux decay like R^{-d/2} with optimal rates.

Sub-Gaussian bounds established for stochastic integrability.

Characterization of fluctuations and bounds on the corrector's growth.

Abstract

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux $(\nabla\phi,a(\nabla \phi+e))$ of the corrector $\phi$, when spatially averaged over a scale $R\gg 1$ decay like the CLT scaling $R^{-\frac{d}{2}}$. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization.