Moment bounds for the corrector in stochastic homogenization of a percolation model

TL;DR

This paper proves the existence and boundedness of all moments of the corrector in a degenerate stochastic homogenization model based on a conditioned Bernoulli percolation, extending previous elliptic results.

Contribution

It establishes the existence and finite moments of the corrector in a degenerate percolation model, generalizing prior results from uniformly elliptic to degenerate cases.

Findings

All finite moments of the corrector are bounded.

Corrector grows sublinearly, slower than any polynomial.

Existence of stationary correctors in the degenerate model.

Abstract

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $\mathbb{Z}^d$, $d>2$. The model is obtained from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result in [GO1], where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows sublinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green's function.