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 of Gloria & Otto, 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 subinearly, 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.
Highlights
We consider the lattice graph (Zd, Bd), d > 2, where Bd denotes the set of nearest-neighbor edges
Given a stationary and ergodic probability measure · on Ω – the space of conductance fields a : Bd → [0, 1] – we study the corrector equation from stochastic homogenization, i.e. the elliptic difference equation
The goal of the present paper is to extend this result to the case of conductances with degenerate ellipticity
Summary
We consider the lattice graph (Zd, Bd), d > 2, where Bd denotes the set of nearest-neighbor edges. While the asymptotic result of stochastic homogenization holds for general stationary and ergodic coefficients (at least in the uniformly elliptic case), the derivation of error estimates requires a quantification of ergodicity. Since the elliptic estimates that we require are less sensitive to the geometry of the graph, and can be obtained by simpler arguments, we opt for a self-contained proof that only relies on elliptic regularity theory Another interesting, and – as we believe – advantageous property of our approach is that (thanks to the Spectral Gap Estimate) probabilistic and deterministic considerations are well separated, e.g. Proposition 1 is pointwise in a and does not involve the ensemble.
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