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

Agnes Lamacz, Stefan Neukamm, and Felix Otto

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Submission date: 22. Jun. 2014
Pages: 31
published in: Electronic journal of probability, 20 (2015), art-no. 106 
DOI number (of the published article): 10.1214/EJP.v20-3618
Bibtex
Keywords and phrases: quantitative stochastic homogenization, Percolation, corrector
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Abstract:
We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on 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 by [GO11], 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.