Electronic Journal of Probability

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

Agnes Lamacz, Stefan Neukamm, and Felix Otto

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 106, 30 pp.

Accepted: 14 October 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39A70: Difference operators [See also 47B39]
Secondary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles

quantitative stochastic homogenization percolation corrector

This work is licensed under aCreative Commons Attribution 3.0 License.


Lamacz, Agnes; Neukamm, Stefan; Otto, Felix. Moment bounds for the corrector in stochastic homogenization of a percolation model. Electron. J. Probab. 20 (2015), paper no. 106, 30 pp. doi:10.1214/EJP.v20-3618. https://projecteuclid.org/euclid.ejp/1465067212

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