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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
71/2014

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

Jonathan Ben-Artzi, Daniel Marahrens and Stefan Neukamm

Abstract

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d\geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.

Received:
Jul 25, 2014
Published:
Aug 7, 2014
MSC Codes:
35B27, 35J08, 60H25, 60F17
Keywords:
stochastic homogenization, corrector equation, quantitative ergodicity

Related publications

inJournal
2017 Repository Open Access
Jonathan Ben-Artzi, Daniel Marahrens and Stefan Neukamm

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

In: Communications in partial differential equations, 42 (2017) 2, pp. 179-234