Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations
Jonathan Ben-Artzi, Daniel Marahrens,and Stefan Neukamm
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Submission date: 25. Jul. 2014 (revised version: September 2014)
published in: Communications in partial differential equations, 42 (2017) 2, p. 179-234
DOI number (of the published article): 10.1080/03605302.2017.1281298
MSC-Numbers: 35B27, 35J08, 60H25, 60F17
Keywords and phrases: stochastic homogenization, corrector equation, quantitative ergodicity
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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 ≥ 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.