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

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coecients. Under the assumption that the coecients 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 d2. Similar estimates have recently been obtained in the special case of diagonal coecients 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. In the critical case d=2, our argument for the estimate on the gradient of the elliptic Green's function uses a Calderon-Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coecients.