Corrector Estimates for Elliptic Systems with Random Periodic Coefficients

We consider an elliptic system of equations on the torus [ −L2 , L 2 )d with random coefficients A, that are assumed to be coercive and stationary. Using two different approaches we obtain moment bounds on the gradient of the corrector, independent of the domain size L. In the first approach we use Green function representation. For that we require A to be locally Hölder continuous and distribution of A to satisfy Logarithmic Sobolev inequality. The second method works for non-smooth (possibly discontinuous) coefficients, and it requires that statistics of A satisfies Spectral Gap estimate.