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We consider the corrector equation for the second order elliptic system on the $d$-dimensional torus of size $L$ ($d \\ge 2$), associated 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's function representation. For that we require $A$ to be locally Hoźlder continuous and the distribution of $A$ to satisfy a logarithmic Sobolev inequality. The second method works for nonsmooth (possibly discontinuous) coefficients, and it requires that the statistic of $A$ satisfies a spectral gap estimate.

Corrector Estimates for Elliptic Systems with Random Periodic Coefficients