Annealed estimates on the Green function

We consider a random, uniformly elliptic coefficient field on the -dimensional integer lattice . We are interested in the spatial decay of the quenched elliptic Green function . Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble . We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, and , have the same decay rates in as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358-390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of , that is, and . As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.