Annealed estimates on the Green function

We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. 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 $\langle\cdot\rangle$. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $\langle|\nabla_x G(x,y)|^p\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|^p\rangle$, have the same decay rates in $|x-y|\gg 1$ as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel \cite{DeuschelDelmotte}, which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of $G$, that is, $\langle|\nabla_x G(x,y)|^2\rangle$ and $\langle|\nabla_x\nabla_y G(x,y)|\rangle$. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

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