Annealed estimates on the Green function

Abstract

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 $⟨\cdot ⟩$. We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, $⟨|{\nabla }_{x}G(x,y){|}^p⟩$ and $⟨|{\nabla }_x{\nabla }_{y}G(x,y){|}^p⟩$, 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, $⟨|{\nabla }_{x}G(x,y){|}^2⟩$ and $⟨|{\nabla }_x{\nabla }_{y}G(x,y)|⟩$. As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

37 pages

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• Noise-stability and central limit theorems for effective resistance of random electric networks

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