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 $$\langle \cdot \rangle $$ . 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 (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 $$G$$ , 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.