Probability Theory and Related Fields

, Volume 163, Issue 3–4, pp 527–573 | Cite as

Annealed estimates on the Green function

  • Daniel MarahrensEmail author
  • Felix Otto


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.


Stochastic homogenization Elliptic equations Green function Annealed estimates 

Mathematics Subject Classification

35B27 35J08 39A70 60H25 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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