Annealed estimates on the Green's function
Daniel Marahrens and Felix Otto
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Submission date: 14. Nov. 2012 (revised version: April 2013)
published in: Probability theory and related fields, 163 (2014) 3, p. 527-573
DOI number (of the published article): 10.1007/s00440-014-0598-0
MSC-Numbers: 35B27, 35J08, 39A70, 60H25
Keywords and phrases: stochastic homogenization, elliptic equations, green's function, annealed estimates
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We consider a random, uniformly elliptic coefficient field a(x) on the d-dimensional cubic lattice ℤd. We are interested in the spatial decay of the quenched elliptic Green’s function G(a;x,y). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently rapidly 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’s function, that is, ⟨|∇xG(x,y)|p⟩ and ⟨|∇x∇yG(x,y)|p⟩, have the same decay rates in |x - y|≫ 1 as for the constant coefficient Green’s function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel [?], which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of G, that is, ⟨|∇xG(x,y)|2⟩ and ⟨|∇x∇yG(x,y)|⟩. As an application, we derive optimal estimates on the random part of the homogenization error.