On annealed elliptic Green function estimates
Daniel Marahrens and Felix Otto
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Submission date: 18. Feb. 2015
published in: Mathematica bohemica, 140 (2015) 4, p. 489-506
MSC-Numbers: 35B27, 35J08, 39A70, 60H25
Keywords and phrases: stochastic homogenization, elliptic equations, Green function, annealed estimates
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We consider a random, uniformly elliptic coefficient field a on the lattice ℤd. The distribution ⟨⋅⟩ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function G(t,x,y) satisfy optimal annealed estimates which are L2 resp. L1 in probability, i.e. they obtained bounds on ⟨|∇xG(t,x,y)|2⟩ and ⟨|∇x∇yG(t,x,y)|⟩, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the ∇ϕ interface model, Probab. Theory Relat. Fields 133 (2005), 358–390. In particular, the elliptic Green function G(x,y) satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. Lp in probability for all p < ∞, see D. Marahrens and F. Otto: Annealed estimates on the Green function, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition ?? below) for ⟨|∇xG(x,y)|2⟩ and ⟨|∇x∇yG(x,y)|⟩.