On annealed elliptic Green function estimates

2014/01/13 by Marahrens, Daniel, Otto, Felix
#35B27 (primary) #35J08 #39A70 #60H25 (secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1401.2859

Abstract

We consider a random, uniformly elliptic coefficient field a on the lattice ℤ^d. 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 ⟨ |∇_x G(t,x,y)|² ⟩^(1)/(2) and ⟨ |∇_x ∇_y G(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. L^p in probability for all p

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