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Title: On annealed elliptic Green's function estimates (English)
Author: Marahrens, Daniel
Author: Otto, Felix
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 140
Issue: 4
Year: 2015
Pages: 489-506
Summary lang: English
Category: math
Summary: We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb Z^d$. The distribution $\langle \cdot \rangle $ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green's function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\smash {\langle |\nabla _x G(t,x,y)|^2\rangle ^{{1}/{2}}}$ and $\langle |\nabla _x \nabla _y G(t,x,y)|\rangle $. In particular, the elliptic Green's function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors extended these elliptic bounds to higher moments, i.e., $L^p$ in probability for all $p<\infty $. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for $\langle |\nabla _x G(x,y)|^2\rangle ^{{1}/{2}}$ and $\langle |\nabla _x \nabla _y G(x,y)|\rangle $. (English)
Keyword: stochastic homogenization
Keyword: elliptic equation
Keyword: Green's function on $\mathbb Z^d$
Keyword: annealed estimate
MSC: 35A01
MSC: 35Q55
idZBL: Zbl 06537679
idMR: MR3432548
DOI: 10.21136/MB.2015.144465
Date available: 2015-11-17T20:55:01Z
Last updated: 2020-07-29
Stable URL:
Reference: [1] Aronson, D. G.: Bounds for the fundamental solution of a parabolic equation.Bull. Am. Math. Soc. 73 (1967), 890-896. Zbl 0153.42002, MR 0217444, 10.1090/S0002-9904-1967-11830-5
Reference: [2] Delmotte, T.: Parabolic Harnack inequality and estimates of Markov chains on graphs.Rev. Mat. Iberoam. 15 (1999), 181-232. Zbl 0922.60060, MR 1681641, 10.4171/RMI/254
Reference: [3] Delmotte, T., Deuschel, J.-D.: On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to $\nabla\varphi$ interface model.Probab. Theory Relat. Fields 133 (2005), 358-390. Zbl 1083.60082, MR 2198017, 10.1007/s00440-005-0430-y
Reference: [4] Lamacz, A., Neukamm, S., Otto, F.: Moment bounds for the corrector in stochastic homogenization of a percolation model.Electron J. Probab. 20 Article 106, 30 pages, (2015). Zbl 1326.39015, MR 3418538
Reference: [5] Marahrens, D., Otto, F.: Annealed estimates on the Green function.(to appear) in Probab. Theory Relat. Fields, MR 3418749, 10.1007/s00440-014-0598-0
Reference: [6] Nash, J. F.: Continuity of solutions of parabolic and elliptic equations.Am. J. Math. 80 (1958), 931-954. Zbl 0096.06902, MR 0100158, 10.2307/2372841


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