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July 2015 Quenched invariance principles for random walks and elliptic diffusions in random media with boundary
Zhen-Qing Chen, David A. Croydon, Takashi Kumagai
Ann. Probab. 43(4): 1594-1642 (July 2015). DOI: 10.1214/14-AOP914

Abstract

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

Citation

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Zhen-Qing Chen. David A. Croydon. Takashi Kumagai. "Quenched invariance principles for random walks and elliptic diffusions in random media with boundary." Ann. Probab. 43 (4) 1594 - 1642, July 2015. https://doi.org/10.1214/14-AOP914

Information

Received: 1 June 2013; Revised: 1 January 2014; Published: July 2015
First available in Project Euclid: 3 June 2015

zbMATH: 1338.60098
MathSciNet: MR3353810
Digital Object Identifier: 10.1214/14-AOP914

Subjects:
Primary: 60F17, 60K37
Secondary: 31C25, 35K08, 82C41

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 4 • July 2015
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