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Quenched invariance principles for random walks and elliptic diffusions in random media with boundary

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Chen, Zhen-Qing, Croydon, David A. and Kumagai, Takashi (2015) Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. The Annals of Probability, 43 (4). pp. 1594-1642. doi:10.1214/14-AOP914

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Official URL: http://dx.doi.org/10.1214/14-AOP914

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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 amongst 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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Statistics
Library of Congress Subject Headings (LCSH): Random walks (Mathematics), Dirichlet forms
Journal or Publication Title: The Annals of Probability
Publisher: Institute of Mathematical Statistics
ISSN: 0091-1798
Official Date: 3 June 2015
Dates:
DateEvent
3 June 2015Published
June 2013Submitted
Volume: 43
Number: 4
Page Range: pp. 1594-1642
DOI: 10.1214/14-AOP914
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Open Access
Funder: National Science Foundation (U.S.) (NSF), Guo jia zi ran ke xue ji jin wei yuan hui (China) [National Natural Science Foundation of China] (NSFC), Nihon Gakujutsu Shinkōkai [Japan Society for the Promotion of Science] (NGS)
Grant number: DMS-1206276 (NSF), 11128101 (NSFC), (B) 22340017 (JSPS), (A) 25247007 (JSPS)
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