Open Access
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


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.


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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.


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

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

Keywords: Dirichlet form , heat kernel , quenched invariance principle , Random conductance model , Supercritical percolation

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 4 • July 2015
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