The Annals of Probability

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

Zhen-Qing Chen, David A. Croydon, and Takashi Kumagai

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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.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1594-1642.

Received: June 2013
Revised: January 2014
First available in Project Euclid: 3 June 2015

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Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles
Secondary: 31C25: Dirichlet spaces 35K08: Heat kernel 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]

Quenched invariance principle Dirichlet form heat kernel supercritical percolation random conductance model


Chen, Zhen-Qing; Croydon, David A.; Kumagai, Takashi. Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. Ann. Probab. 43 (2015), no. 4, 1594--1642. doi:10.1214/14-AOP914.

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