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

Zhen-Qing Chen University of Washington David A. Croydon University of Warwick Takashi Kumagai Kyoto University

Probability mathscidoc:1703.28002

The Annals of Probability, 43, (4), 1594-1642, 2015
Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for ran- dom 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 prin- ciples for domains with more general boundaries.
Quenched invariance principle, Dirichlet form, heat kernel, supercritical percolation, random conductance model
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@inproceedings{zhen-qing2015quenched,
  title={Quenched invariance principles for random walks and elliptic diffusions in random media with boundary},
  author={Zhen-Qing Chen, David A. Croydon, and Takashi Kumagai},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170315043825773360711},
  booktitle={The Annals of Probability},
  volume={43},
  number={4},
  pages={1594-1642},
  year={2015},
}
Zhen-Qing Chen, David A. Croydon, and Takashi Kumagai. Quenched invariance principles for random walks and elliptic diffusions in random media with boundary. 2015. Vol. 43. In The Annals of Probability. pp.1594-1642. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170315043825773360711.
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