## 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.

Original language | English |
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Pages (from-to) | 1594-1642 |

Number of pages | 49 |

Journal | Annals of Probability |

Volume | 43 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

## Keywords

- Dirichlet form
- Heat kernel
- Quenched invariance principle
- Random conductance model
- Supercritical percolation

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty