  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.

Probability