  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.
