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. Let fXng be a transient reversible Markov chain and let f be afunction on the state space with nite Dirichlet energy. We prove crossinginequalities for the process ff(Xn)gn1 and show that it converges almostsurely and in L2. Analogous results are also established for reversible diusionson Riemannian manifolds.x1. Introduction.Benjamini and Schramm (1996) proved that transient planar graphs with bounded degreesupport nonconstant harmonic Dirichlet functions. In the course of...

Crossing Estimates and Convergence of Dirichlet Functions Along Random Walk and Diffusion Paths