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

Let {X n } be a transient reversible Markov chain and let f be a function on the state space with finite Dirichlet energy. We prove crossing inequalities for the process {f(X n )} n≥1 and show that it converges almost surely and in L 2 . Analogous results are also established for reversible diffusions on Riemannian manifolds.