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

摘要

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.