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

Abstract

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 ${f (X _n)}_{n\geq 1}$ and show that it converges almost surely and in $L^2$. Analogous results are also established for reversible diffusions on Riemannian manifolds.