Crossing estimates and convergence of Dirichlet functions along random walk and diffusion paths. (English) Zbl 0945.60063

The authors establish a crossing estimate for a Dirichlet function \(f\) along the path of a transient reversible Markov chain \(\{X_n\}\) and prove that the process \(\{f(X_n)\): \(n\geq 1\}\) converges almost surely and in \(L^2\). Similar results are also established for a transient symmetric diffusion on a Riemannian manifold. Results on crossing estimates are recently generalized to symmetric right Markov processes by Z.-Q. Chen, P. Fitzsimmons and R. Song [“Crossing estimates for symmetric Markov processes”].


60J45 Probabilistic potential theory
60F15 Strong limit theorems
31C25 Dirichlet forms
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