The Annals of Probability

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

Alano Ancona, Russell Lyons, and Yuval Peres

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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.

Article information

Ann. Probab., Volume 27, Number 2 (1999), 970-989.

First available in Project Euclid: 29 May 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]
Secondary: 31C25: Dirichlet spaces 60F15: Strong theorems

Dirichlet energy random walk almost sure convergence Markov chain diffusions manifolds crossing


Ancona, Alano; Lyons, Russell; Peres, Yuval. Crossing Estimates and Convergence of Dirichlet Functions Along Random Walk and Diffusion Paths. Ann. Probab. 27 (1999), no. 2, 970--989. doi:10.1214/aop/1022677392.

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