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Martin capacity for Markov chains. (English) Zbl 0840.60068
Kakutani has proved that a compact set $$\Lambda \subseteq \mathbb{R}^d$$ is visited with positive probability by a $$d$$-dimensional Brownian motion $$(d \geq 3)$$ if and only if $$\Lambda$$ has positive Newtonian capacity. A more quantitative relation holds between this probability and capacity. The probability that a transient Markov chain, or a Brownian path will ever visit a given set $$\Lambda$$ is classically estimated by using the capacity of $$\Lambda$$ with respect to the Green kernel $$G(x,y)$$. The authors show that replacing the Green kernel by the Martin kernel $$G(x,y)/G (0,y)$$ yields improved estimates, which are exact up to a factor of 2. These estimates are applied to random walks on lattices and reveal a connection of Lyons-type between capacity and percolation on trees.

##### MSC:
 60J45 Probabilistic potential theory 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60J65 Brownian motion 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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