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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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