an:00823408
Zbl 0840.60068
Benjamini, Itai; Pemantle, Robin; Peres, Yuval
Martin capacity for Markov chains
EN
Ann. Probab. 23, No. 3, 1332-1346 (1995).
00029463
1995
j
60J45 60J10 60J65 60G50 60K35
Newtonian capacity; Green kernel; random walks on lattices; percolation on trees
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.
S.L.Kalpazidou (Thessaloniki)