The probability that a transient Markov chain, or a Brownian path, will ever visit a given set $\Lambda$ is classically estimated using the capacity of $\Lambda$ with respect to the Green kernel $G(x, y)$. We 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 also to explain a connection found by Lyons between capacity and percolation on trees.
"Martin Capacity for Markov Chains." Ann. Probab. 23 (3) 1332 - 1346, July, 1995. https://doi.org/10.1214/aop/1176988187