Random walks and Kuramochi boundaries of infinite networks

Abstract

In this paper, we study a connected non-parabolic, or transient, network compactified with the Kuramochi boundary, and show that the random walk converges almost surely to a random variable valued in the harmonic boundary, and a function of finite Dirichlet energy converges along the random walk to a random variable almost surely and in $L^{2}$. We also give integral representations of solutions of Poisson equations on the Kuramochi compactification.