Martin Boundaries of Random Walks: Ends of Trees and Groups

Abstract

Consider a transient random walk ${X_n}$ on an infinite tree $T$ whose nonzero transition probabilities are bounded below. Suppose that ${X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, ${X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $T$.) The Martin boundary of ${X_n}$ is shown to coincide with the space $\Omega$ of all ends of $T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $\Omega$ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.