Polar and Nonpolar Sets for a Tree Indexed Process

Abstract

We consider a class of stochastic processes of a type that was first introduced by Dubins and Freedman. These processes are indexed by the lines of descent through an infinite tree and take values in a space of sequences. Our main results concern necessary and sufficient conditions of a potential theoretic type for a subset of the state-space to be hit with positive probability by the sample paths of the process. We examine these conditions in some specific examples and also relate them to conditions expressed in terms of Hausdorff dimension. As well, we use similar techniques to investigate multiple points in the sample paths of the process.