Stable Processes on the Boundary of a Regular Tree

Abstract

We define a class of processes on the boundary of a regular tree that can be viewed as “stable” Lévy processes on $(\mathbb{Z}/n_0\mathbb{Z})^\mathbb{N}$. We show that the range of these processes can be compared with a Bernoulli percolation as in Peres which easily leads to various results on the intersection properties. We develop an alternative approach based on the comparison with a branching random walk. By this method we establish the existence of points of in finite multiplicity when the index of the process equals the dimension of the state space, as for planar Brownian motion.