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.