ANISOTROPIC DIFFUSION ON TOTALLY DISCONNECTED ABELIAN GROUPS

We consider a locally compact, noncompact, totally disconnected, nondis-crete, metrizable abelian group G that is the union of a countable chain of compact subgroups. On G we consider a stationary standard Markov pro-cess defined by a semigroup µt of probability measures, satisfying µs+t = µs ∗ µt and limt→0 µt = δ0, and we consider the Lévy measure associated to the process through the Lévy–Khintchine formula. Under the hypothesis that the Lévy measure is unbounded, we show that the process may be ob-tained as a limit of discrete processes defined on the discrete quotient groups G/Gn, where Gn is a descending chain of compact open subgroups. These discrete processes, in turn, are defined by means of a random walk on a homogeneous tree, naturally associated to G. 1. Introduction an