Anisotropic diffusion on totally disconnected abelian groups

We consider a locally compact, noncompact, totally disconnected, nondiscrete, metrizable abelian group G that is the union of a countable chain of compact subgroups. On G we consider a stationary standard Markov process defined by a semigroup mu(t) of probability measures, satisfying mu(s+t) = mu(s) * mu(t) and lim(t --> 0) mu(t) = delta(0), and we consider the Levy measure associated to the process through the Levy-Khintchine formula. Under the hypothesis that the Levy measure is unbounded, we show that the process may be obtained as a limit of discrete processes defined on the discrete quotient groups G/G(n), where G(n) 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.