Abstract

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 μ_t of probability measures, satisfying μ_s+t = μ_s * μ_t and lim_t→0μ_t = 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 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.

Keywords

tree, ultrametric space, totally disconnected group, diffusion, stationary Markov process

Mathematical Subject Classification

Primary: 43A70

Secondary: 60J60

Authors