Anisotropic diffusion on totally disconnected abelian groups

Mauro Del Muto and Alessandro Figà-Talamanca

Vol. 225 (2006), No. 2, 221–229

DOI: 10.2140/pjm.2006.225.221

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 ${\mu }_t$ of probability measures, satisfying ${\mu }_{s+t}={\mu }_s\ast {\mu }_t$ and $\underset{t\to 0}{lim}{\mu }_t={\delta }_0$, 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 2000

Primary: 43A70

Secondary: 60J60

Milestones

Received: 2 October 2004

Accepted: 1 August 2005

Published: 1 June 2006