Title: Dirichlet forms on totally disconnected spaces and bipartite Markov chains
Author: David Aldous and Steven N. Evans
Date: February, 1998
Abstract: We use Dirichlet form methods to construct
and analyse a general class of reversible Markov processes with
totally disconnected state spaces.
We study in detail the special case of bipartite
Markov chains. The latter processes have a state space
consisting of an ``interior'' with a countable number
of isolated points and a, typically uncountable, ``boundary''.
The equilibrium measure assigns all of its mass to the interior.
When the chain is started at a state in the interior, it holds
for an exponentially distributed amount of time and then
jumps to the boundary. It then instantaneously re-enters the
interior. There is a ``local time on the boundary''.
That is, the set of
times the process is on the boundary is uncountable and
coincides with the points of increase of a continuous
additive functional.
Certain processes with values in the space of
trees and the space of vertices of a fixed tree provide
natural examples of bipartite chains. Moreover,
time--changing a bipartite chain by its
local time on the boundary leads to interesting processes,
including particular L\'evy processes on local fields
(for example, the $p$-adic numbers) that have been considered
elsewhere in the literature.