Dirichlet forms on totally disconnected spaces and bipartite Markov chains

Dirichlet forms on totally disconnected spaces and bipartite Markov chains

Report Number
509
Authors
David Aldous and Steven N. Evans
Citation
Electronic Journal of Probability</em>, Vol. 5 (2000) Paper no. 2, pages 1-18
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.

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