DIRICHLET FORMS ON TOTALLY DISCONNECTED SPACES AND BIPARTITE MARKOV CHAINS

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 Levy processes on local fields (for example, the p-adic numbers) that have been considered elsewhere in the literature