Abstract
We use Dirichlet form methods to construct and analyze a general class of reversible Markov precesses with totally disconnected state space. 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 Lévy processes on local fields (for example, the p-adic numbers) that have been considered elsewhere in the literature.
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Aldous, D., Evans, S.N. Dirichlet Forms on Totally Disconnected Spaces and Bipartite Markov Chains. Journal of Theoretical Probability 12, 839–857 (1999). https://doi.org/10.1023/A:1021640218459
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DOI: https://doi.org/10.1023/A:1021640218459