Dirichlet Forms on Totally Disconnected Spaces and Bipartite Markov Chains |
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Authors: | David Aldous Steven N Evans |
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Institution: | (1) Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, California, 94720-3860 |
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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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Keywords: | Dirichlet form Markov process Markov chain additive functional local time stationary reversible ergodic totally disconnected tree local field p-adic p-series |
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