, Volume 34, Issue 1-2, pp 175-188

Topological conditions enabling use of harris methods in discrete and continuous time

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Abstract

This paper describes the role of continuous components in linking the topological and measuretheoretic (or regenerative) analysis of Markov chains and processes. Under Condition \(\mathcal{T}\) below we show the following parallel results for both discrete and continuous time models:

  1. when the model is open set irreducible it is ϕ-irreducible;

  2. under (i), the measure-theoretic classification of the model as Harris recurrent or positive Harris recurrent is equivalent to a topological classification in terms of not leaving compact sets or of tightness of transition kernels;

  3. under (i), the ‘global’ classification of the model as transient, recurrent or positive recurrent is given by a “local’ classification of any individual reachable point;

  4. under (i), every compact set is a small set, so that through the Nummelin splitting there is pseudo-regeneration within compact sets, and compact sets are ‘test sets’ for stability;

  5. even without irreducibility, there is always a Doeblin decomposition into a countable disjoint collection of Harris sets and a transient set. We conclude with a guide to verifying Condition \(\mathcal{T}\) and indicate that it holds under very mild constraints for a wide range of specific models: in particular a ϕ-irreducible Feller chain satisfies Condition \(\mathcal{T}\) provided only that the support of ϕ has nonempty interior.

Work supported in part by NSF Grant DMS-9205687.