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

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:

• (i) when the model is open set irreducible it is ϕ-irreducible;

• (ii) 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;

• (iii) 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;

• (iv) 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;

• (v) 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.