Doeblin's and Harris' theory of Markov processes

Abstract

Our notation and definitions are taken from (Chung, K. L.: The general theory of Markov processes according to Doeblin. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 2, 230–254 (1964)). A closed set H is called recurrent in the sense of Harris if there exists a σ-finite measure ϕ such that for E=H, ϕ(E) >0 implies Q(x, E)=1 for all tx∃H. Theorem 1. Let X be absolutely essential and indecomposable. Then there exists a closed set B⫅X. such that B contains no acountable disjoint collection of perpetuable sets if and only if X=H+1 where H is recurrent in the sense of Harris and I is either inessential or improperly essential. Theorem 2. If there exists no uncountable disjoint collection of closed sets, then there exists a countable disjoint collection {D_n} _n=1 ^∞ of absolutely essential and indecomposable closed sets such that \(I = X - \sum\nolimits_{n = 1}^\infty {D_n } \). Under the additional assumption that Suslin's Conjecture holds, Theorem 2 was proved by Jamison (Jamison, B.: A Result in Doeblin's Theory of Markov Chains implied by Suslin's Conjecture. Z. Wahrscheinlichkeitstheorie verw. Gebiete 24, 287–293 (1972)).