An Ergodic Decomposition Defined by Transition Probabilities

Abstract

Our main goal in this paper is to prove that any transition probability P on a locally compact separable metric space (X, d) defines a Kryloff-Bogoliouboff-BeboutoffYosida (KBBY) ergodic decomposition of the state space (X, d). Our results extend and strengthen the results of Chap. 5 of Hernández-Lerma and Lasserre (Markov Chains and Invariant Probabilities, 2003) and extend our KBBY-decomposition for Markov-Feller operators that we have obtained in Chap. 2 of our monograph (Zaharopol in Invariant Probabilities of Markov-Feller Operators and Their Supports, 2005). In order to deal with the decomposition that we present in this paper, we had to overcome the fact that the LasotaYorke lemma (Theorem 1.2.4 in our book (op. cit.)) and two results of Lasota and Myjak (Proposition 1.1.7 and Corollary 1.1.8 of our work (op. cit.)) are no longer true in general in the non-Feller case.In the paper, we also obtain a "formula" for the supports of elementary measures of a fairly general type. The result is new even for Markov-Feller operators.We conclude the paper with an outline of the KBBY decomposition for a fairly large class of transition functions. The results for transition functions and transition probabilities seem to us surprisingly similar. However, as expected, the arguments needed to prove the results for transition functions are significantly more involved and are not presented here. We plan to discuss the KBBY decomposition for transition functions with full details in a small monograph that we are currently trying to write.

An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

“…In [32] we obtained a Yosida-type ergodic decomposition for regular Markov operators on Polish spaces, extending results by Hernàndez-Lerma, Lasserre and Zaharopol [17,18,36,37] on locally compact separable metric spaces. In this paper we will show that analogous results hold for regular Markov semigroups.…”

“…Hence we define Γ P cpi := {x ∈ Γ P cp : x is invariant}. Following [17,32,36,37], we define an ergodic measure to be a P -invariant probability measure μ such that μ(E) = 0 or μ(E) = 1 for every P -invariant set E, i.e. a Borel set E ⊂ S such that P δ x (E) = 1 for every x ∈ E.…”

“…Dynkin [10] developed a general (probabilistic) theoretical framework for such decompositions, revolving around so-called (H -) sufficient statistics and σ -algebras. A second line of research concerning ergodic decompositions goes back to pioneering work by Krylov and Bogolioubov [22], Beboutov [5] and Yosida [35] (the 'KBBYapproach'), continued by Hernàndez-Lerma and Laserre [17], Costa and Dufour [7] and Zaharopol [36,37]. It seeks to complement the result of an integral decomposition (1.1) by a decomposition of state space S into subsets related to supports of and concepts of convergence to ergodic measures.…”

Ergodic decompositions associated with regular Markov operators on Polish spaces

“…Regular Markov operators are given by transition probabilities on the underlying space S. Thus the Markov operator associated with a time homogeneous Markov chain is regular. Zaharopol [33,34] managed to extend and strengthen some of their results. In particular, he was able to obtain a partitioning S α of the state space that does not depend on the particular invariant measure.…”

“…In order to arrive at the ergodic decompositions, we need to generalise results of Zaharopol [34] The following result will be crucial in several places where we need to prove convergence of probability measures.…”