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Ergodic decompositions associated with regular Markov operators on Polish spaces

Published online by Cambridge University Press:  26 March 2010

DANIËL T. H. WORM  and
SANDER C. HILLE
DANIËL T. H. WORM
Affiliation:
Mathematical Institute, University Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands (email: [email protected], [email protected])
SANDER C. HILLE
Affiliation:
Mathematical Institute, University Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands (email: [email protected], [email protected])
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Abstract

For any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 2 , April 2011 , pp. 571 - 597
Copyright
Copyright © Cambridge University Press 2010

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