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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: dworm@math.leidenuniv.nl, shille@math.leidenuniv.nl)
SANDER C. HILLE
Affiliation:
Mathematical Institute, University Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands (email: dworm@math.leidenuniv.nl, shille@math.leidenuniv.nl)
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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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References

[1]Aliprantis, C. and Border, K.. Infinite Dimensional Analysis. A Hitchhiker’s Guide, 3rd edn. Springer, Berlin, 2006.Google Scholar
[2]Bauer, H.. Measure and Integration Theory. Walter de Gruyter, Berlin, 2001.CrossRefGoogle Scholar
[3]Deuschel, J.-D. and Stroock, D.. Large Deviations. Academic Press, New York, 1989.Google Scholar
[4]Diestel, J. and Uhl, J. J. Jr. Vector Measures (Mathematical Surveys, 15). American Mathematical Society, Providence, RI, 1977.CrossRefGoogle Scholar
[5]Dudley, R. M.. Convergence of Baire measures. Studia Math. 27 (1966), 251–268.CrossRefGoogle Scholar
[6]Dudley, R. M.. Correction to: ‘Convergence of Baire measures’. Studia Math. 51 (1974), 275.Google Scholar
[7]Dynkin, E. B.. Sufficient statistics and extreme points. Ann. Probab. 6 (1978), 705–730.CrossRefGoogle Scholar
[8]Ethier, S. N. and Kurtz, T. G.. Markov Processes; Characterization and Convergence. Wiley, New York, 1986.CrossRefGoogle Scholar
[9]Foguel, S. R.. The Ergodic Theory of Markov Processes (Van Nostrand Mathematical Studies, 21). Van Nostrand Reinhold Co., New York, 1969.Google Scholar
[10]Gacki, H.. Applications of the Kantorovich–Rubinstein maximum principle in the theory of Markov semigroups, Dissertationes Math. 448 (2007).CrossRefGoogle Scholar
[11]Hernández-Lerma, O. and Lasserre, J. B.. Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math. 54 (1998), 99–119.CrossRefGoogle Scholar
[12]Hernández-Lerma, O. and Lasserre, J. B.. Markov Chains and Invariant Probabilities. Birkhäuser, Basel, 2003.CrossRefGoogle Scholar
[13]Hille, S. C. and Worm, D. T. H.. Embedding of semigroups of Lipschitz maps into positive linear semigroups on ordered Banach spaces generated by measures. Integral Equtions Operator Theory 63 (2009), 351–371.CrossRefGoogle Scholar
[14]Hille, S. C. and Worm, D. T. H.. Continuity properties of Markov semigroups and their restrictions to invariant L 1-spaces. Semigroup Forum 79 (2009), 575–600.CrossRefGoogle Scholar
[15]Horbacz, K., Myjak, J. and Szarek, T.. On stability of some general random dynamical system. J. Stat. Phys. 119 (2005), 35–60.CrossRefGoogle Scholar
[16]Horbacz, K.. Random dynamical systems with jumps. J. Appl. Probab. 41 (2004), 890–910.CrossRefGoogle Scholar
[17]Kakutani, S.. Ergodic theorems and the Markoff process with a stable distribution. Proc. Imp. Acad. 16 (1940), 49–54.Google Scholar
[18]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[19]Kazakevičius, V. and Leipus, R.. A new theorem on the existence of invariant distributions with applications to ARCH processes. J. Appl. Probab. 40(1) (2003), 147–162.CrossRefGoogle Scholar
[20]Lasota, A., Myjak, J. and Szarek, T.. Markov operators with a unique invariant measure. J. Math. Anal. Appl. 276 (2002), 343–356.CrossRefGoogle Scholar
[21]Lant, T. and Thieme, H. R.. Markov transition functions and semigroups of measures. Semigroup Forum 74 (2007), 337–369.CrossRefGoogle Scholar
[22]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer, Berlin, 1987.CrossRefGoogle Scholar
[23]Myjak, J. and Szarek, T.. Attractors of iterated function systems and Markov operators. Abstr. Appl. Anal. 8 (2003), 479–502.CrossRefGoogle Scholar
[24]Myjak, J. and Szarek, T.. On the existence of an invariant measure for Markov–Feller operators. J. Math. Anal. Appl. 294 (2004), 215–222.CrossRefGoogle Scholar
[25]Ollivier, Y.. Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 (2009), 810–864.CrossRefGoogle Scholar
[26]Szarek, T.. Invariant measures for Markov operators with application to function systems. Studia Math. 154 (2003), 207–222.CrossRefGoogle Scholar
[27]Szarek, T.. Invariant measures for nonexpansive Markov operators on Polish spaces, Dissertationes Math. 415 (2003).CrossRefGoogle Scholar
[28]Szarek, T.. The uniqueness of invariant measures for Markov operators. Studia Math. 189(3) (2008), 225–233.CrossRefGoogle Scholar
[29]Varadhan, S. R. S.. Probability Theory (Courant Lecture Notes in Mathematics, 7). American Mathematical Society, Providence, RI, 2001.Google Scholar
[30]Williams, D.. Diffusions, Markov Processes and Martingales. Volume 1: Foundations. Wiley, Chichester, 1979.Google Scholar
[31]Yosida, K.. Simple Markoff process with a locally compact phase space. Math. Japon. 1 (1948), 99–103.Google Scholar
[32]Yosida, K.. Functional Analysis, 6th edn. Springer, Berlin, 1980.Google Scholar
[33]Zaharopol, R.. Invariant Probabilities of Markov–Feller Operators and their Supports. Birkhäuser, Basel, 2005.CrossRefGoogle Scholar
[34]Zaharopol, R.. An ergodic decomposition defined by transition probabilities. Acta. Appl. Math. 104 (2008), 47–81.CrossRefGoogle Scholar
[35]Klünger, M.. Ergodic decomposition of invariant measures, unpublished lecture notes.Google Scholar