Ergodic measures of Markov semigroups with the e-property

We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on...

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Published inErgodic theory and dynamical systems Vol. 32; no. 3; pp. 1117 - 1135
Main Authors SZAREK, TOMASZ, WORM, DANIËL T. H.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.06.2012
Subjects
Dynamical systems
Ergodic processes
Group theory
Invariants
Markov analysis
Markov processes
Mathematical analysis
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Summary:We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on the set of ergodic measures, one of them being the existence of a Borel subset K0 of K with a bijective map from K0 to the ergodic measures, by sending a point in K0 to the weak limit of the Cesàro averages of the Dirac measure on this point. We also give sufficient conditions for the set of ergodic measures to be countable and finite. Finally, we give a quite general condition under which the Cesàro averages of any measure converge to an invariant measure.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385711000022