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ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On finite invariant measures for sets of Markov operators


Author: S. Horowitz
Journal: Proc. Amer. Math. Soc. 34 (1972), 110-114
MSC: Primary 28A70; Secondary 47D99, 60J99
DOI: https://doi.org/10.1090/S0002-9939-1972-0296258-5
MathSciNet review: 0296258
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Abstract: A. Brunel [1] proved that a Markovian operator P has an invariant measure if and only if each convex combination of iterates $ \sum\nolimits_{n = 0}^\infty {{\alpha _n}{P^n}} $ is conservative. In the present paper this result is generalized for any commutative semigroup of Markovian operators: Let II be a semigroup; there exists a common invariant measure for II if and only if each convex combination $ \sum\nolimits_{n = 1}^\infty {{\alpha _n}{P_n}} $ where $ \{ {P_n}\} \subset \Pi $, is conservative.


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  • [1] A. Brunel, New conditions for existence of invariant measures in ergodic theory., Contributions to Ergodic Theory and Probability (Proc. Conf., Ohio State Univ., Columbus, Ohio, 1970) Springer, Berlin, 1970, pp. 7–17. MR 0268355
  • [2] -, Thesis, University of Paris, Paris.
  • [3] Shaul R. Foguel, The ergodic theory of Markov processes, Van Nostrand Mathematical Studies, No. 21, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0261686
  • [4] Michael Lin, Semi-groups of Markov operators, Boll. Un. Mat. Ital. (4) 6 (1972), 20–44 (English, with Italian summary). MR 0320275

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0296258-5
Keywords: Commutative semigroups, Markov operators, invariant measures, conservative processes, dissipative processes, convex combinations, positive combination on $ {L_\infty }$
Article copyright: © Copyright 1972 American Mathematical Society