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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.


References [Enhancements On Off] (What's this?)

  • [1] A. Brunel, New conditions for existence of invariant measures in ergodic theory, Proc. Conference Contributions to Ergodic Theory and Probability (Ohio State Univ., Columbus, Ohio, 1970), Springer, Berlin, 1970, pp. 7-17. MR 42 #3253. MR 0268355 (42:3253)
  • [2] -, Thesis, University of Paris, Paris.
  • [3] S. R. Foguel, The ergodic theory of Markov processes, Van Nostrand Math. Studies, no. 21, Van Nostrand Reinhold, New York, 1969, MR 41 #6299. MR 0261686 (41:6299)
  • [4] M. Lin, Semi-groups of Markov operators. (to appear). MR 0320275 (47:8814)

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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

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