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Non-Markovian reversible Chapman-Kolmogorov measures on subshifts of finite type

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Abstract

We consider shift-invariant probability measures on subshift dynamical systems with a transition matrixA which satisfies the Chapman-Kolmogorov equation for some stochastic matrix Π compatible withA. We call them Chapman-Kolmogorov measures. A nonequilibrium entropy is associated to this class of dynamical systems. We show that ifA is irreducible and aperiodic, then there are Chapman-Kolmogorov measures distinct from the Markov chain associated with Π and its invariant row probability vectorq. If, moreover, (q, Π) is a reversible chain, then we construct reversible Chapman-Kolmogorov measures on the subshift which are distinct from (q, Π).

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References

  1. V. M. Alexeyev, Existence of a bounded function of the maximal spectral type,Ergod. Theory Dynam. Syst. 2:59–261 (1982).

    Google Scholar 

  2. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinai,Ergodic Theory (Springer, 1982).

  3. M. Courbage and G. Nicolis, Markov evolution andH-theorem under coarse graining in conservative dynamical systems,Europhys. Lett. 11:1–6 (1990).

    Google Scholar 

  4. M. Courbage and D. Hamdan, A class of nonmixing dynamical systems with monotonic semigroup property.Lett. Math. Phys. 22:101–106 (1991).

    Google Scholar 

  5. M. Courbage and D. Hamdan, Chapman-Kolmogorov equation for non-Markovian shift invariant measures,Ann. Prob., to appear.

  6. S. R. de Groot and P. Mazur,Non-Equilibrium Thermodynamics (North-Holland, Amsterdam, 1969).

    Google Scholar 

  7. P. Ehrenfest and T. Ehrenfest,The Conceptual Foundations of the Statistical Approach in Mechanics (Cornell University Press, Ithaca, New York, 1959).

    Google Scholar 

  8. M. Kac,Probability and Related Topics in Physical Sciences (Interscience, New York, 1959).

    Google Scholar 

  9. G. E. Uhlenbeck and G. W. Ford,Lectures in Statistical Mechanics (American Mathematical Society, Providence, Rhode Island, 1963).

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Probabilités, Université P. et M. Curie, 75252, Paris Cedex 05, France

    M. Courbage & D. Hamdan

Authors
  1. M. Courbage
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  2. D. Hamdan
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Communicated by Y. Sinai

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Courbage, M., Hamdan, D. Non-Markovian reversible Chapman-Kolmogorov measures on subshifts of finite type. J Stat Phys 74, 1281–1292 (1994). https://doi.org/10.1007/BF02188231

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  • DOI: https://doi.org/10.1007/BF02188231

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