Probability Theory and Related Fields

, Volume 130, Issue 1, pp 109–134 | Cite as

Products of Beta matrices and sticky flows

  • Y. Le JanEmail author
  • S. Lemaire


A discrete model of Brownian sticky flows on the unit circle is described: it is constructed with products of Beta matrices on the discrete torus. Sticky flows are defined by their ‘‘moments’’ which are consistent systems of transition kernels on the unit circle. Similarly, the moments of the discrete model form a consistent system of transition matrices on the discrete torus. A convergence of Beta matrices to sticky kernels is shown at the level of the moments. As the generators of the n-point processes are defined in terms of Dirichlet forms, the proof is performed at the level of the Dirichlet forms. The evolution of a probability measure by the flow of Beta matrices is described by a measure-valued Markov process. A convergence result of its finite dimensional distributions is deduced.


Markov chains with continuous parameter Polya urns Dirichlet laws stochastic flow of kernels Feller semigroups Dirichlet forms convergence of resolvents 


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  1. 1.
    Ethier, S.N., Kurtz, T. G.: Markov processes: Characterization and Convergence. Wiley-Interscience, 1986Google Scholar
  2. 2.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. Walter de Gruyter, 1994Google Scholar
  3. 3.
    Le Jan, Y., Raimond, O.: Flows, coalescence and noise. math.PR/0203221, To appear in The Annals of ProbabilityGoogle Scholar
  4. 4.
    Le Jan, Y., Raimond, O.: Sticky flows on the circle and their noises. To appear in Probab. Theory Relat. FieldsGoogle Scholar
  5. 5.
    Le Jan, Y., Raimond, O.: Sticky flows on the circle. math.PR/0211387, 2002Google Scholar
  6. 6.
    Pitman, E.: The closest estimates of statistical parameters. Proc. Camb. Philos. Soc. 33, 212–222 (1937)zbMATHGoogle Scholar
  7. 7.
    Pitman, J.: Some developments of the Blackwell-MacQueen. In L.S. Shapley T.S. Ferguson and J.B. Macqueen, editors. Statistics, Probability and Game Theory, volume~30, pp. 245–267. IMS Lecture Notes-Monograph, 1996Google Scholar
  8. 8.
    Pitman, J.: Combinatorial stochastic processes. Saint-Flour lecture notes, July 2002Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Université Paris-SudLaboratoire de MathématiqueOrsay cedexFrance

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