Probability Theory and Related Fields

, Volume 101, Issue 2, pp 147–171 | Cite as

On the asymptotic and invariantσ-algebras of random walks on locally compact groups

  • Wojciech Jaworski
Article

Summary

Consider a random walk of law μ on a locally compact second countable groupG. Let the starting measure be equivalent to the Haar measure and denote byQ the corresponding Markov measure on the space of pathsG. We study the relation between the spacesL (G, ℬa,Q) andL (G, ℬi,Q) where ℬa and ℬi stand for the asymptotic and invariant σ-algebras, respectively. We obtain a factorizationL (G, ℬa,Q) ≊L (G, ℬi,Q)⊗L (C) whereC is a cyclic group whose order (finite or infinite) coincides with the period of the Markov shift and is determined by the asymptotic behaviour of the convolution powersμn.

Mathematics Subject Classification

60B15 60J15 60J50 43A05 

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References

  1. 1.
    Azencott, R.: Espaces de Poisson des groupes localement compacts. (Lect. Notes in Math., vol. 148) Berlin Heidelberg New York: Springer 1970Google Scholar
  2. 2.
    Derriennic, Y.: Lois zéro ou deux pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré12, 111–129 (1976)Google Scholar
  3. 3.
    Derriennic, Y.: Entropie, theorems limites et marches aleatoires. In: Heyer, H. (ed.) Probability measures on groups. Proceedings, Oberwolfach 1985. (Lect. Notes in Math., vol. 1210, pp. 241–284) Berlin Heidelberg New York: Springer 1986Google Scholar
  4. 4.
    Derriennic, Y., Lin, M.: Sur la tribu asymptotique des marches aléatoires sur les groupes. Publ. des Seminaires de Math., Inst. Rech. Math. Rennes 1983Google Scholar
  5. 5.
    Derriennic, Y., Lin, M.: Convergence of iterates of averages of certain operator representations and of convolution powers. J. Functional Anal.85, 86–102 (1989)Google Scholar
  6. 6.
    Furstenberg, H.: A Poisson formula for semi-simple Lie groups. Ann. Math.77, 335–386 (1963)Google Scholar
  7. 7.
    Furstenberg, H.: Random walks and discrete subgroups of Lie groups (Adv. Probab. Related Topics, vol. 1, pp. 3–63) New York: Marcel Dekker 1971Google Scholar
  8. 8.
    Furstenberg, H.: Boundary theory and stochastic processes on homogeneous spaces (Proc. Sympos. Pure Math. vol. 26: Harmonic analysis on homogeneous spaces, pp. 193–229) Providence, RI: Am. Math. Soc. 1973Google Scholar
  9. 9.
    Glasner, S.: On Choquet-Deny measures. Ann. Inst. H. Poincare12B, 1–10 (1976)Google Scholar
  10. 10.
    Jamison, B., Orey, S.: Markov chains recurrent in the sense of Harris. Z. Wahrscheinlichkeitstheorie verw. Geb.8, 41–48 (1967)Google Scholar
  11. 11.
    Jaworski, W.: Poisson and Furstenberg boundaries of random walks. Thesis, Queen's University 1991Google Scholar
  12. 12.
    Kaimanovich, V.A., Vershik, A.M.: Random walks on discrete groups: boundary and entropy. Ann Probab.11, 457–490 (1983)Google Scholar
  13. 13.
    Mackey, G.W.: Point realizations of transformation groups. Illinois J. Math.6, 327–335 (1962)Google Scholar
  14. 14.
    Nielsen, O.A.: Direct integral theory. (Lect. Notes in Pure Appl. Math., vol. 61) New York: Mancel Dekker 1980Google Scholar
  15. 15.
    Neveu, J.: Mathematical foundations of the calculus of probability. San Francisco: Holden-Day 1965Google Scholar
  16. 16.
    Ornstein D., Sucheston, L.: An operator theorem onL 1 convergence to zero with application to Markov kernels. Ann. Math. Stat.41, 1631–1639, (1970)Google Scholar
  17. 17.
    Revuz, D.: Markov chains. Amsterdam: North-Holland 1984Google Scholar
  18. 18.
    Rosenblatt, J.: Ergodic and mixing random walks on locally compact groups. Math. Ann.257, 31–42 (1981)Google Scholar
  19. 19.
    Varadarajan, V.S.: Geometry of quantum theory, vol 2. New York: van Nostrand 1970Google Scholar
  20. 20.
    Zimmer, R.J.: Ergodic theory and semisimple groups. Boston: Birkhäuser 1984Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Wojciech Jaworski
    • 1
  1. 1.Department of MathematicsUniversity of OttawaOttawaCanada

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