Inventiones mathematicae

, Volume 187, Issue 1, pp 37–59 | Cite as

Random walks on finite volume homogeneous spaces

  • Yves Benoist
  • Jean-Francois Quint


Extending previous results by A. Eskin and G. Margulis, and answering their conjectures, we prove that a random walk on a finite volume homogeneous space is always recurrent as soon as the transition probability has finite exponential moments and its support generates a subgroup whose Zariski closure is semisimple.


Random Walk Parabolic Subgroup Borel Probability Measure Unipotent Radical Zariski Closure 
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  1. 1.
    Benoist, Y., Quint, J.-F.: Mesures stationnaires et fermés invariants des espaces homogènes,. C. R. Acad. Sci. 347, 9–13 (2009) and preprint (2009) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Benoist, Y., Quint, J.-F.: Lattices in S-adic Lie groups. Preprint (2010) Google Scholar
  3. 3.
    Benoist, Y., Quint, J.-F.: Stationary measures and invariant subsets of homogeneous spaces (in preparation) Google Scholar
  4. 4.
    Benoist, Y., Quint, J.-F.: Random walks on semisimple groups (in preparation) Google Scholar
  5. 5.
    Breuillard, E.: Local limit theorems and equidistribution of random walks on the Heisenberg group. Geom. Funct. Anal. 15, 35–82 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Eskin, A., Margulis, G.: Recurrence properties of random walks on finite volume homogeneous manifolds. In: Random Walks and Geometry, pp. 431–444. de Gruiter, Berlin (2004) Google Scholar
  7. 7.
    Foster, F.: On the stochastic matrices associated with certain queuing processes. Ann. Math. Stat. 24, 355–360 (1953) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Garland, H., Raghunathan, M.S.: Fundamental domains for lattices in ℝ-rank 1 semisimple Lie groups. Ann. Math. 92, 279–326 (1970) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Kempf, G.: Instability in invariant theory. Ann. Math. 108, 299–316 (1978) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Margulis, G.: Discrete Subgroups of Semisimple Lie Groups. Springer, Berlin (1991) zbMATHGoogle Scholar
  11. 11.
    Meyn, S., Tweedie, R.: Markov Chain and Stochastic Stability. Springer, Berlin (1993) Google Scholar
  12. 12.
    Mumford, D.: Geometric Invariant Theory. Springer, Berlin (1965) zbMATHGoogle Scholar
  13. 13.
    Nummelin, E.: General Irreducible Markov Chains and Non-negative Operators. Camb. Univ. Press, Cambridge (1984) CrossRefzbMATHGoogle Scholar
  14. 14.
    Raghunathan, M.: Discrete Subgroups of Lie Groups. Springer, Berlin (1972) zbMATHGoogle Scholar
  15. 15.
    Raghunathan, M.: A note on orbits of reductive groups. J. Indian Math. Soc. 38, 65–70 (1974) MathSciNetGoogle Scholar
  16. 16.
    Ratner, M.: Raghunathan’s conjectures for p-adic Lie groups. Int. Math. Res. Notices 141–146 (1993) Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.CNRS-Université Paris-SudOrsayFrance
  2. 2.CNRS-Université Paris-NordVilletaneuseFrance

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