Asymptotic Properties of Isotropic Brownian Flows

  • Yves le Jan
Part of the Progress in Probability book series (PRPR, volume 19)


Denote by Diff(M) the group of C diffeomorphisms of a Riemannian manifold M. Define a Brownian flow on M as a Diff(M)-valued process Φt, with independent incrementst1, Φt2 ο Φt1 -1, ..., Φt1+1 ο Φti -1 if, t1 < t2 ... < ti). We assume in addition that the increments of the flow are stationarys(law) ~ Φt+s ο Φt -1) and that for any init ial value xM, the one point process Φt(x) is a Brownian motion on M. An analytical characterization of Brownian flows is given in [1].


Entropy Manifold Covariance 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Baxendale, P., Brownian motion on the diffeomorphism group I, Com-positio Mathematica 53 (1984), pp. 19–50.MathSciNetMATHGoogle Scholar
  2. [2]
    Baxendale, P. and Harris, T.E., Isotropic stochastic flows, Annals of Probab. 14 (1986), pp. 1155–1179.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Fukushima, M., “Dirichlet Forms and Markov Processes,” North Holland-Kodansha, 1980.Google Scholar
  4. [4]
    Harris, T.E., Brownian motions on the homeomorphisms of the plane, Ann. Probab. 9 (1981), pp. 232–254.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Kunita, H., “Stochastic Flows and S.D.E.’s,” Cambridge University Press, 1991.Google Scholar
  6. [6]
    Itô, K., Isotropic random currents, “Proc. Third Berkeley Symp. 2,” Univ. Calif. Press, Berkeley, 1956.Google Scholar
  7. [7]
    Ledrappier, F. and Young, L-S., Entropy formula for random transformations, Prob. Th. Rel. Fields 80 (1988), pp. 217–240.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Le Jan, Y., Flots de diffusion dansd, C. R. Acad Sci. Paris Serie I 294 (1982), pp. 697–699.MATHGoogle Scholar
  9. [9]
    Le Jan, Y., On isotropic Brownian motions, Zeit. für Wahrsch 70 (1985), pp. 609–620.MATHCrossRefGoogle Scholar
  10. [10]
    Le Jan, Y. and Watanabe, S., Stochastic flows of diffeomorphisms, “Proc. of the Taniguchi Symposium, (edited by K. Itô),” North Holland, 1984.Google Scholar
  11. [11]
    Le Jan, Y., Equilibre statistique pour les produits de difféomorphismes aléatoires indépendants, Ann. Inst. Henri Poincaré 23 (1987), pp. 111–120.MATHGoogle Scholar
  12. [12]
    Le Jan, Y., Propriétés asymptotiques des flots browniens isotropes, C. R. Acad Sci. Paris Serie I 309 (1989), pp. 63–65.MATHGoogle Scholar
  13. [13]
    Yaglom, A., Some classes of randonm fields in n-dimensional space, Theory Probab. Appl. 2 (1957), pp. 273–320.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Yves le Jan
    • 1
  1. 1.Laboratoire de Probabilités (CNRS)ParisFrance

Personalised recommendations