Stochastic Diffeology and Homotopy

  • Rémi Léandre
Part of the Progress in Probability book series (PRPR, volume 50)


There are two stochastic cohomology theories of the loop space: In the first, we endow the loop space with the Brownian bridge measure if the underlying manifold is Riemannian. Sobolev spaces of forms are defined because a Hilbert tangent space is used, although the Brownian loop is only continuous ([8], [2]). To a random form is associated a series of numbers, that is its Sobolev norms, and the stochastic exterior derivative acts continuously over the intersection of Sobolev spaces ([10], [11], [13]). It is shown that if the manifold is simply connected, then the Sobolev stochastic cohomology of forms in this sense (a metric invariant) is equal to the Hochschild cohomology of forms over the manifold ([11], [13]) and therefore to the de Rham cohomology of the smooth loop space (a differentiable invariant). This result is a stochastic generalization of a result of Adams ([1]) and Chen ([4]), which says that the Hochschild cohomology is equal to the cohomology of smooth loops.


Loop Space Brownian Bridge Hochschild Cohomology Stochastic Form Abstract Wiener Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Adams J.F., On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42(1956), 346–373.CrossRefGoogle Scholar
  2. [2]
    Bismut J.M., Large deviations and the Malliavin Calculus, Progress in Math 45, Birkhäuser, 1984.MATHGoogle Scholar
  3. [3]
    Bott R., Tu L.W., Differential forms in algebraic topology, Springer, 1986.Google Scholar
  4. [4]
    Chen K.T., Iterated path integrals of differential forms and loop space homology, Ann. Math., 107 (1973), 213–237.Google Scholar
  5. [5]
    Getzler E., Jones J.D.S., Petrack S., Differential forms on loop spaces and the cyclic bar complex, Topology, 30 (1991), 339–373.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Iglesias P., Thesis, Université de Provence, 1985.Google Scholar
  7. [7]
    Iglesias P., La trilogie du moment, Ann. Institut Fourier, 45:3 (1995), 825–857.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Jones J.D.S., Léandre R., Lp Chen forms on loop spaces, in Stochastic Analysis, M. Barlow, N. Bingham eds., Cambridge University Press, 1991, 104–162.Google Scholar
  9. [9]
    Kusuoka S., De Rham Cohomology of Wiener-Riemannian manifolds. preprint, 1990.Google Scholar
  10. [10]
    Léandre R., Cohomologie de Bismut-Nualart-Pardoux et cohomologie de Hochschild entière, in: Séminaire de Probabilités XXX in honour of P.A. Meyer and J. Neveu, J. Azéma, M. Emery, M. Yor, eds., Lectures Notes Math. 1626, 1996, 68–100.Google Scholar
  11. [11]
    Léandre R., Brownian cohomology of an homogeneous manifold, in New trends in Stochastic Analysis, K.D. Elworthy, S. Kusuoka, I. Shigekawa, eds., World Scientific, 1997, 305–347.Google Scholar
  12. [12]
    Léandre R., Singular integral homology of the stochastic loop space, Infinite dimensional analysis, quantum probability and related topics, 1:1 (1998), 17–31.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Léandre R., Stochastic Adams theorem for a general compact manifold, to appear Reviews in Math. Physics.Google Scholar
  14. [14]
    Léandre R., Stochastic cohomology of Chen-Souriau and line bundle over the Brownian bridge, to appear Probability Theory and Related Fields.Google Scholar
  15. [15]
    Léandre R., Stochastic plots and universal cover of the loop space, to appear Potential Analysis.Google Scholar
  16. [16]
    Léandre R., Smolyanov O., Stochastic homology of the loop space, in: Analysis on Infinite-Dimentional Lie Groups and Algebras, H. Heyer, J. Marion, eds., World Scientific, 1999, 229–235.Google Scholar
  17. [17]
    Ramer R., On the de Rham complex of finite codimensional forms on infinite dimensional manifold, Thesis, University of Warwick, 1974.Google Scholar
  18. [18]
    Shigekawa I., De Rham-Hodge-Kodaira decomposition on an abstract Wiener space, J. Math., Kyoto Univ. 26 (1986), 191–202.MathSciNetMATHGoogle Scholar
  19. [19]
    Smolyanov O., De Rham’s current’s and Stoke’s formula in a Hilbert space, Sov. Math. Dokl., 33 (1986), 140–144.MATHGoogle Scholar
  20. [20]
    Souriau J.M., Un algorithme générateur de structures quantiques, in: Elie Cartan et les Mathématiques d’aujourd’hui, Astérisque, 1985, 341–399.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Rémi Léandre
    • 1
  1. 1.Institut Elie Cartan, Département de MathématiquesUniversité Henri PoincaréVandoeuvre-les-NancyFrance

Personalised recommendations