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Lévy’s stochastic area formula and Brownian motion on compact Lie groups

  • Shinzo Watanabe

Abstract

Let (W, P) be the d-dimensional Wiener space: W be the space of continuous paths W = {w ∈ C([0, ∞) → R d )|w(0) = 0} and P be the standard d-dimensional Wiener measure on W. Then w = (w k(t)) k=1 d in W is a canonical realization of a d-dimensional Wiener process. Lévy’s stochatic area is defined on the two-dimensional Wiener space by Itô’s stochastic integral as follows:
$$S\left( t.\omega \right)=\frac{1}{2}\int{_{0}^{t}}{{w}^{1}}\left( s \right)d{{w}^{2}}\left( s \right)-{{w}^{2}}\left( s \right)d{{w}^{1}}\left( s \right)$$
.

Keywords

Brownian Motion Riemannian Manifold Stochastic Differential Equation Heat Kernel Index Theorem 
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References

  1. 1.
    M. T. Arede. Géométrie du noyau de la chaleur sur les variétés, Thése de Doctorat de 3ème Cycle, Séptembre 1983, Université d’Aix-Marseille II.Google Scholar
  2. 2.
    A.-I. Benabdallah. Noyau de diffusion sur les espaces homogènes compacts, Bull. Soc. math. France 101 (1973), 265–283.MathSciNetMATHGoogle Scholar
  3. 3.
    M. Berger, P. Gauduchon and E. Mazet. Le Spéctre d’une Variété Riemannienne,LNM 194,Springer,1971Google Scholar
  4. 4.
    N. Berline, E. Getzler and M. Vergne. Heat Kernels and Dirac Operators, Springer, 1991Google Scholar
  5. 5.
    J.-M. Bismut. The Atiyah-Singer theorems: a probabilistic approach, I. index theorem, II. the Lefschetz fixed point formulas, Jour. Funct. Anal. 57(1984), 55–99 and 329–348Google Scholar
  6. 6.
    J.-M. Bismut. Formule de localisation et formules de Paul Lévy, Astérisque 157–158,Société Math. France(1988), 37–58Google Scholar
  7. 7.
    D. K. Elworthy. Stochastic Differential Equations on Manifolds,Lecture Note Series 70 London Math. Soc.,1982Google Scholar
  8. 8.
    D. K. Elworthy and A. Truman. The diffusion equation and classical mechanics: an elementary formula, Stochastic Processes in Quantum Physics,LNP 173 Springer(1982), 136–146Google Scholar
  9. 9.
    L. D. Eskin. The heat equation and the Weierstrass transform on certain symmetric Riemannian spaces, AMS Translation, 75 (1968), 239–254MATHGoogle Scholar
  10. 10.
    S. Fang. Rotations et quasi-invariance sur l’espace de chemins, Potential Analysis 4 (1995), 67–77MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    K. Helmes and A. Schwane. Lévy’s stochastic area formula in higher dimensions, Jour. Funct. Anal. 54 (1983), 177–192MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    N. Ikeda. Probabilistic methods in the study of asymptotics, Ecole d’Été de Probabilités de Saint-Flour XVIII-1988,LNM 1427 Springer(1990), 197–325Google Scholar
  13. 13.
    N. Ikeda and S. Watanabe. Stochastic Diffential Equations and Diffusion Processes, Second Edition, North-Holland/Kodansha, 1988Google Scholar
  14. 14.
    K. Itô. Stochastic differential equations in a differentiable manifold, Nagoya Math. J. 1 (1950), 35–47MATHGoogle Scholar
  15. 15.
    K. Itô. Brownian motions in a Lie group, Proc. Japan Acad. 26 (1950), 4–10MATHCrossRefGoogle Scholar
  16. 16.
    P. Malliavin. Géométrie Différentielle Stochastique, Les Presse de l’Université Montréal, 1978Google Scholar
  17. 17.
    S. A. Molchanov. Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975), 1–63MATHCrossRefGoogle Scholar
  18. 18.
    S. Takanobu and S. Watanabe. Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations, Asymptotic problems in probability theory: Wiener functionals and asymptotics, Proc. Taniguchi Symp. 1990,Pitman Research Notes in Math. Ser. 284 Longman(1993), 194241Google Scholar
  19. 19.
    S. Watanabe. Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Annals of Probab. 15 (1987), 1–39MATHCrossRefGoogle Scholar
  20. 20.
    S. Watanabe. Short time asymptotic problems in Wiener functional integration theory. Applications to heat kernels and index theorems, Stochastic Analysis and Related Topics II, Proc. Silivri 1988,LNM 1444 Springer(1990), 1–62.Google Scholar
  21. 21.
    M. Yor. The law of some Brownian functionals, Proc. ICM, Kyoto 1990, II, Springer(1991), 1105–1112Google Scholar

Copyright information

© Springer-Verlag Tokyo 1996

Authors and Affiliations

  • Shinzo Watanabe
    • 1
  1. 1.Department of Mathematics, Graduate School of ScienceKyoto UniversityKyotoJapan

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