Skip to main content
SpringerLink
Log in

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

  • Chapter
Itô’s Stochastic Calculus and Probability Theory

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)) d k=1 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:

$$S\left( t.\omega \right)=\frac{1}{2}\int{_{0}^{t}}{{w}^{1}}\left( s \right)d{{w}^{2}}\left( s \right)-{{w}^{2}}\left( s \right)d{{w}^{1}}\left( s \right)$$

.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. A.-I. Benabdallah. Noyau de diffusion sur les espaces homogènes compacts, Bull. Soc. math. France 101 (1973), 265–283.

    MathSciNet  MATH  Google Scholar 

  3. M. Berger, P. Gauduchon and E. Mazet. Le Spéctre d’une Variété Riemannienne,LNM 194,Springer,1971

    Google Scholar 

  4. N. Berline, E. Getzler and M. Vergne. Heat Kernels and Dirac Operators, Springer, 1991

    Google Scholar 

  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 329–348

    Google Scholar 

  6. J.-M. Bismut. Formule de localisation et formules de Paul Lévy, Astérisque 157–158,Société Math. France(1988), 37–58

    Google Scholar 

  7. D. K. Elworthy. Stochastic Differential Equations on Manifolds,Lecture Note Series 70 London Math. Soc.,1982

    Google Scholar 

  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), 136–146

    Google Scholar 

  9. L. D. Eskin. The heat equation and the Weierstrass transform on certain symmetric Riemannian spaces, AMS Translation, 75 (1968), 239–254

    MATH  Google Scholar 

  10. S. Fang. Rotations et quasi-invariance sur l’espace de chemins, Potential Analysis 4 (1995), 67–77

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Helmes and A. Schwane. Lévy’s stochastic area formula in higher dimensions, Jour. Funct. Anal. 54 (1983), 177–192

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Ikeda. Probabilistic methods in the study of asymptotics, Ecole d’Été de Probabilités de Saint-Flour XVIII-1988,LNM 1427 Springer(1990), 197–325

    Google Scholar 

  13. N. Ikeda and S. Watanabe. Stochastic Diffential Equations and Diffusion Processes, Second Edition, North-Holland/Kodansha, 1988

    Google Scholar 

  14. K. Itô. Stochastic differential equations in a differentiable manifold, Nagoya Math. J. 1 (1950), 35–47

    MATH  Google Scholar 

  15. K. Itô. Brownian motions in a Lie group, Proc. Japan Acad. 26 (1950), 4–10

    Article  MATH  Google Scholar 

  16. P. Malliavin. Géométrie Différentielle Stochastique, Les Presse de l’Université Montréal, 1978

    Google Scholar 

  17. S. A. Molchanov. Diffusion processes and Riemannian geometry, Russian Math. Surveys 30 (1975), 1–63

    Article  MATH  Google Scholar 

  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), 194241

    Google Scholar 

  19. S. Watanabe. Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Annals of Probab. 15 (1987), 1–39

    Article  MATH  Google Scholar 

  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. M. Yor. The law of some Brownian functionals, Proc. ICM, Kyoto 1990, II, Springer(1991), 1105–1112

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-01, Japan

    Shinzo Watanabe

Authors
  1. Shinzo Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Ritsumeikan University, 525-77, Kusatsu, Shiga, Japan

    Nobuyuki Ikeda (Professor) (Professor)

  2. Department of Mathematics, Graduate School of Science, Kyoto University, 606-01, Kyoto, Japan

    Shinzo Watanabe (Professor) (Professor)

  3. Department of Mathematics, Faculty of Fundamental Engineering, Osaka University, Toyonaka, Osaka, 560, Japan

    Masatoshi Fukushima (Professor) (Professor)

  4. Graduate School of Mathematics, Kyushu University, Fukuoka, 812, Japan

    Hiroshi Kunita (Professor) (Professor)

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Springer-Verlag Tokyo

About this chapter

Cite this chapter

Watanabe, S. (1996). Lévy’s stochastic area formula and Brownian motion on compact Lie groups. In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds) Itô’s Stochastic Calculus and Probability Theory. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68532-6_26

Download citation

  • DOI: https://doi.org/10.1007/978-4-431-68532-6_26

  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-68534-0

  • Online ISBN: 978-4-431-68532-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation