Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Fractional order Sobolev spaces on Wiener space
Download PDF
Download PDF
  • Published: June 1993

Fractional order Sobolev spaces on Wiener space

  • Shinzo Watanabe1 

Probability Theory and Related Fields volume 95, pages 175–198 (1993)Cite this article

  • 418 Accesses

  • 41 Citations

  • Metrics details

Summary

Fractional order Sobolev spaces are introduced on an abstract Wiener space and Donsker's delta functions are defined as generalized Wiener functionals belonging to Sobolev spaces with negative differentiability indices. By using these notions, the regularity in the sense of Hölder continuity of a class of conditional expectations is obtained.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Adams, R.A.: Sobolev spaces. New York: Academic Press 1978

    Google Scholar 

  2. Airault, H., Malliavin, P.: Intégration géométrique sur l'espace de Wiener. Bull. Sci. Math.112, 3–52 (1988)

    Google Scholar 

  3. Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener space. Berlin New York: de Gruyter 1991

    Google Scholar 

  4. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. 2nd edn. Amsterdam New York: North-Holland/Kodansha 1988

    Google Scholar 

  5. Kuo, H.-H.: Donsker's delta function as a generalized Brownian functionals and its application. In: Theory and applications of random fields. Proc. IFIP Conf. Bangalore 1982. Kallianpur, G. (ed.),LNCI 49, 167–178, (1983)

  6. Kusuoka, S.: On the foundation of Wiener-Riemannian manifolds. In: Elworthy, K.D., Zambrini, J.-C. (eds.) Stochastic analysis, path integration and dynamics, pp. 130–164 Harlow, Essex: Longman 1989

    Google Scholar 

  7. Kusuoka, S., Stroock, D.W.: Applications of the Malliavin calculus, I. In: Ito, K. (ed.), Stochastic analysis, pp. 271–306. Proc. Taniguchi Symp. Katata and Kyoto 1982. Tokyo: Kinokuniya 1984

    Google Scholar 

  8. Lions, J.-L.: Sur les espaces d'interpolation; dualit'e, Math. Scand.9, 147–177 (1961)

    Google Scholar 

  9. Malliavin, P.: Implicit functions in finite corank on the Wiener space. In: Ito K. (ed.), Stochastic analysis, pp. 369–386. Proc. Taniguchi Symp. Katata and Kyoto 1982. Tokyo: Kinokuniya 1984

    Google Scholar 

  10. Meyer, P.A.: Retour sur la théorie de Littlewood-Paley. In: Azéma, J., Yor, M. (eds.), Séminaire de Prob. XV, 1979/1980LNM 850, 151–166 (1981)

  11. Shigekawa, I.: Derivatives of Wiener functionals, and absolute continuity of induced measures. J. Math. Kyoto Univ.20, 263–289 (1980)

    Google Scholar 

  12. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, N.J.: Princeton University Press 1970

    Google Scholar 

  13. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton, N.J.: Princeton University Press 1971

    Google Scholar 

  14. Sugita, H.: Sobolev space of Wiener functionals and Malliavin calculus. J. Math. Kyoto Univ.25, 31–48 (1985)

    Google Scholar 

  15. Sugita, H.: On a characterization of the Sobolev spaces over an abstract Wiener space. J. Math. Kyoto Univ.25, 717–757 (1985)

    Google Scholar 

  16. Sugita, H.: Positive generalized Wiener functionals and potential theory over abstract Wiener spaces. Osaka J. Math.25, 665–698 (1988)

    Google Scholar 

  17. Watanabe, S.: Malliavin's calculus in terms of generalized Wiener functionals. In: Kalliapur, G. (ed.) Theory and application of random fields. Proc. IFIP Conf. Bangalore 1982,LNCI 49, 284–290 (1983)

  18. Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Tata Institute of Fundamental Research. Berlin Heidelberg New York: Springer 1984

    Google Scholar 

  19. Watanabe, S.: Donsker's δ-functions in the Malliavin calculus. In: Mayer-Wolf, E., Merzbach, E., Shwartz, A. (eds.) Stochastic analysis, liber amicorum for Moshe Zakai, pp. 495–502. New York: Academic Press 1991

    Google Scholar 

  20. Watanabe, S.: Some refinement of conditional expectations on Wiener space by means of the Malliavin calculus. Proc. 6-th USSR-Japan Symp. on Probability Theory, pp. 414–421. Singapore: World Scientific 1992

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Kyoto University, 606, Kyoto, Japan

    Shinzo Watanabe

Authors
  1. Shinzo Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Watanabe, S. Fractional order Sobolev spaces on Wiener space. Probab. Th. Rel. Fields 95, 175–198 (1993). https://doi.org/10.1007/BF01192269

Download citation

  • Received: 29 January 1992

  • Revised: 15 July 1992

  • Issue Date: June 1993

  • DOI: https://doi.org/10.1007/BF01192269

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification

  • 60H07
  • 60H10
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature