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
The divergence of Banach space valued random variables on Wiener space
Download PDF
Download PDF
  • Published: 27 December 2004

The divergence of Banach space valued random variables on Wiener space

  • E. Mayer-Wolf1 &
  • M. Zakai2 

Probability Theory and Related Fields volume 132, pages 291–320 (2005)Cite this article

  • 108 Accesses

  • 8 Citations

  • Metrics details

An Erratum to this article was published on 20 November 2007

Abstract.

The domain of definition of the divergence operator δ on an abstract Wiener space (W,H,μ) is extended to include W–valued and – valued “integrands”. The main properties and characterizations of this extension are derived and it is shown that in some sense the added elements in δ’s extended domain have divergence zero. These results are then applied to the analysis of quasiinvariant flows induced by W-valued vector fields and, among other results, it turns out that these divergence-free vector fields “are responsible” for generating measure preserving flows.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bogachev, V., Mayer-Wolf, E.: Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions. J. Func. Anal. 167, 1–68 (1999)

    Article  Google Scholar 

  2. Cipriano, F., Cruzeiro, A.B.: Flows associated to tangent processes on Wiener space. J. Funct. Anal. 166, 310–331 (1999)

    Article  Google Scholar 

  3. Cruzeiro, A.B.: Équations differentielles sur l’espace de Wiener et formules de Cameron–Martin non linéaires. J. Funct. Anal. 54, 206–227 (1983)

    Article  Google Scholar 

  4. Cruzeiro, A.B., Malliavin, P.: Renormalized differential geometry on path space, structural equation, curvature. J. Funct. Anal. 139, 119–181 (1996)

    Article  Google Scholar 

  5. Cruzeiro, A.B., Malliavin, P.: A class of anticipative tangent processes on the Wiener space. C.R. Acad. Sci. Paris, 333 (1), 353–358 (2001)

    Google Scholar 

  6. Driver, B.K.: A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact manifold. J. Funct. Anal. 110, 272–376 (1992)

    Article  Google Scholar 

  7. Driver, B.K.: Towards calculus and geometry on path spaces. Symp. Proc. Pure Math. 57, 405–422 (1995)

    Google Scholar 

  8. Feyel, D., de La Pradelle: Espaces de Sobolev gaussiens. Ann. Inst. Fourier 41, 49–76 (1991)

    Google Scholar 

  9. Hu, Y., Üstünel, A.S., Zakai, M.: Tangent processes on Wiener space. J. Funct. Anal. 192, 234–270 (2002)

    Article  Google Scholar 

  10. Kusuoka, S.: Nonlinear transformations containing rotation and Gaussian measure. J. Math. Sci. Univ. Tokyo, 10, 1–40 (2003)

    Google Scholar 

  11. Lions, P-L.: Sur les équations différentielles ordinaires et les équations de transport. C.R. Acad. Sci Paris 326 (1), 833–838 (1998)

    Google Scholar 

  12. Ledoux, M., Talagland, M.: Probability in Banach spaces. Springer, 1991

  13. Malliavin, P.: Stochastic Analysis. Springer-Verlag, Berlin/New York, 1997

  14. Malliavin, P., Nualart, D.: Quasi sure analysis of stochastic flows and Banach space valued smooth functionals on the Wiener space. J. Funct. Anal. 112, 287–317 (1993)

    Article  Google Scholar 

  15. Mayer-Wolf, E., Zakai, M.: The Clark-Ocone formula for vector valued Wiener functionals. Preprint, (arXiv:math.PR/0409451)

  16. Nualart, D., Zakai, M.: A summary of some identities of the Malliavin calculus. In: G. Da Prato, L. Tubaro, (eds.), Stochastics Partial Differential Equations and Applications II, Lecture Notes in Mathematics 1390, Springer, 1989, pp. 192–196

  17. Peters, G.: Anticipating flows on the Wiener space generated by vector fields of low regularity. J. Funct. Anal. 142, 129–192 (1996)

    Article  Google Scholar 

  18. Shigekawa, I.: De Rham-Hodge-Kodaira’s decomposition on an abstract Wiener space. J. Math. Kyoto Univ. 26, 191–202 (1986)

    Google Scholar 

  19. Shigekawa, I.: Sobolev spaces of Banach-valued functions associated with a Markov process. Probab. Theory Relat. Fields 99, 425–441 (1994)

    Article  Google Scholar 

  20. Üstünel, A.S., An introduction to analysis of Wiener space. Lecture Notes in Math. 1610, Springer, 1996

  21. Üstünel, A.S., Zakai, M.: The construction of filtrations on abstract Wiener space. J. Funct. Anal. 143, 10–32 (1997)

    Article  Google Scholar 

  22. Üstünel, A.S., Zakai, M.: Transformation of Measure on Wiener Space. Springer-Verlag, New York/Berlin, 1999

  23. Zakai, M.: Rotations and tangent processes on Wiener space, Séminaire de Probabilités, XXXVIII, Lect. Notes Math 1857, 2004 to appear, (arXiv:math.PR/0301351)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion I.I.T., Haifa, Israel

    E. Mayer-Wolf

  2. Department of Electrical Engineering, Technion I.I.T., Haifa, Israel

    M. Zakai

Authors
  1. E. Mayer-Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Zakai
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Mathematics Subject Classification (2000): Primary 60H07, Secondary 60H05

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-007-0111-0.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mayer-Wolf, E., Zakai, M. The divergence of Banach space valued random variables on Wiener space. Probab. Theory Relat. Fields 132, 291–320 (2005). https://doi.org/10.1007/s00440-004-0397-0

Download citation

  • Received: 06 January 2004

  • Revised: 23 September 2004

  • Published: 27 December 2004

  • Issue Date: June 2005

  • DOI: https://doi.org/10.1007/s00440-004-0397-0

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

Keywords

  • Abstract Wiener Space
  • Divergence
  • Flows
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