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
Large deviation for stochastic line integrals as L p-currents
Download PDF
Download PDF
  • Published: 13 May 2009

Large deviation for stochastic line integrals as L p-currents

  • Shigeo Kusuoka1,
  • Kazumasa Kuwada2 &
  • Yozo Tamura3 

Probability Theory and Related Fields volume 147, pages 649–674 (2010)Cite this article

  • 170 Accesses

  • 8 Citations

  • Metrics details

Abstract

The large deviation principle for stochastic line integrals along Brownian paths on a compact Riemannian manifold is studied. We regard them as a random map on a Sobolev space of 1-forms. We show that the differentiability order of the Sobolev space can be chosen to be almost independent of the dimension of the underlying space by assigning higher integrability on 1-forms. The large deviation is formulated for the joint distribution of stochastic line integrals and the empirical distribution of a Brownian path. As the result, the rate function is given explicitly.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Dembo A., Zeitouni O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics, vol. 38. Springer, New York (1998)

    Google Scholar 

  2. Deuschel J.-D., Stroock D.W.: Large Deviations. Academic Press, Boston (1989)

    MATH  Google Scholar 

  3. Flandoli F., Giaquinta M., Gubinelli M., Tortorelli V.M.: Stochastic currents. Stoch. Process. Appl. 115(9), 1583–1601 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Flandoli F., Gubinelli M., Russo F.: On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model. Ann. Inst. H. Poincaré 45(2), 545–576 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hsu E.P.: Stochastic Analysis on Manifolds. Graduate Studies in Mathematics, vol. 38. American Mathematical Society, Providence (2002)

    Google Scholar 

  6. Ikeda, N.: Limit Theorems for a Class of Random Currents. Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985), pp. 181–193. Academic Press, Boston (1987)

  7. Ikeda N., Manabe S.: Integral of differential forms along the path of diffusion processes. Publ. RIMS, Kyoto Univ. 15, 827–852 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ikeda N., Ochi Y.: Central limit theorems and random currents. Lect. Notes Contr. Inform. Sci. 78, 195–205 (1986)

    Article  MathSciNet  Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North-Holland Mathematical Library, vol. 24. North-Holland, Amsterdam (1989)

  10. Kuwada K.: Sample path large deviations for a class of random currents. Stoch. Process. Appl. 108, 203–228 (2003)

    MATH  MathSciNet  Google Scholar 

  11. Kuwada K.: On large deviations for random currents induced from stochastic line integrals. Forum Math. 18, 639–676 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Manabe S.: Stochastic intersection number and homological behavior of diffusion processes on Riemannian manifolds. Osaka J. Math. 19, 429–454 (1982)

    MATH  MathSciNet  Google Scholar 

  13. Manabe S.: Large deviation for a class of current-valued processes. Osaka J. Math. 29, 89–102 (1992)

    MATH  MathSciNet  Google Scholar 

  14. Ochi Y.: Limit theorems for a class of diffusion processes. Stochastics 15, 251–269 (1985)

    MATH  MathSciNet  Google Scholar 

  15. Revuz D., Yor M.: Continuous martingales and Brownian motion, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 293. Springer, Berlin (1999)

    Google Scholar 

  16. Shigekawa I.: L p contraction semigroups for vector valued functions. J. Funct. Anal. 147, 69–108 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Sion M.: On general minimax theorems. Pac. J. Math. 8, 171–176 (1958)

    MATH  MathSciNet  Google Scholar 

  18. Yoshida N.: Sobolev spaces on a Riemannian manifold and their equivalence. J. Math. Kyoto univ. 32(3), 621–654 (1992)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Graduate School of Mathematics, The University of Tokyo, Tokyo, 153-8914, Japan

    Shigeo Kusuoka

  2. Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo, 112-8610, Japan

    Kazumasa Kuwada

  3. Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, 223-0061, Japan

    Yozo Tamura

Authors
  1. Shigeo Kusuoka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kazumasa Kuwada
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yozo Tamura
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kazumasa Kuwada.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kusuoka, S., Kuwada, K. & Tamura, Y. Large deviation for stochastic line integrals as L p-currents. Probab. Theory Relat. Fields 147, 649–674 (2010). https://doi.org/10.1007/s00440-009-0219-5

Download citation

  • Received: 22 July 2008

  • Revised: 25 March 2009

  • Published: 13 May 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0219-5

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

  • Large deviation
  • Stochastic line integral
  • Empirical distribution
  • Current-valued process

Mathematics Subject Classification (2000)

  • 60F10
  • 60B12
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