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
Stochastic heat equation driven by fractional noise and local time
Download PDF
Download PDF
  • Published: 04 January 2008

Stochastic heat equation driven by fractional noise and local time

  • Yaozhong Hu1 &
  • David Nualart1 

Probability Theory and Related Fields volume 143, pages 285–328 (2009)Cite this article

  • 736 Accesses

  • 97 Citations

  • Metrics details

Abstract

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bass R.F. and Chen X. (2004). Self-intersection local time: Critical exponent, large deviations and laws of the iterated logarithm. Ann. Probab. 32: 3221–3247

    Article  MATH  MathSciNet  Google Scholar 

  2. Buckdahn R. and Nualart D. (1994). Linear stochastic differential equations and Wick products. Probab. Theory Relat. Fields 99: 501–526

    Article  MATH  MathSciNet  Google Scholar 

  3. Duncan T.E., Maslowski B. and Pasik-Duncan B. (2002). Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn. 2: 225–250

    Article  MATH  MathSciNet  Google Scholar 

  4. Gubinelli M., Lejay A., Tindel S.: Young integrals and SPDE. Pot. Anal. 25, 307–326 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Hu Y. (2001). Heat equation with fractional white noise potentials. Appl. Math. Optim. 43: 221–243

    Article  MATH  MathSciNet  Google Scholar 

  6. Hu Y. and Nualart D. (2005). Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33: 948–983

    Article  MATH  MathSciNet  Google Scholar 

  7. Le Gall, J.-F.: Exponential moments for the renormalized self-intersection local time of planar Brownian motion. Séminaire de Probabilités, XXVIII, Lecture Notes in Mathematics, vol. 1583, pp. 172–180. Springer, Berlin (1994)

  8. Lyons T. and Qian Z. (2002). System control and rough paths. Oxford Mathematical Monographs. Oxford Science Publications, Oxford University Press, Oxford

    Google Scholar 

  9. Maslowski B. and Nualart D. (2003). Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202: 277–305

    Article  MATH  MathSciNet  Google Scholar 

  10. Memin J., Mishura Y. and Valkeila E. (2001). Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist. Probab. Lett. 51: 197–206

    Article  MATH  MathSciNet  Google Scholar 

  11. Mueller C. and Tribe R. (2004). A singular parabolic Anderson model. Electron. J. Probab. 9: 98–144

    MathSciNet  Google Scholar 

  12. Muirhead R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York

    Google Scholar 

  13. Nualart D. (2006). The Malliavin Calculus and Related Topics, 2nd edn. Springer, Berlin

    Google Scholar 

  14. Nualart D. and Rozovskii B. (1997). Weighted stochastic Sobolev spaces and bilinear SPDEs driven by space-time white noise. J. Funct. Anal. 149: 200–225

    Article  MathSciNet  Google Scholar 

  15. Nualart D. and Zakai M. (1989). Generalized Brownian functionals and the solution to a stochastic partial differential equation. J. Funct. Anal. 84: 279–296

    Article  MATH  MathSciNet  Google Scholar 

  16. Pipiras V. and Taqqu M. (2000). Integration questions related to fractional Brownian motion. Probab. Theory Relat. Fields 118: 251–291

    Article  MATH  MathSciNet  Google Scholar 

  17. Russo F. and Vallois P. (1993). Forward, backward and symmetric stochastic integration. Probab. Theory Relat. Fields 97(3): 403–421

    Article  MATH  MathSciNet  Google Scholar 

  18. Tindel S., Tudor C. and Viens F. (2003). Stochastic evolution equations with fractional Brownian motion. Probab. Theory Relat. Fields 127: 186–204

    Article  MATH  MathSciNet  Google Scholar 

  19. Tudor C. (2004). Fractional bilinear stochastic equations with the drift in the first fractional chaos. Stoch. Anal. Appl. 22: 1209–1233

    Article  MATH  MathSciNet  Google Scholar 

  20. Walsh, J.B.: An introduction to stochastic partial differential equations. In: Ecole d’Ete de Probabilites de Saint Flour XIV, Lecture Notes in Mathematics, vol. 1180, pp. 265–438 (1986)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS, 66045-2142, USA

    Yaozhong Hu & David Nualart

Authors
  1. Yaozhong Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Nualart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaozhong Hu.

Additional information

Y. Hu is supported by the National Science Foundation under DMS0504783.

D. Nualart is supported by the National Science Foundation under DMS0604207.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hu, Y., Nualart, D. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143, 285–328 (2009). https://doi.org/10.1007/s00440-007-0127-5

Download citation

  • Received: 16 February 2007

  • Revised: 26 November 2007

  • Published: 04 January 2008

  • Issue Date: January 2009

  • DOI: https://doi.org/10.1007/s00440-007-0127-5

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

Mathematics Subject Classification (2000)

  • 60H15
  • 60H07
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