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
A version of Hörmander’s theorem for the fractional Brownian motion
Download PDF
Download PDF
  • Published: 16 June 2007

A version of Hörmander’s theorem for the fractional Brownian motion

  • Fabrice Baudoin1 &
  • Martin Hairer2 

Probability Theory and Related Fields volume 139, pages 373–395 (2007)Cite this article

  • 484 Accesses

  • 53 Citations

  • Metrics details

Abstract

It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy Hörmander’s condition. The main new ingredient of the proof is an extension of Norris’ lemma to this situation.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Arous G.B. (1989). Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale. Ann l’institut Fourier 39: 73–99

    MATH  Google Scholar 

  2. Baudoin F. (2005). An Introduction to the Geometry of Stochastic Flows. Imperial College Press, London

    Google Scholar 

  3. Baudoin, F., Coutin, L.: Self-similarity and fractional brownian motions on lie groups (2006) (preprint)

  4. Bismut J.-M. (1981). Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z. Wahrsch. Verw. Gebiete 56(4): 469–505

    Article  MATH  Google Scholar 

  5. Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence, RI (1998)

    Google Scholar 

  6. Cirel′son, B.S., Ibragimov, I.A., Sudakov, V.N.: Norms of Gaussian sample functions. In: Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), pp. 20–41. Lecture Notes in Mathematics., vol. 550. Springer, Berlin (1976)

  7. Coutin L., Qian Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Relat Fields. 122(1): 108–140

    Article  MATH  Google Scholar 

  8. Friz, P., Victoir, N.: Euler estimates for rough differential equations (2006) (preprint)

  9. Hu, Y., Nualart, D.: Differential equations driven by hölder continuous functions of order greater than 1/2 (2006) (preprint)

  10. Hörmander L. (1967). Hypoelliptic second order differential equations. Acta Math. 119: 147–171

    Article  MATH  Google Scholar 

  11. Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. I. In: Stochastic analysis (Katata/Kyoto, 1982 pp. 271–306). North-Holland Mathematics. Library, vol. 32. North-Holland, Amsterdam (1984)

  12. Kusuoka S., Stroock D. (1985). Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32(1): 1–76

    MATH  Google Scholar 

  13. Kusuoka S., Stroock D. (1987). Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34(2): 391–442

    MATH  Google Scholar 

  14. Lyons T.J. (1994). Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1(4): 451–464

    MATH  Google Scholar 

  15. Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Symp. Stoch. Diff. Equations, Kyoto 1976 147–171

  16. Norris, J.: Simplified Malliavin calculus. In: Séminaire de Probabilités, XX, 1984/85, pp. 101–130. Lecture Notes in Mathematics,vol. 1204. Springer, Berlin Heidelberg Newyork (1986)

  17. Nualart D., Răşcanu A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53(1): 55–81

    MATH  Google Scholar 

  18. Nualart, D., Saussereau, B.: Malliavin calculus for stochastic differential equations driven by a fractional brownian motion (2005) (preprint)

  19. Nourdin I., Simon T. (2006). On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Stat. Prob Lett. 76: 907–912

    Article  MATH  Google Scholar 

  20. Nualart, D.: The Malliavin Calculus and Related Topics. In:Probability and its Applications (New York). Springer, Berlin Heidelberg New York (1995)

  21. Pipiras V., Taqqu M.S. (2000). Integration questions related to fractional Brownian motion. Prob. Theory Relat. Fields 118(2): 251–291

    Article  MATH  Google Scholar 

  22. Rothschild L.P., Stein E.M. (1976). Hypoelliptic differential operators and nilpotent groups. Acta Math. 137(3–4): 247–320

    Article  Google Scholar 

  23. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993). Theory and applications, Edited and with a foreword by S. M. Nikol′skiĭ, Translated from the 1987 Russian original, Revised by the authors

  24. Talagrand M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81: 73–205

    MATH  Google Scholar 

  25. Young L.C. (1936). An inequality of Hölder type connected with Stieltjes integration. Acta Math. 67: 251–282

    Article  MATH  Google Scholar 

  26. Zähle M. (2001). Integration with respect to fractal functions and stochastic calculus. II. Mathe. Nachrich. 225: 145–183

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, 118 Route de Narbonne, Toulouse, France

    Fabrice Baudoin

  2. Mathematics Institute, The University of Warwick, Coventry, CV4 7AL, UK

    Martin Hairer

Authors
  1. Fabrice Baudoin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Hairer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Hairer.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Baudoin, F., Hairer, M. A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Relat. Fields 139, 373–395 (2007). https://doi.org/10.1007/s00440-006-0035-0

Download citation

  • Received: 26 May 2006

  • Revised: 05 October 2006

  • Published: 16 June 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-006-0035-0

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

  • Stochastic Differential Equation
  • Fractional Brownian Motion
  • Gaussian Measure
  • Carnot Group
  • Hurst Parameter
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