Skip to main content
SpringerLink
Log in

The Malliavin calculus and its application to second order parabolic differential equations: Part I

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

The first part of this paper contains a rigorous and detailed development of the Malliavin calculus and its relation to stochastic integral equations. The second part is devoted to examples of applications of this machinery to the study of solutions to the Fokker-Planck equation, associated with diffusions. The applications given are by no means exhaustive, but instead they have been chosen to demonstrate the scope of Malliavin's ideas in the hope of stimulating further investigations into this subject.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. B. Folland, “Subelliptic Estimates and Function Spaces on Nilpotent Lie Groups,”Arkiv. f. Mat. 13 (1975), pp. 161–207.

    Google Scholar 

  2. I. I. Gihman, and A. V. Skorokod,Stochastic Differential Equations, Springer-Verlag, Berlin (1972).

    Google Scholar 

  3. L. Hörmander, “Hypoelliptic Second Order Differential Equations,”Acta Math, 119 (1967), pp. 147–171.

    Google Scholar 

  4. N. Ikada, and S. Watenabe,Stochastic Differential Equations and Diffusion Processes (to appear).

  5. M. Kac, “Distribution of Certain Wiener Functionals,” T.A.M.S., vol. 65 (1949), pp. 1–13.

    Google Scholar 

  6. P. Malliavin, “C k-Hypoelliplicity with Degeneracy” (Parts I and II),Stochastic Analysis, edited by Friedman and Pinsky; Academic Press (1978), pp. 199–214 and pp. 327–340.

  7. P. Malliavin, “Stochastic Calculus of Variation and Hypoelliptic Operators,” Proc. of the International Symposium on Stochastic Differential Equations (Kyoto 1976) Tokyo, 1978.

  8. H. P. McKean,Stochastic Integrals, Academic Press, N.Y. (1969).

    Google Scholar 

  9. L. P. Rochchild, and E. M. Stein, “Hypoelliptic Differential Operators and Nilpotent Groups,”Acta Math vol. 137 (1976), pp. 247–320.

    Google Scholar 

  10. D. Stroock, and R. Holley, “Generalized Ornstein-Uhlenbeck Processes and Infinite Particle Branching Brownian Motions,”Publ. of R.I.M.S., Kyoto Univ., vol. 14 no. 3, pp. 741–788.

  11. D. Stroock, and S. R. S. Varadhan,Multidimensional Diffusion Processes, Springer-Verlag, New York (1979).

    Google Scholar 

  12. D. Stroock, and M. Yor, On Extremal Solutions of Martingale Problems,Ann. Ecole Norm. Sup., to appear.

  13. H. Kunita, and S. Watanabe, “On Square Integrable Martingales,”Nagoya Math. J., 30 (1967), pp. 209–245.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics, University of Colorado, 80309, Boulder, CO, USA

    Daniel W. Stroock (Guggenheim fellow)

Authors
  1. Daniel W. Stroock
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research partially supported by N.S.F. Grant MCS 77-14881 A01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stroock, D.W. The Malliavin calculus and its application to second order parabolic differential equations: Part I. Math. Systems Theory 14, 25–65 (1981). https://doi.org/10.1007/BF01752389

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01752389

Keywords

Search

Navigation