Skip to main content
SpringerLink
Log in

Smoothing properties of nonlinear stochastic equations in Hilbert spaces

  • Published:
Nonlinear Differential Equations and Applications NoDEA Aims and scope Submit manuscript

Abstract

A nonlinear stochastic equation in a Hilbert space is considered, with constant but possibly degenerate diffusion term. Some smoothing properties for the associated transition semigroup are studied. In particular, strong Feller property and irreducibility are proved. The main tools are Malliavin calculus and Girsanov transformation.

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. DA PRATO, A. LUNARDI, On the Ornstein-Uhlenbeck operator in spaces of continuous functions, to appear inJ. Functional Anal.

  2. G. DA PRATO, D. NUALART, J. ZABCZYK, Strong Feller property for infinite-dimensional stochastic equations, preprint, Scuola Normale Superiore di Pisa, 1994

  3. G. DA PRATO, J. ZABCZYK, “Stochastic equations in infinite dimensions,” Encyclopaedia of mathematics and its applications 44, Cambridge University Press, 1992

  4. K. D. ELWORTHY, Stochastic flows in Riemannian manifolds,inM.A. Pinsky and V. Vihstutz editors, “Diffusion processes and related problems in Analysis,” vol. II, 37–72, Birkhäuser, Boston, Basel, Berlin, (1992)

    Google Scholar 

  5. B. GAVEAU, J.M. MOULINIER, Régularité des mesures et perturbations stochastiques de champs de vecteurs sur des espaces de dimension infinie,Publ. RIMS Kyoto Univ. 21, 593–616 (1985)

    Google Scholar 

  6. L. HÖRMANDER, Hypoelliptic differential equations of second order,Acta Math. 119, 147–171 (1967)

    Google Scholar 

  7. J. M. LASRY, P. L. LIONS, A remark on regularization in Hilbert spaces,Israel J. Math. 55, 3, 257–266 (1986)

    Google Scholar 

  8. A. LUNARDI, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients inR n, preprint, Scuola Normale Superiore di Pisa, 1994

  9. B. MASLOWSKI, On probability distributions of solutions of semilinear stochastic evolution equations,Stochastics Stochastics Rep. 45, 17–44 (1993)

    Google Scholar 

  10. A. S. NEMIROVSKI, S. M. SEMENOV, The polynomial approximation of functions in Hilbert spaces,Mat. Sb. (N.S)92, 134, 257–281 (1973)

    Google Scholar 

  11. D. NUALART, “The Malliavin calculus and related topics,” book to appear

  12. S. PESZAT, J. ZABCZYK, Strong Feller property and irreducibility for diffusions on Hilbert spaces, to appear inAnnals of Probability.

  13. T. SEIDMAN, How violent are fast controls?,Math. Control Signals Systems 1, 89–95 (1988)

    Google Scholar 

  14. I. SHIGEKAWA, Existence of invariant measures of diffusions on an abstract Wiener space,Osaka J. Math. 24, 37–59 (1987)

    Google Scholar 

  15. J. ZABCZYK, “Mathematical control theory: an introduction,” Birkhäuser, Boston, Basel, Berlin (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci, 32, 20133, Milano, Italia

    Marco Fuhrman

Authors
  1. Marco Fuhrman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by Italian MURST 60% research funds.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fuhrman, M. Smoothing properties of nonlinear stochastic equations in Hilbert spaces. NoDEA 3, 445–464 (1996). https://doi.org/10.1007/BF01193830

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01193830

Keywords

Search

Navigation