Existence of an invariant measure for stochastic evolutions driven by an eventually compact semigroup

  • Joris Bierkens
  • Onno van Gaans
  • Sjoerd Verduyn Lunel
Open Access


It is shown that for an SDE in a Hilbert space, eventual compactness of the driving semigroup together with compact perturbations can be used to establish the existence of an invariant measure. The result is applied to stochastic functional differential equations and the heat equation perturbed by delay and noise, which are both shown to be driven by an eventually compact semigroup.

Mathematics Subject Classification (2000)

Primary 60H10 Secondary 47D06 


Delay equation Eventually compact semigroup Invariant measure Reaction diffusion equation Stochastic evolution equation Tightness 


Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    A. Bátkai and S. Piazzera. Semigroups for Delay Equations. AK Peters, Ltd., 2005.Google Scholar
  2. 2.
    V.I. Bogachev. Gaussian Measures. American Mathematical Society, 1998.Google Scholar
  3. 3.
    Da Prato G., Gatarek D., Zabczyk J.: Invariant measures for semilinear stochastic equations. Stochastic Analysis and Applications 10(4), 387–408 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Da Prato and J. Zabczyk. Ergodicity for Infinite Dimensional Systems. Cambridge University Press, 1996.Google Scholar
  5. 5.
    Delfour M.C.: The largest class of hereditary systems defining a C 0 semigroup on the product space. Canad. J. Math. 32(4), 969–978 (1980)zbMATHMathSciNetGoogle Scholar
  6. 6.
    O. Diekmann, S.A. Van Gils, S.M. Verduyn Lunel, and H.O. Walther. Delay Equations: Functional-, Complex-, and Nonlinear Analysis. Springer Verlag, 1995.Google Scholar
  7. 7.
    Dyson J.A., Villella-Bressan R., Webb G.F.: Asynchronous exponential growth in an age structured population of proliferating and quiescent cells. Mathematical Biosciences 177-178, 73–83 (2002)CrossRefMathSciNetGoogle Scholar
  8. 8.
    K.J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer Verlag, 2000.Google Scholar
  9. 9.
    J. Hale. Theory of functional differential equations. Springer-Verlag, New York, second edition, 1977. Applied Mathematical Sciences, Vol. 3.Google Scholar
  10. 10.
    Li W.-T., Wang Z.-C., Wu J.: Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity. J. Differential Equations 245(1), 102–129 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J.R. Munkres. Topology, Second Edition. Prentice-Hall, 2000.Google Scholar
  12. 12.
    W. Rudin. Functional analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Inc., New York, second edition, 1991.Google Scholar
  13. 13.
    C. C. Travis and G. F. Webb. Existence and stability for partial functional differential equations. In Dynamical systems (Proc. Internat. Sympos., Brown Univ., Providence, R.I., 1974), Vol. II, pages 147–151. Academic Press, New York, 1976.Google Scholar
  14. 14.
    van Gaans O., Verduyn Lunel S.M.: Long Term Behavior Of Dichotomous Stochastic Differential Equations In Hilbert Spaces. Communications in Contemporary Mathematics 6(3), 349–376 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Joris Bierkens
    • 1
  • Onno van Gaans
    • 1
  • Sjoerd Verduyn Lunel
    • 1
  1. 1.Mathematical InstituteLeidenThe Netherlands

Personalised recommendations