Journal of Evolution Equations

, Volume 11, Issue 4, pp 959–993 | Cite as

Ornstein–Uhlenbeck equations with time-dependent coefficients and Lévy noise in finite and infinite dimensions



We solve a time-dependent linear SPDE with additive Lévy noise in the mild and weak sense. Existence of a generalized invariant measure for the associated transition semigroup is established and the generator is studied on the corresponding L2-space. The square field operator is characterized, allowing to derive a Poincaré and a Harnack inequality.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio S., Rüdiger B.: Stochastic Integrals and the Lévy–Ito Decomposition Theorem on Separable Banach Spaces. Stochastic Analyis and Applications 23, 217–253 (2005)MATHCrossRefGoogle Scholar
  2. 2.
    Applebaum D.: Martingale-valued measures, Ornstein–Uhlenbeck processes with jumps and operator self-decomposability in Hilbert space. Séminaire de Probabilités 39, 171–196 (2006)MathSciNetGoogle Scholar
  3. 3.
    Arendt W. in: One-parameter Semigroups of Positive Operators. Springer, Heidelberg (1986)Google Scholar
  4. 4.
    Chicone C., Latushkin Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Providence, American Mathematical Society (1999)MATHGoogle Scholar
  5. 5.
    Chojnowska-Michalik A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Daleckii Ju.L., Krein M.G.: Stability of solutions of differential equations in Banach space. Providence, American Mathematical Society (1974)Google Scholar
  7. 7.
    Da Prato G., Lunardi A.: Ornstein–Uhlenbeck operators with time periodic coefficients. Journal of Evolution Equations 7, 587–614 (2007)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Dudley R.: Real Analysis and Probability. Wadsworth, Brooks/Cole (1989)MATHGoogle Scholar
  9. 9.
    Friedman A.: Partial Differential Equations. Holt, Reinhart and Winston (1969)MATHGoogle Scholar
  10. 10.
    Fuhrmann M., Röckner M.: Generalized Mehler Semigroups: The Non-Gaussian Case. Potential Analysis 12, 1–47 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lescot P., Röckner M.: Generators of Mehler-Type Semigroups as Pseudo-Differential Operators. Infinite Dimensional Analysis, Quantum Probability and Related Topics 5, 297–316 (2002)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Lescot P., Röckner M.: Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Lévy Noise and Singular Drift. Potential Analysis 20, 317–344 (2004)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Linde W.: Infinitely Divisible and Stable Measures on Banach Spaces. Leipzig, Teubner (1983)MATHGoogle Scholar
  14. 14.
    Ma Z., Röckner M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Heidelberg (1992)MATHCrossRefGoogle Scholar
  15. 15.
    Neidhardt, H. and Zagrebnov, V.: Linear non-autonomous Cauchy problems and evolution semigroups - arXiv:0711.0284v1 [math-ph] 2007.Google Scholar
  16. 16.
    Nickel, G.: On evolution semigroups and wellposedness of nonautonomous Cauchy problems - Ph.D. thesis, Tübingen: Univ. Tübingen, 1996.Google Scholar
  17. 17.
    Peszat, S. and Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise - Cambridge University Press, 2007.Google Scholar
  18. 18.
    Röckner M., Wang F.: Harnack and functional inequalities for generalized Mehler semigroups. Journal of Functional Analysis 203, 237–261 (2003)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Sato, K.: Lévy processes and infinitely divisible distributions - Cambridge University Press, 1999.Google Scholar
  20. 20.
    Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures - Oxford University Press, 1973.Google Scholar
  21. 21.
    Veraar, M.: Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations arXiv:0806.4439v3 [math.PR] 18 Nov 2008.Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Universität BielefeldFakultät für MathematikBielefeldGermany

Personalised recommendations