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

  • Florian Knäble


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 L 2-space. The square field operator is characterized, allowing to derive a Poincaré and a Harnack inequality.


Invariant Measure Mild Solution Harnack Inequality Evolution Family Strong Continuity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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)zbMATHCrossRefGoogle 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)zbMATHGoogle Scholar
  5. 5.
    Chojnowska-Michalik A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dudley R.: Real Analysis and Probability. Wadsworth, Brooks/Cole (1989)zbMATHGoogle Scholar
  9. 9.
    Friedman A.: Partial Differential Equations. Holt, Reinhart and Winston (1969)zbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Linde W.: Infinitely Divisible and Stable Measures on Banach Spaces. Leipzig, Teubner (1983)zbMATHGoogle Scholar
  14. 14.
    Ma Z., Röckner M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Heidelberg (1992)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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