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

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  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  3. Arendt W. in: One-parameter Semigroups of Positive Operators. Springer, Heidelberg (1986)

  4. Chicone C., Latushkin Y.: Evolution Semigroups in Dynamical Systems and Differential Equations. Providence, American Mathematical Society (1999)

    MATH  Google Scholar 

  5. Chojnowska-Michalik A.: On processes of Ornstein–Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daleckii Ju.L., Krein M.G.: Stability of solutions of differential equations in Banach space. Providence, American Mathematical Society (1974)

    Google Scholar 

  7. Da Prato G., Lunardi A.: Ornstein–Uhlenbeck operators with time periodic coefficients. Journal of Evolution Equations 7, 587–614 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dudley R.: Real Analysis and Probability. Wadsworth, Brooks/Cole (1989)

    MATH  Google Scholar 

  9. Friedman A.: Partial Differential Equations. Holt, Reinhart and Winston (1969)

    MATH  Google Scholar 

  10. Fuhrmann M., Röckner M.: Generalized Mehler Semigroups: The Non-Gaussian Case. Potential Analysis 12, 1–47 (2000)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Linde W.: Infinitely Divisible and Stable Measures on Banach Spaces. Leipzig, Teubner (1983)

    MATH  Google Scholar 

  14. Ma Z., Röckner M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Heidelberg (1992)

    Book  MATH  Google Scholar 

  15. Neidhardt, H. and Zagrebnov, V.: Linear non-autonomous Cauchy problems and evolution semigroups - arXiv:0711.0284v1 [math-ph] 2007.

  16. Nickel, G.: On evolution semigroups and wellposedness of nonautonomous Cauchy problems - Ph.D. thesis, Tübingen: Univ. Tübingen, 1996.

  17. Peszat, S. and Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise - Cambridge University Press, 2007.

  18. Röckner M., Wang F.: Harnack and functional inequalities for generalized Mehler semigroups. Journal of Functional Analysis 203, 237–261 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sato, K.: Lévy processes and infinitely divisible distributions - Cambridge University Press, 1999.

  20. Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures - Oxford University Press, 1973.

  21. Veraar, M.: Non-autonomous stochastic evolution equations and applications to stochastic partial differential equations arXiv:0806.4439v3 [math.PR] 18 Nov 2008.

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Correspondence to Florian Knäble.

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Knäble, F. Ornstein–Uhlenbeck equations with time-dependent coefficients and Lévy noise in finite and infinite dimensions. J. Evol. Equ. 11, 959–993 (2011).

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