Probability Theory and Related Fields

, Volume 149, Issue 1–2, pp 97–137 | Cite as

Structural properties of semilinear SPDEs driven by cylindrical stable processes

  • Enrico PriolaEmail author
  • Jerzy Zabczyk


We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise. We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results are applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.


Stochastic PDEs with jumps Strong Feller property Regularity of trajectories 

Mathematics Subject Classification (2000)

60H15 60J75 47D07 35R60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albeverio S., Wu J.L., Zhang T.S.: Parabolic SPDEs driven by Poisson white noise. Stoch. Process Appl. 74, 21–36 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Applebaum, D.: Lévy processes and stochastic calculus. In: Cambridge Studies in Advanced Mathematics, vol. 93. Cambridge University Press, Cambridge (2004)Google Scholar
  3. 3.
    Bogachev V.I., Röckner M., Schmuland B.: Generalized Mehler semigroups and applications. Probab. Theory Relat. Fields 105, 193–225 (1996)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chojnowska-Mikhalik A.: On Processes of Ornstein-Uhlenbeck type in Hilbert space. Stochastics 21, 251–286 (1987)MathSciNetGoogle Scholar
  5. 5.
    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. In: Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)Google Scholar
  6. 6.
    Da Prato, G., Zabczyk, J.: Ergodicity for infinite-dimensional systems. In: London Mathematical Society. Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996)Google Scholar
  7. 7.
    Dawson D.A., Li Z., Schmuland B., Sun W.: Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal. 21(1), 75–97 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Engel K., Nagel R.: One-parameter Semigroups for Linear Evolution Equations. In: Springer Graduate Texts in Mathematics 194. Springer, Berlin (2000)Google Scholar
  9. 9.
    Ethier S.N., Kurtz T.G.: Markov Processes: Characterization and Convergence. Wiley, NY (1986)zbMATHGoogle Scholar
  10. 10.
    Feller W.: An Introduction to Probability Theory and its Applications, vol. II, 2nd edn. Wiley, NY (1971)zbMATHGoogle Scholar
  11. 11.
    Fuhrman M., Röckner M.: Generalized Mehler semigroups: the non-Gaussian case. Potential Anal. 12(1), 1–47 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Haagerup, U.: The best constants in the Khintchine inequality. Studia Math. 70 (1981), no. 3, 231–283 (1982)Google Scholar
  13. 13.
    Hairer M., Mattingly J.C.: Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. Math. (2) 164(2), 993–1032 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Hairer, M., Mattingly, J.C.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Probab. (2009, in press)Google Scholar
  15. 15.
    Hairer, M., Mattingly, J.C.: A Theory of Hypoellipticity and Unique Ergodicity for Semilinear Stochastic PDEs. Preprint. (2008)
  16. 16.
    Kallenberg, O.: Foundations of modern probability, 2nd edn. In: Probability and its Applications (New York). Springer, New York (2002)Google Scholar
  17. 17.
    Kwapień S., Woyczyński W.A.: Random Series and Stochastic Integrals: Single And Multiple. Probability and its Applications. Birkhäuser Boston, Inc., Boston (1992)Google Scholar
  18. 18.
    Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy noise. Cambridge (2007)Google Scholar
  19. 19.
    Priola, E., Zabczyk, J.: Harmonic functions for generalized Mehler semigroups. In: Stochastic Partial Differential Equations and Applications-VII, pp. 243-256. Lect. Notes Pure Appl. Math., vol. 245. Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
  20. 20.
    Priola E., Zabczyk J.: Densities for Ornstein-Uhlenbeck processes with jumps, Bull. Lond. Math. Soc. 41, 41–50 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Priola E., Zabczyk J.: Liouville theorems for non local operators. J. Funct. Anal. 216, 455–490 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Priola, E., Zabczyk, J.: Structural Properties of Semilinear SPDEs Driven by Cylindrical Stable Processes, Preprint.
  23. 23.
    Röckner M., Wang F.Y.: Harnack and Functional Inequalities for Generalised Mehler Semigroups. J. Funct. Anal. 203, 237–261 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Rosiński J., Woyczyński W.A.: On Itô stochastic integration with respect to p-stable motion: inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14, 271–286 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Samorodnitsky G., Taqqu M.S.: Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman & Hall, New York (1994)Google Scholar
  26. 26.
    Sato K.I.: Lévy processes and infinite divisible distributions. Cambridge University Press, London (1999)Google Scholar
  27. 27.
    Simon T.: Sur les petites déviations d’un processus de Lévy. (French) [On the small deviations of a Lévy process]. Potential Anal. 14, 155–173 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Triebel H.: Interpolation theory, function spaces, differential operators, 2nd edn. Johann Ambrosius Barth, Heidelberg (1995)zbMATHGoogle Scholar
  29. 29.
    Zolotarev V.M.: One-dimensional stable distributions. Translated from the Russian by H.H. McFaden. Translations of Mathematical Monographs, 65. American Mathematical Society, Providence (1986)Google Scholar
  30. 30.
    Zinn J.: Admissible translates of stable measures. Studia Math. 54, 245–257 (1976)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di TorinoTurinItaly
  2. 2.Instytut MatematycznyPolskiej Akademii NaukWarsawPoland

Personalised recommendations