Transition Semigroup

  • Viorel Barbu
  • Giuseppe Da Prato
  • Michael Röckner
Part of the Lecture Notes in Mathematics book series (LNM, volume 2163)


This chapter is devoted to existence of invariant measures for transition semigroups associated with stochastic porous media equations with additive noise studied in previous chapters.


Markov Transition Semigroup Stochastic Porous Media Equations Invariant Measure Nonlinear Stochastic Partial Differential Equations Approximate Controllability Result 
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.


  1. 6.
    V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces (Springer, New York, 2010)CrossRefzbMATHGoogle Scholar
  2. 10.
    V. Barbu, G. Da Prato, The two phase stochastic Stefan problem. Probab. Theory Relat. Fields 124, 544–560 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 11.
    V. Barbu, G. Da Prato, Invariant measures and the Kolmogorov equation for the stochastic fast diffusion equation. Stoch. Process. Appl. 120 (7), 1247–1266 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 12.
    V. Barbu, G. Da Prato, Ergodicity for the phase-field equations perturbed by Gaussian noise. Infinite Dimen. Anal. Quantum Probab. Relat. Top. 14 (1), 35–55 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 20.
    V. Barbu, V. Bogachev, G. Da Prato, M. Röckner, Weak solutions to the stochastic porous media equation via Kolmogorov equations: the degenerate case. J. Funct. Anal. 237, 54–75 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 43.
    S. Cerrai, A Hille–Yosida theorem for weakly continuous semigroups. Semigroup Forum 49, 349–367 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 46.
    G. Da Prato, Kolmogorov Equations for Stochastic PDEs (Birkhäuser, Basel, 2004)CrossRefzbMATHGoogle Scholar
  8. 49.
    G. Da Prato, M. Röckner, Well posedness of Fokker–Planck equations for generators of time-inhomogeneous Markovian transition probabilities. Rend. Lincei Mat. Appl. 23 (4), 361–376 (2012)MathSciNetzbMATHGoogle Scholar
  9. 50.
    G. Da Prato, J. Zabczyk, Ergodicity for Infinite Dimensional Systems. London Mathematical Society Lecture Notes, vol. 229 (Cambridge University Press, Cambridge, 1996)Google Scholar
  10. 51.
    G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications, 2nd edn. (Cambridge University Press, Cambridge, 2014)Google Scholar
  11. 53.
    G. Da Prato, M. Röckner, B.L. Rozovskii, F. Wang, Strong solutions of stochastic generalized porous media equations: existence, uniqueness, and ergodicity. Commun. Partial Differ. Equ. 31 (1–3), 277–291 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 76.
    W. Liu, M. Röckner, Stochastic Partial Differential Equations: An Introduction. Universitext. (Springer, Cham, 2015)Google Scholar
  13. 77.
    W. Liu, J. Tölle, Existence and uniqueness of invariant measures for stochastic evolution equations with weakly dissipative drifts. Electronic Commun. Probab. 16, 447–457 (2011)MathSciNetCrossRefGoogle Scholar
  14. 83.
    E. Priola, On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Stud. Math. 136, 271–295 (1999)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Viorel Barbu
    • 1
  • Giuseppe Da Prato
    • 2
  • Michael Röckner
    • 3
  1. 1.Department of MathematicsAl. I. Cuza University & Octav Mayer Institute of Mathematics of the Romanian AcademyIasiRomania
  2. 2.Classe di ScienzeScuola Normale Superiore di PisaPisaItaly
  3. 3.Department of MathematicsUniversity of BielefeldBielefeldGermany

Personalised recommendations