Journal of Evolution Equations

, Volume 17, Issue 3, pp 1049–1062 | Cite as

Uniqueness for a class of stochastic Fokker–Planck and porous media equations

  • Michael Röckner
  • Francesco Russo


The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker–Planck equation under very general assumptions. In particular, the second-order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space.


Stochastic partial differential equations Infinite volume Porous media type equation Multiplicative noise Stochastic Fokker–Planck type equation 

Mathematics Subject Classification

35R60 60H15 82C31 


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  1. 1.
    V. Barbu, G. Da Prato, and M. Röckner. Stochastic Porous Media Equation, volume 1905 of Lecture Notes in Mathematics. Springer, Berlin, 2016, in Press.Google Scholar
  2. 2.
    V. Barbu, M. Roeckner, and F. Russo. Doubly probabilistic representation for the stochastic porous media type equation. Annales de l’Institut Henri Poincare. Section Probabilites et Statistiques, to appear.Google Scholar
  3. 3.
    V. Barbu, M. Röckner, and F. Russo. Stochastic porous media equations in \({\mathbb{R}}^{d}\). J. Math. Pures Appl. (9), 103(4):1024–1052, 2015.Google Scholar
  4. 4.
    N. Belaribi and F. Russo. Uniqueness for Fokker-Planck equations with measurable coefficients and applications to the fast diffusion equation. Electron. J. Probab., 17:no. 84, 28, 2012.Google Scholar
  5. 5.
    P. Blanchard, M. Röckner, and F. Russo. Probabilistic representation for solutions of a porous media type equation. Ann. Probab., 38(5):1870–1900, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    V.I. Bogachev, N.V. Krylov, M. Röckner, and Shaposhnikov. Fokker-Planck-Kolmogorov equations. Izhewsk Institute of Computer Science, 2015. Russian version. English version in preparation.Google Scholar
  7. 7.
    H. Brezis and M. G. Crandall. Uniqueness of solutions of the initial-value problem for \(u_{t}-\Delta {\varphi } (u)=0\). J. Math. Pures Appl. (9), 58(2):153–163, 1979.Google Scholar
  8. 8.
    G. Da Prato and J. Zabczyk. Stochastic equations in infinite dimensions, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1992.Google Scholar
  9. 9.
    W. Liu and M. Röckner. Stochastic partial differential equations: an introduction. Universitext. Springer, Cham, 2015.Google Scholar
  10. 10.
    C. Prévôt and M. Röckner. A concise course on stochastic partial differential equations, volume 1905 of Lecture Notes in Mathematics. Springer, Berlin, 2007.Google Scholar
  11. 11.
    J. Ren, M. Röckner, and F.-Y. Wang. Stochastic generalized porous media and fast diffusion equations. J. Differential Equations, 238(1):118–152, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    M. Röckner and F.-Y. Wang. Non-monotone stochastic generalized porous media equations. J. Differential Equations, 245(12):3898–3935, 2008.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.ENSTA ParisTechUniversité Paris-SaclayPalaiseauFrance

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