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


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.

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  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.

  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.

  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.

  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.

  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.

    MathSciNet  Article  MATH  Google Scholar 

  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.

  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.

  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.

  9. W. Liu and M. Röckner. Stochastic partial differential equations: an introduction. Universitext. Springer, Cham, 2015.

  10. C. Prévôt and M. Röckner. A concise course on stochastic partial differential equations, volume 1905 of Lecture Notes in Mathematics. Springer, Berlin, 2007.

  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.

    MathSciNet  Article  MATH  Google Scholar 

  12. M. Röckner and F.-Y. Wang. Non-monotone stochastic generalized porous media equations. J. Differential Equations, 245(12):3898–3935, 2008.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Francesco Russo.

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Röckner, M., Russo, F. Uniqueness for a class of stochastic Fokker–Planck and porous media equations. J. Evol. Equ. 17, 1049–1062 (2017).

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Mathematics Subject Classification

  • 35R60
  • 60H15
  • 82C31


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