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Nonlinear Fokker–Planck–Kolmogorov Equations for Measures

  • Stanislav V. ShaposhnikovEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

Existence and uniqueness of solutions of the Cauchy problem for nonlinear Fokker–Planck–Kolmogorov equations for measures are investigated. We consider the difficult case when the diffusion matrix depends on the solution. Moreover we give a short survey of the known results connected with these problems.

Keywords

Nonlinear Fokker–Planck–Kolmogorov equation McKean–Vlasov equation Existence and uniqueness of solutions to the Cauchy problem for nonlinear parabolic equations 

AMS subject classification.

35K55 35Q84 35Q83 

Notes

Acknowledgements

The author is grateful to Prof. V.I. Bogachev and Prof. M. Röckner for fruitful discussions and valuable remarks. The work was supported by the President Grant MD-207.2017.1, the RFBR grant 17-01-00622, the Simons Foundation, and the SFB 1283 at Bielefeld University.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005)Google Scholar
  2. 2.
    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Global regularity and bounds for solutions of parabolic equations for probability measures. Teor. Verojatn. Primen. 50(4), 652–674 (2005) (in Russian); English transl. Theory Probab. Appl. 50(4), 561–581 (2006)Google Scholar
  3. 3.
    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Nonlinear evolution and transport equations for measures. Doklady Math. 80(3), 785–789 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bogachev, V.I., Da Prato, G., Röckner, M., Shaposhnikov, S.V.: Nonlinear evolution equations for measures on infinite dimensional spaces. Stochastic Partial Differential Equations and Applications. Quaderni di Matematica, Dipartimento di Matematica Seconda Universita di Napoli Napoli 25, 51–64 (2010)Google Scholar
  5. 5.
    Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker–Planck–Kolmogorov Equations. American Mathematical Society, Providence, Rhode Island (2015)CrossRefGoogle Scholar
  6. 6.
    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations. J. Funct. Anal. 271, 1262–1300 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carrillo, J.A., Difrancesco, M., Figalli, A., Laurent, T., Slepcev, D.: Global-in-time weak measure solutions and finite-time aggregation for non-local interaction equations. Duke Math. J. 156(2), 229–271 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    DiPerna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)MathSciNetCrossRefGoogle Scholar
  9. 9.
    DiPerna, R.J., Lions, P.L.: On the Fokker–Planck–Boltzmann equation. Commun. Math. Phys. 120(1), 1–23 (1988)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Frank, T.D.: Nonlinear Fokker-Planck Equations. Fundamentals and Applications. Springer, Berlin (2005)Google Scholar
  11. 11.
    Funaki, T.: A certain class of duffusion processes associated with nonlinear parabolic equations. Z. Wahrscheinlichkeitstheorie verw. Geb. 67, 331–348 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kolmogorov, A.N.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104, 415–458 (1931)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Le Bris, C., Lions, P.L.: Existence and uniqueness of solutions to Fokker–Planck type equations with irregular coefficients. Commun. Partial Differ. Equ. 33, 1272–1317 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Manita, O.A.: Nonlinear Fokker–Planck–Kolmogorov equations in Hilbert spaces. J. Math. Sci. (New York) 216(1), 120–135 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Manita, O.A., Shaposhnikov, S.V.: Nonlinear parabolic equations for measures. St. Petersb. Math. J. 25(1), 43–62 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Manita, O.A., Romanov, M.S., Shaposhnikov, S.V.: On uniqueness of solutions to nonlinear Fokker–Planck–Kolmogorov equations. Nonlinear Anal. Theory Methods Appl. 128, 199–226 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56, 1907–1911 (1966)MathSciNetCrossRefGoogle Scholar
  19. 19.
    McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. In: Lecture Series in Differential Equations, Session, 7, pp. 177–194. Catholic University (1967)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.St.-Tikhon’s Orthodox Humanitarian UniversityMoscowRussia

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