Advertisement

Convergence to Equilibrium in Fokker–Planck Equations

  • Min Ji
  • Zhongwei Shen
  • Yingfei Yi
Article
  • 64 Downloads

Abstract

The present paper is devoted to the investigation of long-time behaviors of global probability solutions of Fokker–Planck equations with rough coefficients. In particular, we prove the convergence of probability solutions under a Lyapunov condition in terms of the Markov semigroup associated to the stationary one. A generalization of earlier results on the existence and uniqueness of global probability solutions is also given.

Keywords

Fokker–Planck equation Probability solutions Convergence 

Mathematics Subject Classification

Primary 37C40 37C75 34F05 60H10 Secondary 35Q84 60J60 

Notes

Acknowledgements

We would like to thank Professors Wen Huang and Zhenxin Liu for some preliminary discussions.

References

  1. 1.
    Arapostathis, A., Borkar, V.S., Ghosh, M.K.: Ergodic Control of Diffusion Processes. Encyclopedia of Mathematics and Its Applications, vol. 143. Cambridge University Press, Cambridge (2012)zbMATHGoogle Scholar
  2. 2.
    Bogachev, V.I., Da Prato, G., Röckner, M.: Existence of solutions to weak parabolic equations for measures. Proc. Lond. Math. Soc. 88(3), 753–774 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogachev, V.I., Da Prato, G., Röckner, M.: On parabolic equations for measures. Commun. Partial Differ. Equ. 33(1–3), 397–418 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bogachev, V.I., Da Prato, G., Röckner, M., Stannat, W.: Uniqueness of solutions to weak parabolic equations for measures. Bull. Lond. Math. Soc. 39(4), 631–640 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bogachev, V.I., Krylov, N.V., Röckner, M.: On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Commun. Partial Differ. Equ. 26(11–12), 2037–2080 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bogachev, V.I., Krylov, N.V., Rökner, M.: Elliptic and parabolic equations for measures. (Russian) Uspekhi Mat. Nauk 646(390), 5–116 (2009); translation. Russian Math. Surveys 64(6), 973–1078 (2009)Google Scholar
  7. 7.
    Bogachev, V.I., Krylov, N.V., Röckner, M., Shaposhnikov, S.V.: Fokker–Planck–Kolmogorov Equations. Mathematical Surveys and Monographs, vol. 207. American Mathematical Society, Providence, RI (2015)CrossRefGoogle Scholar
  8. 8.
    Bogachev, V.I., Rökner, M.A.: A generalization of Khas’minskiǐ’s theorem on the existence of invariant measures for locally integrable drifts. (Russian) Teor. Veroyatnost. i Primenen. 45(3), 417–436 (2000); translation in Theory Probab. Appl. 45(3), 363–378 (2002)Google Scholar
  9. 9.
    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: On uniqueness problems related to elliptic equations for measures. Problems in mathematical analysis. No. 58. J. Math. Sci. (N. Y.) 176(6), 759–773 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures. Problems in mathematical analysis. No. 61. J. Math. Sci. (N. Y.) 179(1), 7–47 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bogachev, V.I., Rökner, M., Shaposhnikov, S.V.: On positive and probability solutions of the stationary Fokker-Planck-Kolmogorov equation. (Russian) Dokl. Akad. Nauk 444(3), 245–249 (2012); translation in Dokl. Math. 85(3), 350–354 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bogachev, V.I., Rökner, M., Stannat, V.: Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. (Russian) Mat. Sb. 193(7), 3–36 (2002); translation in Sb. Math. 193(7–8), 945–976 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Doob, J.L.: Asymptotic properties of Markoff transition prababilities. Trans. Am. Math. Soc. 63, 393–421 (1948)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Denisov, V.N.: On the behavior of solutions of parabolic equations for large time values. (Russian) Uspekhi Mat. Nauk 60(4)(364), 145–212 (2005); translation. Russian Math. Surveys 60(4), 721–790 (2005)Google Scholar
  15. 15.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series, vol. 229. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  16. 16.
    Gyöngy, I., Krylov, N.: Existence of strong solutions for Itô’s stochastic equations via approximations. Probab. Theory Relat. Fields 105(2), 143–158 (1996)CrossRefGoogle Scholar
  17. 17.
    Hairer, M.: Convergence of Markov Processes. http://www.hairer.org/notes/Convergence.pdf
  18. 18.
    Has’minskiǐ, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. (Russian) Teor. Verojatnost. i Primenen 5, 196–214 (1960)MathSciNetGoogle Scholar
  19. 19.
    Huang, W., Ji, M., Liu, Z., Yi, Y.: Integral identity and measure estimates for stationary Fokker–Planck equations. Ann. Probab. 43(4), 1712–1730 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Huang, W., Ji, M., Liu, Z., Yi, Y.: Steady states of Fokker–Planck equations: I. Existence. J. Dyn. Differ. Equ. 27(3–4), 721–742 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Il’in, A.M., Khasminskii, R.: Asymptotic behavior of solutions of parabolic equations and an ergodic property of non-homogeneous diffusion processes. (Russian) Mat. Sb. (N.S.) 60(102), 366–392 (1963)MathSciNetGoogle Scholar
  22. 22.
    Khasminskii, R.: Stochastic stability of differential equations. In: Milstein, G.N., Nevelson, M.B. (eds.) With Contributions, Completely Revised and Enlarged Second Edition. Stochastic Modelling and Applied Probability, vol. 66. Springer, Heidelberg (2012)Google Scholar
  23. 23.
    Krylov, N.V.: Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces. J. Funct. Anal. 183(1), 1–41 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Manita, O.A., Shaposhnikov, S.V.: On the Cauchy problem for Fokker–Planck–Kolmogorov equations with potential terms on arbitrary domains. J. Dyn. Differ. Equ. 28(2), 493–518 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Meyn, S., Tweedie, R.L.: Markov Chains and Stochastic Stability. With a Prologue by Peter W. Glynn, 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  26. 26.
    Shaposhnikov, S.V.: On the uniqueness of the probabilistic solution of the Cauchy problem for the Fokker–Planck–Kolmogorov equation. (Russian) Teor. Veroyatn. Primen. 56(1), 77–99 (2011); translation in Theory Probab. Appl. 56(1), 96–115 (2012)Google Scholar
  27. 27.
    Stannat, W.: (Nonsymmetric) Dirichlet operators on \(L^{1}\): existence, uniqueness and associated Markov processes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1), 99–140 (1999)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Veretennikov, AYu.: On polynomial mixing bounds for stochastic differential equations. Stoch. Process. Appl. 70(1), 115–127 (1997)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Academy of Mathematics and System SciencesChinese Academy of Sciences and University of Chinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

Personalised recommendations