Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps

  • 181 Accesses


In this paper, we study mean-field stochastic differential equations with jumps. By Malliavin calculus for Wiener–Poisson functionals, sharp gradient estimates are derived. Based on the gradient estimates, exponential convergence to the unique invariant measure in total variation distance is also obtained under a dissipative condition.

This is a preview of subscription content, log in to check access.


  1. 1.

    Barczy, M., Li, Z.H., Pap, M.: Yamada–Watanabe results for stochastic differential equations with jumps. Int. J. Stoch. Anal. Art. ID 460472 (2015)

  2. 2.

    Benachour, S., Roynette, B., Talay, D., Vallois, P.: Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos. Stoch. Process. Appl. 75, 173–201 (1998)

  3. 3.

    Bismut, J.M.: Calcul des variations stochastiques et processus de sauts. Z. Wahrsch. Verw. Gebiete 63, 147–235 (1983)

  4. 4.

    Bitchtler, K., Jacod, J.J.: Gravereaux, Malliavin Calculus for Processes with Jumps. Gordan and Breach Science Publishers, New York (1987)

  5. 5.

    Buckdahn, R., Li, J., Peng, S., Rainer, C.: Mean-field stochastic differential equations and associated PDEs. Ann. Prob. 45, 824–878 (2017)

  6. 6.

    Butkovsky, O.A.: On ergodic properties of nonlinear Markov chains and stochastic Mckean–Vlasov equations. Theory Probab. Appl. 58, 661–674 (2014)

  7. 7.

    Cattiaux, P., Guillin, A., Malrieu, F.: Probability approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140, 19–40 (2008)

  8. 8.

    Cardaliaguet, P.: Notes on mean filed games (from P.L. Lions’ lectures at Collège de France) (2013)

  9. 9.

    Carmona, R., Delarue, F.: Forward–backward stochastic differential equations and controlled McKean–Vlasov dynamics. Ann. Probab. 43, 2647–2700 (2015)

  10. 10.

    Chan, T.: Dynamics of the McKean–Vlasov equation. Ann. Probab. 22, 431–441 (1994)

  11. 11.

    Dawson, D.: Stochastic McKean–Vlasov equations. Nonlinear Differ. Equ. Appl. 2, 199–229 (1995)

  12. 12.

    Graham, C.: McKean–Vlasov Ito–Skorohod equations, and nonlinear diffusions with discrete jump sets. Stoch. Process. Appl. 40, 69–82 (1992)

  13. 13.

    Hao, T., Li, J.: Mean-field SDEs with jumps and nonlocal integral-PDEs. Nonlinear Differ. Equ. Appl. 23, 17 (2016)

  14. 14.

    Huang, X., Röckner, M., Wang, F.-Y.: Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. arXiv:1709.00556v1 (2017)

  15. 15.

    Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197. University California Press, Berkeley (1956)

  16. 16.

    Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations. Cambridge University Press, Cambridge (2010)

  17. 17.

    Kotelenez, P., Kurtz, T.: Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type. Probab. Theory Relat. Fields 146, 189–222 (2010)

  18. 18.

    Lions, P.: Cours au Collège de France: Théorie des jeuàchamps moyens. Available at http://www.college-de-france.fr/default/EN/all/equ[1]der/audiovideo.jsp

  19. 19.

    McKean, H.P.: Propagation of chaos for a class of nonlinear parabolic equations. In: Lecture Series in Differential Equations, vol. 7, pp. 41–57 (1967)

  20. 20.

    Protter, E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2004)

  21. 21.

    Nualart, D.: The Malliavin Calsulus and Related Topics. Springer, New York (2006)

  22. 22.

    Song, Y.: Gradient estimates and coupling property for semilinear SDEs driven by jump processes. Sci. China Math. 58, 447–458 (2015)

  23. 23.

    Song, Y., Xie, Y.: Existence of density functions for the running maximum of a Lévy–Itô diffusion. Potential Anal. 48(1), 35–48 (2018)

  24. 24.

    Song, Y., Zhang, X.: Regularity of density for SDEs driven by degenrate Lévy noises. Electron. J. Probab. 20(21), 1–27 (2015)

  25. 25.

    Wang, F.-Y.: Gradient estimate for Ornstein–Uhlenbeck jump processes. Stoch. Process. Appl. 121, 466–478 (2011)

  26. 26.

    Wang, F.-Y.: Distribution dependent SDEs for Landau type equations. Stoch. Process. Appl. 128, 595–621 (2018)

  27. 27.

    Wang, F.-Y., Xu, L., Zhang, X.: Gradient estimates for SDEs driven by multiplicative Lévy noise. J. Funct. Anal. 269, 3195–3219 (2015)

  28. 28.

    Wang, L., Xie, L., Zhang, X.: Derivative formulae for SDEs driven by multiplicative \(\alpha \)-stable-like processes. Stoch. Process. Appl. 125, 867–885 (2015)

  29. 29.

    Veretennikov, A.Y.: On ergodic measures for Mckean-Vlasov stochastic equations. In: Harald, N., Denis, T. (eds.) Monte-Carlo and Quasi-Monte Carlo Methods, pp. 471–486. Springer, Berlin (2006)

Download references


The author is very grateful to the editor and referee for detailed reports and corrections. He also would like to thank Professors Zhao Dong, Renming Song, and Fengyu Wang for their valuable discussions. This work is supported by National Natural Science Foundation of China (Nos. 11501286, 11790272), Natural Science Foundation of Jiangsu Province (No. BK20150564) and CSC.

Author information

Correspondence to Yulin Song.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Song, Y. Gradient Estimates and Exponential Ergodicity for Mean-Field SDEs with Jumps. J Theor Probab 33, 201–238 (2020). https://doi.org/10.1007/s10959-018-0845-x

Download citation


  • Malliavin calculus
  • Gradient estimates
  • Exponential ergodicity
  • Density functions
  • McKean–Vlasov equations

Mathematics Subject Classification (2010)

  • 60H07
  • 60H10