Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

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 via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

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

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Google Scholar 

Download references

Acknowledgements

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

Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, Jiangsu Province, People’s Republic of China

    Yulin Song

Authors
  1. Yulin Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yulin Song.

Rights and permissions

Reprints and permissions

About this article

Check for updates. 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

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-018-0845-x

Keywords

Mathematics Subject Classification (2010)

Search

Navigation