Skip to main content
SpringerLink
Log in

Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

By using the coupling method and the localization technique, we establish non-uniform gradient estimates for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Lévy noises, where the coefficients only satisfy local monotonicity conditions.

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. Bertoldi, M., Fornaro, S.: Gradient estimates in parabolic problems with unbounded coefficients. Studia Math., 165, 221–254 (2004)

    Article  MathSciNet  Google Scholar 

  2. Bertoldi, M., Lorenzi, L.: Estimates for the derivatives for parabolic operators with unbounded coefficients. Trans. Amer. Math. Soc., 357, 2627–2664 (2005)

    Article  MathSciNet  Google Scholar 

  3. Cerrai, S.: Second Order PDEs in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Math., vol. 1762, Springer, Berlin, 2001

    MATH  Google Scholar 

  4. Cranston, M.: A probabilistic approach to gradient estimates. Canad. Math. Bull., 35, 46–55 (1992)

    Article  MathSciNet  Google Scholar 

  5. Da Prato, G., Priola, E.: Gradient estimates for SDEs without monotonicity type conditions. J. Differential Equations, 265, 1984–2012 (2018)

    Article  MathSciNet  Google Scholar 

  6. Elworthy, K. D., Li, X. M.: Formulae for the derivatives of heat semigroups. J. Funct. Anal., 125, 252–286 (1994)

    Article  MathSciNet  Google Scholar 

  7. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, First Ed., North-Holland, Amsterdam, 1981

    MATH  Google Scholar 

  8. Liang, M., Wang, J.: Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling. Stochastic Process. Appl., 130, 3053–3094 (2020)

    Article  MathSciNet  Google Scholar 

  9. Luo, D., Wang, J.: Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises. Stochastic Process. Appl., 129, 3129–3173 (2019)

    Article  MathSciNet  Google Scholar 

  10. Priola, E., Wang, F. Y.: Gradient estimates for diffusion semigroups with singular coefficients. J. Funct. Anal., 236, 244–264 (2006)

    Article  MathSciNet  Google Scholar 

  11. Stroock, D. W., Varadhan, S. R. S.: Multidimensional Diffusion Processes, Springer, Berlin, 1979

    MATH  Google Scholar 

  12. Wang, F. Y.: Gradient estimates on Rd. Canad. Math. Bull., 37, 560–570 (1994)

    Article  MathSciNet  Google Scholar 

  13. Wang, F. Y., Xu, L., Zhang, X. C.: Gradient estimates for SDEs driven by multiplicative Lévy noises. J. Funct. Anal., 269, 3135–3219 (2015)

    Article  Google Scholar 

  14. Zhang, X. C.: Stochastic flows and Bismut formulas for stochastic Hamiltonian systems. Stochastic Process. Appl., 120, 1929–1949 (2010)

    Article  MathSciNet  Google Scholar 

  15. Zhang, X. C.: Derivative formula and gradient estimate for SDEs driven by α-stable processes. Stochastic Process. Appl., 123, 1213–1228 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the two referees for numerous corrections.

Author information

Authors and Affiliations

  1. College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA) & Center for Applied Mathematics of Fujian Province (FJNU), Fujian Normal University, Fuzhou, 350007, P. R. China

    Jian Wang

  2. College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350007, P. R. China

    Bing Yao Wu

Authors
  1. Jian Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bing Yao Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Wang.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 11831014), the Program for Probability and Statistics: Theory and Application (Grant No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, J., Wu, B.Y. Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions. Acta. Math. Sin.-English Ser. 37, 458–470 (2021). https://doi.org/10.1007/s10114-020-9365-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-9365-6

Keywords

MR(2010) Subject Classification

Search

Navigation