IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v146y2022icp114-142.html
   My bibliography  Save this article

Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises

Author

Listed:
  • Bao, Jianhai
  • Wang, Jian

Abstract

We establish exponential ergodicity for the stochastic Hamiltonian system (Xt,Vt)t≥0 on R2d with Lévy noises dXt=(aXt+bVt)dt,dVt=U(Xt,Vt)dt+dLt,where a≥0, b>0, U:R2d→Rd and (Lt)t≥0 is an Rd-valued pure jump Lévy process. The approach is based on a new refined basic coupling for Lévy processes and a Lyapunov function for stochastic Hamiltonian systems. In particular, we can handle the case that U(x,v)=−v−∇U0(x) with double well potential U0 which is super-linear growth at infinity such as U0(x)=c1(1+|x|2)l−c2|x|2 with l>1 or U0(x)=c1e(1+|x|2)l−c2|x|2 with l>0 for any c1,c2>0, and also deal with the case that the Lévy measure ν of (Lt)t≥0 is degenerate in the sense that ν(dz)≥c|z|d+θ01{00 and θ0∈(0,2), where z1 is the first component of the vector z∈Rd.

Suggested Citation

  • Bao, Jianhai & Wang, Jian, 2022. "Coupling approach for exponential ergodicity of stochastic Hamiltonian systems with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 114-142.
    • Handle: RePEc:eee:spapps:v:146:y:2022:i:c:p:114-142
    DOI: 10.1016/j.spa.2021.12.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414921002246
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2021.12.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.
    2. Komorowski, Tomasz & Walczuk, Anna, 2012. "Central limit theorem for Markov processes with spectral gap in the Wasserstein metric," Stochastic Processes and their Applications, Elsevier, vol. 122(5), pages 2155-2184.
    3. Majka, Mateusz B., 2017. "Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 4083-4125.
    4. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    5. Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.
    6. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bao, Jianhai & Fang, Rongjuan & Wang, Jian, 2024. "Exponential ergodicity of Lévy driven Langevin dynamics with singular potentials," Stochastic Processes and their Applications, Elsevier, vol. 172(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Huang, Lu-Jing & Majka, Mateusz B. & Wang, Jian, 2022. "Strict Kantorovich contractions for Markov chains and Euler schemes with general noise," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 307-341.
    2. Bao, Jianhai & Fang, Rongjuan & Wang, Jian, 2024. "Exponential ergodicity of Lévy driven Langevin dynamics with singular potentials," Stochastic Processes and their Applications, Elsevier, vol. 172(C).
    3. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    4. Song, Renming & Xie, Longjie, 2020. "Well-posedness and long time behavior of singular Langevin stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1879-1896.
    5. Liang, Mingjie & Wang, Jian, 2020. "Gradient estimates and ergodicity for SDEs driven by multiplicative Lévy noises via coupling," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3053-3094.
    6. Jianhai Bao & Jian Wang, 2023. "Coupling methods and exponential ergodicity for two‐factor affine processes," Mathematische Nachrichten, Wiley Blackwell, vol. 296(5), pages 1716-1736, May.
    7. Xie, Longjie & Yang, Li, 2022. "The Smoluchowski–Kramers limits of stochastic differential equations with irregular coefficients," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 91-115.
    8. Anna Melnykova, 2020. "Parametric inference for hypoelliptic ergodic diffusions with full observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(3), pages 595-635, October.
    9. Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    10. Dexheimer, Niklas & Strauch, Claudia, 2022. "Estimating the characteristics of stochastic damping Hamiltonian systems from continuous observations," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 321-362.
    11. P. Cattiaux & José R. León & C. Prieur, 2015. "Recursive estimation for stochastic damping hamiltonian systems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 401-424, September.
    12. Quentin Clairon & Adeline Samson, 2020. "Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 105-127, April.
    13. Ankit Kumar & Manil T. Mohan, 2023. "Large Deviation Principle for Occupation Measures of Stochastic Generalized Burgers–Huxley Equation," Journal of Theoretical Probability, Springer, vol. 36(1), pages 661-709, March.
    14. Qiu Lin & Ruisheng Qi, 2023. "Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise," Mathematics, MDPI, vol. 12(1), pages 1-29, December.
    15. Gao, Shuaibin & Li, Xiaotong & Liu, Zhuoqi, 2023. "Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    16. Kulik, Alexey M., 2011. "Asymptotic and spectral properties of exponentially [phi]-ergodic Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1044-1075, May.
    17. ur Rahman, Ghaus & Badshah, Qaisar & Agarwal, Ravi P. & Islam, Saeed, 2021. "Ergodicity & dynamical aspects of a stochastic childhood disease model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 738-764.
    18. Casella, Bruno & Roberts, Gareth O. & Stramer, Osnat, 2011. "Stability of Partially Implicit Langevin Schemes and Their MCMC Variants," MPRA Paper 95220, University Library of Munich, Germany.
    19. Jianhai Bao & Feng‐Yu Wang & Chenggui Yuan, 2020. "Ergodicity for neutral type SDEs with infinite length of memory," Mathematische Nachrichten, Wiley Blackwell, vol. 293(9), pages 1675-1690, September.
    20. Luo, Dejun & Wang, Jian, 2019. "Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3129-3173.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:146:y:2022:i:c:p:114-142. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.