IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v125y2015i9p3636-3656.html
   My bibliography  Save this article

Hypercontractivity for functional stochastic differential equations

Author

Listed:
  • Bao, Jianhai
  • Wang, Feng-Yu
  • Yuan, Chenggui

Abstract

An explicit sufficient condition on the hypercontractivity is derived for the Markov semigroup associated with a class of functional stochastic differential equations. Consequently, the semigroup Pt converges exponentially to its unique invariant probability measure μ in both L2(μ) and the totally variational norm ‖⋅‖var, and it is compact in L2(μ) for sufficiently large t>0. This provides a natural class of non-symmetric Markov semigroups which are compact for large time but non-compact for small time. A semi-linear model which may not satisfy this sufficient condition is also investigated. As the associated Dirichlet form does not satisfy the log-Sobolev inequality, the standard argument using functional inequalities does not work.

Suggested Citation

  • Bao, Jianhai & Wang, Feng-Yu & Yuan, Chenggui, 2015. "Hypercontractivity for functional stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3636-3656.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:9:p:3636-3656
    DOI: 10.1016/j.spa.2015.04.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414915001027
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2015.04.001?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. Reiß, M. & Riedle, M. & van Gaans, O., 2006. "Delay differential equations driven by Lévy processes: Stationarity and Feller properties," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1409-1432, October.
    2. Wang, Feng-Yu & Yuan, Chenggui, 2011. "Harnack inequalities for functional SDEs with multiplicative noise and applications," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2692-2710, November.
    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. Wujun Lv & Xing Huang, 2021. "Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 34(2), pages 827-851, June.

    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. Wujun Lv & Xing Huang, 2021. "Harnack and Shift Harnack Inequalities for Degenerate (Functional) Stochastic Partial Differential Equations with Singular Drifts," Journal of Theoretical Probability, Springer, vol. 34(2), pages 827-851, June.
    2. Li, Zhi & Long, Qinyi & Xu, Liping & Wen, Xueqi, 2022. "h-stability for stochastic Volterra–Levin equations," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    3. Carsten Chong, 2017. "Lévy-driven Volterra Equations in Space and Time," Journal of Theoretical Probability, Springer, vol. 30(3), pages 1014-1058, September.
    4. Zhang, Shao-Qin, 2013. "Harnack inequality for semilinear SPDE with multiplicative noise," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1184-1192.
    5. Li, Zhi & Zhang, Wei, 2017. "Stability in distribution of stochastic Volterra–Levin equations," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 20-27.
    6. Bachmann, Stefan, 2020. "On the strong Feller property for stochastic delay differential equations with singular drift," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4563-4592.
    7. Bao, Jianhai & Wang, Feng-Yu & Yuan, Chenggui, 2019. "Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4576-4596.
    8. 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.
    9. Wang, Ya & Wu, Fuke & Yin, George & Zhu, Chao, 2022. "Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality," Stochastic Processes and their Applications, Elsevier, vol. 149(C), pages 1-38.
    10. Wang, Feng-Yu & Zhang, Tusheng, 2014. "Log-Harnack inequality for mild solutions of SPDEs with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 124(3), pages 1261-1274.
    11. Wang, Zhaojuan & Liu, Meng, 2023. "Periodic measure of a stochastic single-species model in periodic environments," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    12. Zong, Gaofeng & Chen, Zengjing, 2013. "Harnack inequality for mean-field stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1424-1432.
    13. Yong-Hua Mao & Tao Wang, 2022. "Convergence Rates in Uniform Ergodicity by Hitting Times and $$L^2$$ L 2 -Exponential Convergence Rates," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2690-2711, December.
    14. Xi, Fubao & Yin, George, 2013. "The strong Feller property of switching jump-diffusion processes," Statistics & Probability Letters, Elsevier, vol. 83(3), pages 761-767.
    15. Yang, Jiangtao, 2022. "Periodic measure of a stochastic non-autonomous predator–prey system with impulsive effects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 202(C), pages 464-479.
    16. Nguyen, Dang H. & Nguyen, Nhu N. & Yin, George, 2021. "Stochastic functional Kolmogorov equations, I: Persistence," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 319-364.

    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:125:y:2015:i:9:p:3636-3656. 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.