IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v83y1999i1p39-59.html
   My bibliography  Save this article

Asymptotics and spectral results for random walks on p-adics

Author

Listed:
  • Albeverio, Sergio
  • Karwowski, Witold
  • Zhao, Xuelei

Abstract

Markov semigroups on the field of p-adic numbers which are symmetric with respect to a measure [rho](x) dx absolutely continuous relative to the Haar measure dx on are constructed. The corresponding Dirichlet forms and associated Markov processes are exhibited and shown to be of the jump type. A detailed description of the spectrum of the generator (eigenvalues and eigenfunctions) is provided. Necessary and sufficient conditions for reducibility, recurrence and transience are found. Results about exit times from balls are also presented, as well as a proof of null recurrence.

Suggested Citation

  • Albeverio, Sergio & Karwowski, Witold & Zhao, Xuelei, 1999. "Asymptotics and spectral results for random walks on p-adics," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 39-59, September.
  • Handle: RePEc:eee:spapps:v:83:y:1999:i:1:p:39-59
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(99)00016-2
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Albeverio, Sergio & Karwowski, Witold, 1994. "A random walk on p-adics--the generator and its spectrum," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 1-22, September.
    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. Kaneko, Hiroshi, 2000. "A class of spatially inhomogeneous Dirichlet spaces on the p-adic number field," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 161-174, July.
    2. D. A. Dawson & L. G. Gorostiza, 2018. "Transience and Recurrence of Random Walks on Percolation Clusters in an Ultrametric Space," Journal of Theoretical Probability, Springer, vol. 31(1), pages 494-526, March.
    3. Hiroshi Kaneko & Xuelei Zhao, 2006. "Transition Semi–groups on a Local Field Induced by Galois Group and their Representation," Journal of Theoretical Probability, Springer, vol. 19(1), pages 221-234, January.

    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. Kumi Yasuda, 2010. "Markov Processes on the Adeles and Representations of Euler Products," Journal of Theoretical Probability, Springer, vol. 23(3), pages 748-769, September.
    2. Cruz-López, Manuel & Estala-Arias, Samuel & Murillo-Salas, Antonio, 2016. "A random walk on the profinite completion of Z," Statistics & Probability Letters, Elsevier, vol. 109(C), pages 130-138.
    3. Kaneko, Hiroshi & Tsuzuki, Yoichi, 2011. "A lift of spatially inhomogeneous Markov process to extensions of the field of p-adic numbers," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 394-405, February.
    4. Yasuda, Kumi, 2013. "Markov processes on the adeles and Chebyshev function," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 238-244.
    5. David Aldous & Steven N. Evans, 1999. "Dirichlet Forms on Totally Disconnected Spaces and Bipartite Markov Chains," Journal of Theoretical Probability, Springer, vol. 12(3), pages 839-857, July.
    6. Urban, Roman, 2012. "Markov processes on the adeles and Dedekind’s zeta function," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1583-1589.
    7. Hiroshi Kaneko & Xuelei Zhao, 2006. "Transition Semi–groups on a Local Field Induced by Galois Group and their Representation," Journal of Theoretical Probability, Springer, vol. 19(1), pages 221-234, January.
    8. Erik Bakken & David Weisbart, 2019. "Continuous Time p-Adic Random Walks and Their Path Integrals," Journal of Theoretical Probability, Springer, vol. 32(2), pages 781-805, June.
    9. David Weisbart, 2022. "Estimates of Certain Exit Probabilities for p-Adic Brownian Bridges," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1878-1897, September.
    10. Kumi Yasuda, 2000. "On Infinitely Divisible Distributions on Locally Compact Abelian Groups," Journal of Theoretical Probability, Springer, vol. 13(3), pages 635-657, July.
    11. Kaneko, Hiroshi, 2000. "A class of spatially inhomogeneous Dirichlet spaces on the p-adic number field," Stochastic Processes and their Applications, Elsevier, vol. 88(1), pages 161-174, July.
    12. Sergio Albeverio & Xuelei Zhao, 2001. "A Decomposition Theorem for Lévy Processes on Local Fields," Journal of Theoretical Probability, Springer, vol. 14(1), pages 1-19, January.
    13. Claude Dellacherie & Servet Martinez & Jaime San Martin, 2009. "Ultrametric and Tree Potential," Journal of Theoretical Probability, Springer, vol. 22(2), pages 311-347, June.
    14. Cruz-López, Manuel & Estala-Arias, Samuel, 2018. "A random walk on the profinite completion of a finitely generated group," Statistics & Probability Letters, Elsevier, vol. 143(C), pages 7-16.

    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:83:y:1999:i:1:p:39-59. 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.