IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v33y2020i4d10.1007_s10959-019-00938-5.html
   My bibliography  Save this article

Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2

Author

Listed:
  • Endre Csáki

    (Hungarian Academy of Sciences)

  • Antónia Földes

    (CUNY)

Abstract

We study the path behavior of the simple symmetric walk on some comb-type subsets of $${{\mathbb {Z}}}^2$$ Z 2 which are obtained from $${{\mathbb {Z}}}^2$$ Z 2 by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences.

Suggested Citation

  • Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00938-5
    DOI: 10.1007/s10959-019-00938-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-019-00938-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-019-00938-5?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. Nane, Erkan, 2009. "Laws of the iterated logarithm for a class of iterated processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1744-1751, August.
    2. Arkhincheev, V.E., 2010. "Unified continuum description for sub-diffusion random walks on multi-dimensional comb model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 1-6.
    3. Shuler, Kurt E., 1979. "Random walks on sparsely periodic and random lattices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 95(1), pages 12-34.
    4. Zahran, M.A. & Abulwafa, E.M. & Elwakil, S.A., 2003. "The fractional Fokker–Planck equation on comb-like model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 323(C), pages 237-248.
    5. Arkhincheev, V.E, 2000. "Anomalous diffusion and charge relaxation on comb model: exact solutions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 280(3), pages 304-314.
    6. Weiss, George H. & Havlin, Shlomo, 1986. "Some properties of a random walk on a comb structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 134(2), pages 474-482.
    7. Reynolds, A.M, 2004. "On anomalous transport on comb structures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 334(1), pages 39-45.
    8. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
    Full references (including those not matched with items on IDEAS)

    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. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 2011. "On the local time of random walk on the 2-dimensional comb," Stochastic Processes and their Applications, Elsevier, vol. 121(6), pages 1290-1314, June.
    2. Endre Csáki & Antónia Földes, 2022. "Strong Approximation of the Anisotropic Random Walk Revisited," Journal of Theoretical Probability, Springer, vol. 35(4), pages 2879-2895, December.
    3. Pottier, N., 1994. "Analytic study of a model of biased diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 208(1), pages 91-123.
    4. Arkhincheev, V.E., 2020. "The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 550(C).
    5. Balakrishnan, V. & Van den Broeck, C., 1995. "Transport properties on a random comb," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 217(1), pages 1-21.
    6. Gajda, Janusz & Magdziarz, Marcin, 2014. "Large deviations for subordinated Brownian motion and applications," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 149-156.
    7. Yimin Xiao, 1998. "Local Times and Related Properties of Multidimensional Iterated Brownian Motion," Journal of Theoretical Probability, Springer, vol. 11(2), pages 383-408, April.
    8. Baskin, Emmanuel & Iomin, Alexander, 2011. "Electrostatics in fractal geometry: Fractional calculus approach," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 335-341.
    9. Pece Trajanovski & Petar Jolakoski & Ljupco Kocarev & Trifce Sandev, 2023. "Ornstein–Uhlenbeck Process on Three-Dimensional Comb under Stochastic Resetting," Mathematics, MDPI, vol. 11(16), pages 1-28, August.
    10. Dzhanoev, A.R. & Sokolov, I.M., 2018. "The effect of the junction model on the anomalous diffusion in the 3D comb structure," Chaos, Solitons & Fractals, Elsevier, vol. 106(C), pages 330-336.
    11. Iomin, A. & Zaburdaev, V. & Pfohl, T., 2016. "Reaction front propagation of actin polymerization in a comb-reaction system," Chaos, Solitons & Fractals, Elsevier, vol. 92(C), pages 115-122.
    12. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 1997. "On the occupation time of an iterated process having no local time," Stochastic Processes and their Applications, Elsevier, vol. 70(2), pages 199-217, October.
    13. Valerii M Sukhorukov & Jürgen Bereiter-Hahn, 2009. "Anomalous Diffusion Induced by Cristae Geometry in the Inner Mitochondrial Membrane," PLOS ONE, Public Library of Science, vol. 4(2), pages 1-14, February.
    14. Nane, Erkan, 2009. "Laws of the iterated logarithm for a class of iterated processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1744-1751, August.
    15. Sandev, Trifce & Schulz, Alexander & Kantz, Holger & Iomin, Alexander, 2018. "Heterogeneous diffusion in comb and fractal grid structures," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 551-555.
    16. Casse, Jérôme & Marckert, Jean-François, 2016. "Processes iterated ad libitum," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3353-3376.
    17. Aslangul, C. & Pottier, N. & Chvosta, P., 1994. "Analytic study of a model of diffusion on a random comblike structure," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 203(3), pages 533-565.
    18. Trifce Sandev & Viktor Domazetoski & Alexander Iomin & Ljupco Kocarev, 2021. "Diffusion–Advection Equations on a Comb: Resetting and Random Search," Mathematics, MDPI, vol. 9(3), pages 1-24, January.
    19. Endre Csáki & Antónia Földes, 2022. "On the Local Time of the Half-Plane Half-Comb Walk," Journal of Theoretical Probability, Springer, vol. 35(2), pages 1247-1261, June.
    20. Iksanov, Alexander & Kabluchko, Zakhar & Marynych, Alexander & Shevchenko, Georgiy, 2017. "Fractionally integrated inverse stable subordinators," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 80-106.

    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:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00938-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.