IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i6p613-d516971.html
   My bibliography  Save this article

Fractal Stochastic Processes on Thin Cantor-Like Sets

Author

Listed:
  • Alireza Khalili Golmankhaneh

    (Department of Physics, Urmia Branch, Islamic Azad University, Urmia 57169-63896, Iran)

  • Renat Timergalievich Sibatov

    (Laboratory of Diffusion Processes, Ulyanovsk State University, 432017 Ulyanovsk, Russia
    Department of Theoretical Physics, Moscow Institute of Physics and Technology, 141701 Dolgoprudny, Russia)

Abstract

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

Suggested Citation

  • Alireza Khalili Golmankhaneh & Renat Timergalievich Sibatov, 2021. "Fractal Stochastic Processes on Thin Cantor-Like Sets," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
  • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:613-:d:516971
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/6/613/pdf
    Download Restriction: no

    File URL: https://www.mdpi.com/2227-7390/9/6/613/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Zunino, L. & Pérez, D.G. & Kowalski, A. & Martín, M.T. & Garavaglia, M. & Plastino, A. & Rosso, O.A., 2008. "Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(24), pages 6057-6068.
    2. Rocco, Andrea & West, Bruce J., 1999. "Fractional calculus and the evolution of fractal phenomena," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 265(3), pages 535-546.
    3. dos Santos, Maike A.F., 2019. "Analytic approaches of the anomalous diffusion: A review," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 86-96.
    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. Khalili Golmankhaneh, Alireza & Ontiveros, Lilián Aurora Ochoa, 2023. "Fractal calculus approach to diffusion on fractal combs," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    2. Ayse Gertik & Aykut Karaman, 2023. "The Fractal Approach in the Biomimetic Urban Design: Le Corbusier and Patrick Schumacher," Sustainability, MDPI, vol. 15(9), pages 1-21, May.
    3. Takahashi, Hiroshi & Tamura, Yozo, 2023. "Diffusion processes in Brownian environments on disconnected selfsimilar fractal sets in R," Statistics & Probability Letters, Elsevier, vol. 193(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. dos Santos, M.A.F. & Colombo, E.H. & Anteneodo, C., 2021. "Random diffusivity scenarios behind anomalous non-Gaussian diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    2. Fernandes, Leonardo H.S. & de Araújo, Fernando H.A. & Silva, Igor E.M. & Neto, Jusie S.P., 2021. "Macroeconophysics indicator of economic efficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 573(C).
    3. Zhao, Xiaojun & Ji, Mengfan & Zhang, Na & Shang, Pengjian, 2020. "Permutation transition entropy: Measuring the dynamical complexity of financial time series," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    4. Shi, Hong-Da & Du, Lu-Chun & Huang, Fei-Jie & Guo, Wei, 2022. "Collective topological active particles: Non-ergodic superdiffusion and ageing in complex environments," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    5. Maike A. F. dos Santos, 2019. "Mittag–Leffler Memory Kernel in Lévy Flights," Mathematics, MDPI, vol. 7(9), pages 1-13, August.
    6. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2009. "Forbidden patterns, permutation entropy and stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2854-2864.
    7. Rosso, Osvaldo A. & Craig, Hugh & Moscato, Pablo, 2009. "Shakespeare and other English Renaissance authors as characterized by Information Theory complexity quantifiers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(6), pages 916-926.
    8. Potirakis, Stelios M. & Zitis, Pavlos I. & Eftaxias, Konstantinos, 2013. "Dynamical analogy between economical crisis and earthquake dynamics within the nonextensive statistical mechanics framework," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(13), pages 2940-2954.
    9. Zhang, Yongping & Shang, Pengjian, 2018. "Refined composite multiscale weighted-permutation entropy of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 496(C), pages 189-199.
    10. Ahmed, E. & Elgazzar, A.S., 2007. "On fractional order differential equations model for nonlocal epidemics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(2), pages 607-614.
    11. Zunino, Luciano & Zanin, Massimiliano & Tabak, Benjamin M. & Pérez, Darío G. & Rosso, Osvaldo A., 2010. "Complexity-entropy causality plane: A useful approach to quantify the stock market inefficiency," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(9), pages 1891-1901.
    12. Wei, Q. & Yang, S. & Zhou, H.W. & Zhang, S.Q. & Li, X.N. & Hou, W., 2021. "Fractional diffusion models for radionuclide anomalous transport in geological repository systems," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    13. Olivares, Felipe & Zunino, Luciano, 2020. "Multiscale dynamics under the lens of permutation entropy," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    14. Eftaxias, K., 2010. "Footprints of nonextensive Tsallis statistics, selfaffinity and universality in the preparation of the L’Aquila earthquake hidden in a pre-seismic EM emission," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(1), pages 133-140.
    15. Rosso, Osvaldo A. & De Micco, Luciana & Plastino, A. & Larrondo, Hilda A., 2010. "Info-quantifiers’ map-characterization revisited," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(21), pages 4604-4612.
    16. Wang, Peng & Huo, Jie & Wang, Xu-Ming & Wang, Bing-Hong, 2022. "Diffusion and memory effect in a stochastic process and the correspondence to an information propagation in a social system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    17. Liu, Zhengli & Shang, Pengjian & Wang, Yuanyuan, 2020. "Characterization of time series through information quantifiers," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    18. Mateos, Diego M. & Zozor, Steeve & Olivares, Felipe, 2020. "Contrasting stochasticity with chaos in a permutation Lempel–Ziv complexity — Shannon entropy plane," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 554(C).
    19. Ngueuteu Mbouna, S.G. & Banerjee, Tanmoy & Yamapi, René & Woafo, Paul, 2022. "Diverse chimera and symmetry-breaking patterns induced by fractional derivation effect in a network of Stuart-Landau oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    20. Serrano, Alfredo Blanco & Allen-Perkins, Alfonso & Andrade, Roberto Fernandes Silva, 2022. "Efficient approach to time-dependent super-diffusive Lévy random walks on finite 2D-tori using circulant analogues," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 592(C).

    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:gam:jmathe:v:9:y:2021:i:6:p:613-:d:516971. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.