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Fractal Stochastic Processes on Thin Cantor-Like Sets

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  • 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
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    References listed on IDEAS

    as
    1. 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.
    2. dos Santos, Maike A.F., 2019. "Analytic approaches of the anomalous diffusion: A review," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 86-96.
    3. 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.
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    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).

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