Fractal Stochastic Processes on Thin Cantor-Like Sets

Alireza Khalili Golmankhaneh and Renat Timergalievich Sibatov
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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

Mathematics, 2021, vol. 9, issue 6, 1-13

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.

Keywords: fractal calculus; fractional Brownian motion; fractal derivative; fractal stochastic process; Brownian motion (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

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