Fractal Stochastic Processes on Thin Cantor-Like Sets

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.

[1]  N. Laskin.  Fractional Poisson process , 2003 .

[2]  Rudolf Hilfer,et al.  EXACT SOLUTIONS FOR A CLASS OF FRACTAL TIME RANDOM WALKS , 1995 .

[3]  Vladimir V. Uchaikin,et al.  REVIEWS OF TOPICAL PROBLEMS: Self-similar anomalous diffusion and Levy-stable laws , 2003 .

[4]  Patrick Flandrin,et al.  On the spectrum of fractional Brownian motions , 1989, IEEE Trans. Inf. Theory.

[5]  Seema Satin,et al.  CALCULUS ON FRACTAL CURVES IN Rn , 2009, 0906.0676.

[6]  Arran Fernandez,et al.  Random Variables and Stable Distributions on Fractal Cantor Sets , 2019, Fractal and Fractional.

[7]  A. Zhokh,et al.  Non-Fickian Transport in Porous Media: Always Temporally Anomalous? , 2018, Transport in Porous Media.

[8]  Alfred Hanssen,et al.  Spectral correlations of fractional Brownian motion. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  Manuel Duarte Ortigueira,et al.  A Fractional Linear System View of the Fractional Brownian Motion , 2004 .

[10]  C. Tunç,et al.  Stochastic differential equations on fractal sets , 2020 .

[11]  M. D. dos Santos.  A fractional diffusion equation with sink term , 2020, Indian Journal of Physics.

[12]  M. D. Paola,et al.  A physically based connection between fractional calculus and fractal geometry , 2014, 1508.04652.

[13]  Karmina K. Ali,et al.  Fractal Kronig-Penney model involving fractal comb potential , 2020 .

[14]  A. D. Gangal,et al.  Fokker Planck Equation on Fractal Curves , 2010, 1004.4422.

[15]  A. Golmankhaneh,et al.  Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review , 2021 .

[16]  F. Tatom.  THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS , 1995 .

[17]  A. D. Gangal,et al.  CALCULUS ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH ORDINARY CALCULUS , 2011 .

[18]  Wei Zhang,et al.  Accurate relationships between fractals and fractional integrals: New approaches and evaluations , 2017 .

[19]  Bruce J. West,et al.  Fractional Calculus and the Evolution of Fractal Phenomena , 1999 .

[20]  R. T. Sibatov,et al.  Fractional Processes: from Poisson to Branching One , 2008, Int. J. Bifurc. Chaos.

[21]  Georgiy Shevchenko,et al.  Fractional Brownian motion in a nutshell , 2014, 1406.1956.

[22]  E. Barkai,et al.  Fractional Fokker-Planck equation, solution, and application. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[24]  M. D. dos Santos.  Analytic approaches of the anomalous diffusion: A review , 2019, Chaos, Solitons & Fractals.

[25]  Alireza Khalili Golmankhaneh.  On the Fractal Langevin Equation , 2019, Fractal and Fractional.

[26]  Igor M. Sokolov,et al.  ANOMALOUS TRANSPORT IN EXTERNAL FIELDS : CONTINUOUS TIME RANDOM WALKS AND FRACTIONAL DIFFUSION EQUATIONS EXTENDED , 1998 .

[27]  Michael F. Shlesinger,et al.  FRACTAL TIME IN CONDENSED MATTER , 1988 .

[28]  D. Baleanu,et al.  Non-local Integrals and Derivatives on Fractal Sets with Applications , 2017, 1701.01054.

[29]  H. Kantz,et al.  Anomalous diffusion on a fractal mesh. , 2016, Physical review. E.

[30]  Hilfer,et al.  Fractional master equations and fractal time random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[31]  Hongguang Sun,et al.  Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder , 2019, Fractal and Fractional.

[32]  Spectral density of a fractional Brownian process , 2013 .

[33]  A. D. Gangal,et al.  Calculus on fractal subsets of real line - I: formulation , 2003 .

[34]  J. A. Tenreiro Machado,et al.  Fractional order description of DNA , 2015 .

[35]  Andrey G. Cherstvy,et al.  Underdamped scaled Brownian motion: (non-)existence of the overdamped limit in anomalous diffusion , 2016, Scientific Reports.

[36]  Roman S. Rutman,et al.  On the paper by R. R. Nigmatullin “fractional integral and its physical interpretation” , 1994 .

[37]  One-dimensional stochastic Levy-lorentz gas , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[38]  A. Zhokh,et al.  Relationship between the anomalous diffusion and the fractal dimension of the environment , 2018 .

[39]  A. M. Kowalski,et al.  Fractional Brownian motion, fractional Gaussian noise, and Tsallis permutation entropy , 2008 .

[40]  C. Cattani,et al.  Fractal Logistic Equation , 2019, Fractal and Fractional.

[41]  A. Golmankhaneh,et al.  On fractional and fractal Einstein’s field equations , 2021 .

[42]  Seema Satin,et al.  Langevin Equation on Fractal Curves , 2014, 1404.6896.

[43]  Alireza Khalili Golmankhaneh,et al.  Sub- and super-diffusion on Cantor sets: Beyond the paradox , 2018 .

[44]  Siswanto,et al.  The Relation between Hölder Continuous Function of Order α ∈ (0,1) and Function of Bounded Variation , 2020, Journal of Physics: Conference Series.

[45]  R. Nigmatullin.  The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry , 1986, January 1.