Fractal Stochastic Processes on Thin Cantor-Like Sets

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.

Noise analysis of electrical circuits on fractal set

“…Note also that RV(Sκbα(τ)+Sκbα(x)−Sκbα(y)) stands for the autocorrelation function of V ( t ) in terms of Sκbα(τ)+Sκbα(x)−Sκbα(y). Since V ( t ) = σ N ( t ), it can be seen that where δκb(τ)is a delta function on κ b (Golmankhaneh and Sibatov, 2021). Based on (7) and (8), R Δ I ( τ ) can be obtained as follows …”

Nonlocal fractal calculus based analyses of electrical circuits on fractal set

“…Definition 11: For arbitrary κ b , its Hausdorff dimension can be given by (Golmankhaneh and Sibatov, 2021): …”

“…Here, we let i(0) = 10 mA and assume that L = 1 H and C = 1 F for simplicity. Before we proceed further, it is worthy to introduce the following definition and lemma: Definition 11: For arbitrary k b , its Hausdorff dimension can be given by (Golmankhaneh and Sibatov, 2021):…”

The Generic Nonlocal Fractal Calculus

“…The local fractional derivative has been successfully applied to the studies of the abovementioned electromagnetism in fractal time/space [12], the vibration in fractal media [13] and the analyses of electrical circuits defined on fractal set [14]- [19]. On the other hand, the local fractal derivative has also been widely applied to many fractal concept related issues e.g., the fractal Fokker-Planck equation [20], the fractal stochastic processes [21], the diffusion in fractal structure [22] and of course the analyses of fractal set defined electrical circuits [23]- [25].…”

“…Since the nonlocality like that of the fractional calculus is necessary for modelling such memory effect, the nonlocal fractal derivatives have been introduced by Golmankhaneh and Baleanu [31], [32] based on the classical Riemann-Liouville and Caputo fractional derivatives which employ a power law-based kernel. These nonlocal fractal derivatives have been successfully applied to various applications e.g., the mathematical modelling of fractional Brownian motion with fractal support [21] and the analysis of fractional electrical circuits defined on fractal set [23], [33] etc. However, they are inconsistent with the local fractal derivative as will be shown later unlike the fractional derivative that is consistent with the conventional operator.…”