The Fractional Chapman-Kolmogorov Equation

TL;DR

This paper derives a fractional generalization of the Chapman-Kolmogorov equation using fractional integrals to model fractal media, leading to a fractional Fokker-Planck equation.

Contribution

It introduces a novel fractional Chapman-Kolmogorov equation that accounts for fractal media via fractional integrals, extending classical stochastic process models.

Findings

Derived the fractional Chapman-Kolmogorov equation.

Obtained the fractional Fokker-Planck equation from it.

Showed fractional integrals approximate integrals on fractals.

Abstract

The Chapman-Kolmogorov equation with fractional integrals is derived. An integral of fractional order is considered as an approximation of the integral on fractal. Fractional integrals can be used to describe the fractal media. Using fractional integrals, the fractional generalization of the Chapman-Kolmogorov equation is obtained. From the fractional Chapman-Kolmogorov equation, the Fokker-Planck equation is derived.