Stochastic Processes and Mean Square Calculus on Fractal Curves

TL;DR

This paper develops a framework for defining and analyzing stochastic processes on fractal curves using fractal calculus, including new concepts like fractal mean square derivatives and integrals, and solves related stochastic equations.

Contribution

It introduces a generalized mean square calculus on fractal curves, extending stochastic process analysis to fractal geometries for the first time.

Findings

Defined stochastic processes on fractal curves using fractal calculus

Formulated a new fractal mean square calculus including derivatives and integrals

Derived and visualized solutions to fractal stochastic equations

Abstract

In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define cumulative distribution function, probability density function, moments, variance and correlation function of stochastic process on fractal curve. A new framework which is a generalization of mean square calculus is formulated. Sequence of random variable on fractal curve, fractal mean square continuity, mean square $F^{\alpha}$-derivative, and fractal mean square integral. The mean square solution of a fractal stochastic equation is derived and plotted in order to show the details.