Stochastic processes and mean square calculus on fractal curves

Abstract

Abstract In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define the cumulative distribution function, probability density function, moments, variance, and correlation function of stochastic processes on fractal curves. A new framework, which is a generalization of mean square calculus, is formulated. The sequence of random variables on the fractal curve, fractal mean square continuity, mean square <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>F</m:mi> <m:mi>α</m:mi> </m:msup> </m:math> {F^{\alpha}} -derivative, and fractal mean square integral are discussed. The mean square solution of a fractal stochastic equation is derived and plotted to illustrate the details.