On the fractal Langevin equation

Abstract

In this paper, fractal stochastic Langevin equations are suggested, providing a mathematical model for random walks on the middle-τ Cantor set. The fractal mean square displacement of different random walks on the middle-τ Cantor set are presented. Fractal under-damped and over-damped Langevin equations, fractal scaled Brownian motion, and ultra-slow fractal scaled Brownian motion are suggested and the corresponding fractal mean square displacements are obtained. The results are plotted to show the details.