Langevin Equation on Fractal Curves

TL;DR

This paper develops a Langevin equation framework for modeling particle motion on fractal curves, incorporating fractal geometry via $F^eta$-Calculus, and provides solutions for this novel stochastic process.

Contribution

It introduces a Langevin equation tailored for fractal curves using $F^eta$-Calculus, linking fractal geometry with stochastic dynamics in a new way.

Findings

Derived a Langevin equation specific to fractal curves

Solved the equation using $F^eta$-Calculus techniques

Highlighted the influence of fractal geometry on particle motion

Abstract

We analyse a random motion of a particle on a fractal curve, using Langevin approach. This involves defining a new velocity in terms of mass of the fractal curve, as defined in recent work. The geometry of the fractal curve, hence plays an important role in this analysis. A Langevin equation with a particular noise model is thus proposed and solved using techniques of the newly developed $F^\alpha$-Calculus .