Random Walk and Broad Distributions on Fractal Curves

TL;DR

This paper investigates random walks on fractal curves using fractal calculus, revealing Gaussian-like distributions with deviations and analyzing Levy distributions, highlighting the influence of fractal dimension on distribution laws.

Contribution

It introduces a method to analyze random walks on fractal curves with fractal calculus and explores how fractal dimension affects distribution behaviors.

Findings

Gaussian-like probability distributions with deviations on fractal curves

Levy distributions are influenced by fractal dimension

Moments depend on Euclidean distance and fractal properties

Abstract

In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves.