Winding Number of Fractional Brownian Motion

TL;DR

This paper derives the exact distribution of the winding number for fractional Brownian motion and related fractal time processes, revealing a Cauchy-type distribution similar to classical Brownian motion, for large times.

Contribution

It provides the first exact large-time winding number distribution for Riemann-Liouville fractional Brownian motion and fractal time processes, extending classical results.

Findings

Winding number distribution is of Cauchy type for fractional Brownian motion.

Distribution matches classical Brownian motion in form for large times.

Results apply to processes with finite size winding centers.

Abstract

We find the exact winding number distribution of Riemann-Liouville fractional Brownian motion for large times in two dimensions using the propagator of a free particle. The distribution is similar to the Brownian motion case and it is of Cauchy type. In addition we find the winding number distribution of fractal time process, i.e., time fractional Fokker-Planck equation, in the presence of finite size winding center.