Quaternionic Brownian windings

TL;DR

This paper investigates the winding behavior of 3D Brownian motion in quaternionic spaces, revealing Gaussian asymptotics in flat and spherical geometries and a novel Cauchy-related law in hyperbolic space.

Contribution

It introduces the concept of quaternionic windings in various geometries and characterizes their long-term asymptotic laws, including a new distribution in hyperbolic space.

Findings

Gaussian asymptotics in Euclidean and spherical geometries

Hyperbolic winding follows a new Cauchy-related distribution

Distinct long-term behaviors depending on geometry

Abstract

We define and study the 3-dimensional windings along Brownian paths in the quaternionic Euclidean, projective and hyperbolic spaces. In particular, the asymptotic laws of these windings are shown to be Gaussian for the flat and spherical geometries while the hyperbolic winding exhibits a different long time-behavior. The corresponding asymptotic law seems to be new and is related to the Cauchy relativistic distribution.