Octonionic Brownian Windings

TL;DR

This paper investigates the winding behavior of Brownian motion in octonionic geometries, revealing Gaussian asymptotics in flat and spherical cases, and distinct behavior in hyperbolic space.

Contribution

It introduces the concept of octonionic Brownian windings and analyzes their asymptotic laws across different octonionic geometries, highlighting new geometric probabilistic phenomena.

Findings

Gaussian windings in flat and spherical geometries

Distinct long-term behavior in hyperbolic geometry

Extension of winding analysis to octonionic spaces

Abstract

We define and study the windings along Brownian paths in the octonionic Euclidean, projective and hyperbolic spaces which are isometric to 8-dimensional Riemannian model 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.