Stochastic areas, Winding numbers and Hopf fibrations

TL;DR

This paper investigates stochastic area processes and winding numbers for Brownian motions on complex symmetric spaces, deriving characteristic functions, limit theorems, and analyzing geometric influences of Hopf and anti-de Sitter fibrations.

Contribution

It introduces new stochastic processes on complex symmetric spaces and computes their characteristic functions and limit distributions, highlighting geometric effects.

Findings

Derived characteristic functions for stochastic areas on $ ext{CP}^n$ and $ ext{CH}^n$

Established limit theorems for these processes

Analyzed winding distributions for $n=1$ cases

Abstract

We define and study stochastic areas processes associated with Brownian motions on the complex symmetric spaces $\mathbb{CP}^n$ and $\mathbb{CH}^n$. The characteristic functions of those processes are computed and limit theorems are obtained. In the case $n=1$, we also study windings of the Brownian motion on those spaces and compute the limit distributions. For $\mathbb{CP}^n$ the geometry of the Hopf fibration plays a central role, whereas for $\mathbb{CH}^n$ it is the anti-de Sitter fibration.