Stochastic areas, Winding numbers and Hopf fibrations

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.

arXiv commentFinal version, To appear in Probability Theory and Related Fields