Generalized stochastic areas, Winding numbers, and hyperbolic Stiefel fibrations

TL;DR

This paper investigates Brownian motion on a complex hyperbolic Grassmann manifold, utilizing matrix stochastic calculus and hyperbolic fibrations to analyze generalized stochastic areas and windings, revealing connections to the Maass Laplacian.

Contribution

It introduces a matrix diffusion framework for Brownian motion on hyperbolic Grassmannians and explores its relation to stochastic areas and windings via hyperbolic Stiefel fibrations.

Findings

Connection to the generalized Maass Laplacian in complex hyperbolic space

Analysis of Brownian windings in the Lie group U(n-k,k)

Development of a matrix diffusion approach for hyperbolic stochastic processes

Abstract

We study the Brownian motion on the non-compact Grassmann manifold $\frac{\mathbf{U}(n-k,k)} {\mathbf{U}(n-k)\mathbf{U}(k)}$ and some of its functionals. The key point is to realize this Brownian motion as a matrix diffusion process, use matrix stochastic calculus and take advantage of the hyperbolic Stiefel fibration to study a functional that can be understood in that setting as a generalized stochastic area process. In particular, a connection to the generalized Maass Laplacian of the complex hyperbolic space is presented and applications to the study of Brownian windings in the Lie group $\mathbf{U}(n-k,k)$ are then given.