Generalized Stochastic Areas, Winding Numbers, and Hyperbolic Stiefel Fibrations

Abstract

Abstract We study the Brownian motion on the non-compact Grassmann manifold $\frac {\textbf {U}(n-k,k)} {\textbf {U}(n-k)\textbf {U}(k)}$ 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 $\textbf {U}(n-k,k)$ are then given.