Generalized Stochastic Areas, Winding Numbers, and Hyperbolic Stiefel Fibrations

We study the Brownian motion on the non-compact Grassmann manifold U(n-k,k)/U(n-k)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 U(n - k, k) are then given.