Generalized Stochastic Areas, Winding Numbers, and Hyperbolic Stiefel Fibrations

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