  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.
