  We study several matrix diffusion processes constructed from a unitary
Brownian motion. In particular, we use the Stiefel fibration to lift the
Brownian motion of the complex Grassmannian to the complex Stiefel manifold and
deduce a skew-product decomposition of the Stiefel Brownian motion. As an
application, we prove asymptotic laws for the determinants of the block entries
of the unitary Brownian motion.
