Quaternionic stochastic areas on quaternionic full flag manifolds and applications

We show that a Brownian motion on the quaternionic full flag manifold can be represented as a matrix-valued diffusion obtained in a simple way from a symplectic Brownian motion. By relating its radial dynamics to the Brownian motion on the quaternionic sphere, an explicit formula for the characteristic function of the joint distribution of the quaternionic stochastic areas is obtained. The limit law of these quaternionic stochastic areas is shown to be a multivariate normal distribution with zero mean and non-diagonal covariance matrix. These results are subsequently applied to establish new results about simultaneous quaternionic windings on the quaternionic spheres.