Quaternionic stochastic areas

We define and study quaternionic stochastic areas processes associated with Brownian motions on the rank-one symmetric spaces HHn HPn. The characteristic functions of fixed-time marginals these are computed allow for explicit description their corresponding large-time limits. also obtain exact formulas semigroup densities area using a Doob transform in former case density circular Jacobi process latter. For HHn, geometry anti-de Sitter fibration plays central role, whereas HPn, this role is played by Hopf fibration.