Quaternionic stochastic areas

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