Quaternionic stochastic areas

2019/03/02 by Baudoin, Fabrice, Demni, Nizar, Wang, Jing
#Differential Geometry (math.DG) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1903.00727

Abstract

We 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 allows 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.

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