Limiting behaviors of the Brownian motions on hyperbolic spaces

Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. Using the representations, we show that the almost sure convergence of the Brownian motions and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the Poisson kernels by combining the representations with some results on the distributions of the random variables which are defined by the perpetual (infinite) integrals of the usual geometric Brownian motions with negative drifts.