Poisson kernel and Green function of the ball in real hyperbolic spaces

Let $(X_t)_{t\geq0}$ be the $n$-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space $\D^n$ having the Laplace-Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process $(X_t)_{t\geq0}$. Under some additional hypotheses we give the formulas for the Poisson kernel itself. In particular, we provide formulas in $\D^4$ and $\D^6$ spaces for the Poisson kernel and the Green function as well.