Hitting spheres on hyperbolic spaces

For a hyperbolic Brownian motion on the Poincar'e half-plane ${\mathbb{H}}^2$, starting from a point of hyperbolic coordinates $z=(\eta ,\alpha )$ inside a hyperbolic disc $U$ of radius $\bar{\eta }$, we obtain the probability of hitting the boundary $\mathrm{\partial }U$ at the point $(\bar{\eta },\bar{\alpha })$. For $\bar{\eta }\to \mathrm{\infty }$ we derive the asymptotic Cauchy hitting distribution on $\mathrm{\partial }{\mathbb{H}}^2$ and for small values of $\eta$ and $\bar{\eta }$ we obtain the classical Euclidean Poisson kernel. The exit probabilities ${\mathbb{P}}_z\{T_{{\eta }_1}<T_{{\eta }_2}\}$ from a hyperbolic annulus in ${\mathbb{H}}^2$ of radii ${\eta }_1$ and ${\eta }_2$ are derived and the transient behaviour of hyperbolic Brownian motion is considered. Similar probabilities are calculated also for a Brownian motion on the surface of the three dimensional sphere. For the hyperbolic half-space ${\mathbb{H}}^n$ we obtain the Poisson kernel of a ball in terms of a series involving Gegenbauer polynomials and hypergeometric functions. For small domains in ${\mathbb{H}}^n$ we obtain the $n$-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the $n$-dimensional case.