HITTING SPHERES ON HYPERBOLIC SPACES

For a hyperbolic Brownian motion on the Poincare half-plane H-2, starting from a point z = (eta, alpha) inside a hyperbolic disc U of radius (eta) over bar, we obtain the probability of hitting the boundary partial derivative U at the point ((eta) over bar, (alpha) over bar). For (eta) over bar -> infinity we derive the asymptotic Cauchy hitting distribution on partial derivative H-2. In particular, it follows that the hyperbolic Brownian motion starting at (x, y) is an element of H-2 "hits" the boundary of H-2 at a point which is Cauchy distributed with scale parameter y' = y/(x(2) + y(2)) and position parameter x' = x/(x(2) + y(2)). For small values of eta and (eta) over bar. we obtain the classical Euclidean Poisson kernel. The exit probabilities from a hyperbolic annulus in H-2 of radii eta(1) and eta(2) are derived and the transient behavior 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 H-n we obtain, with a proof based on the method of separation of variables, the Poisson kernel of a ball. For small domains in H-n we obtain the n-dimensional Euclidean Poisson kernel. The exit probabilities from an annulus are derived also in the n-dimensional case.