On the Asymptotic Behavior of the Hyperbolic Brownian Motion

The main results in this paper concern large and moderate deviations for the radial component of a n-dimensional hyperbolic Brownian motion (for n >= 2) on the Poincare half-space. We also investigate the asymptotic behavior of the hitting probability P-eta(Tau((n))(eta 1) < infinity) of a ball of radius eta(1), as the distance eta of the starting point of the hyperbolic Brownian motion goes to infinity