The main results in this paper concern large and moderate deviations for the radial component of a $n$-dimensional hyperbolic Brownian motion (for $n\geq 2$) on the Poincar\'{ e} half-space. We also investigate the asymptotic behavior of the hitting probability $P_\eta(T_{ \eta_1}^{ (n)}<\infty)$ of a ball of radius $\eta_1$, as the distance $\eta$ of the starting point of the hyperbolic Brownian motion goes to infinity.
Categories: math.PR