HITTING HYPERBOLIC HALF-SPACE

Let X (mu) = {X (t)(mu) t; t >= 0}, mu > 0, be the n-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic space H-n having the Laplace-Beltrami operator with drift as its generator. We prove the reflection principle for X (mu) which enables us to study the process X (mu) killed when exiting the hyperbolic half-space, that is the set D = {x is an element of H-n : x(1) > 0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X (mu). Finally, we derive formula for the lambda-Poisson kernel of the set D.