Hitting hyperbolic half-space

Abstract

Abstract Let X ( μ ) = { X t ( μ ) ; t ≥ 0 } $X^{(\mu )} = \{ X_t^{(\mu )} ;t \ge 0\} $ , μ > 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 ( μ ) which enables us to study the process X ( μ ) killed when exiting the hyperbolic half-space, that is the set D = {x ∈ 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 ( μ ). Finally, we derive formula for the λ-Poisson kernel of the set D.