On potential theory of hyperbolic Brownian motion with drift

Consider the $\lambda$-Green function and the $\lambda$-Poisson kernel of a Lipschitz domain $U\subset {\mathbb{H}}^n=\{x\in {\mathbb{R}}^n:x_n>0\}$ for hyperbolic Brownian motion with drift. We provide several relationships that facilitate studying those objects and explain somehow theirs nature. As an application, we yield uniform estimates in case of sets of the form $S_{a,b}=\{x\in {\mathbb{H}}^n:x_n>a,x_1\in (0,b)\}$, $a,b>0$, $a,b>0$, which covers and extends existing results of that kind.