Abstract:

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

2010 AMS Mathematics Subject Classification: Primary 60J60; Secondary 58J65.

Keywords and phrases: hyperbolic space, hyperbolic Brownian motion with drift, \(\lambda\)-Poisson kernel, \(\lambda\)-Green function.