On potential theory of hyperbolic Brownian motion with drift

TL;DR

This paper investigates the potential theory related to hyperbolic Brownian motion with drift, focusing on Green functions and Poisson kernels in Lipschitz domains, providing new relationships and uniform estimates for specific sets.

Contribution

It introduces new relationships for Green functions and Poisson kernels of hyperbolic Brownian motion with drift, extending existing estimates to broader classes of sets.

Findings

Derived relationships for Green functions and Poisson kernels.

Provided uniform estimates for specific Lipschitz domains.

Extended previous results to new geometric configurations.

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 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.