Hitting hyperbolic half-space

TL;DR

This paper establishes the reflection principle for hyperbolic Brownian motion with drift, enabling analysis of the process in hyperbolic half-space and deriving explicit formulas and asymptotics for associated Green functions and kernels.

Contribution

It introduces the reflection principle for hyperbolic Brownian motion with drift and derives explicit formulas and asymptotic behaviors for Green functions and Poisson kernels in hyperbolic half-space.

Findings

Reflection principle for hyperbolic Brownian motion with drift proved.

Explicit formulas and estimates for Green function and Poisson kernel obtained.

Asymptotic behavior of the Green function and Poisson kernel described.

Abstract

Let X^\mu={X_t^\mu;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\in 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.