On the asymptotic behavior of the hyperbolic Brownian motion

TL;DR

This paper studies the large and moderate deviation behaviors of the radial component of hyperbolic Brownian motion in high dimensions, and analyzes the asymptotic hitting probabilities as the starting point recedes to infinity.

Contribution

It provides new asymptotic results for the radial component's deviations and hitting probabilities in hyperbolic Brownian motion, extending understanding of its long-term behavior.

Findings

Large deviation principles established for the radial component.

Asymptotic behavior of hitting probabilities analyzed for distant starting points.

Results applicable to high-dimensional hyperbolic spaces.

Abstract

The main results in this paper concern large and moderate deviations for the radial component of a $n$-dimensional hyperbolic Brownian motion (for $n\geq 2$) on the Poincar\'{e} half-space. We also investigate the asymptotic behavior of the hitting probability $P_\eta(T_{\eta_1}^{(n)}<\infty)$ of a ball of radius $\eta_1$, as the distance $\eta$ of the starting point of the hyperbolic Brownian motion goes to infinity.