Limiting behaviors of the Brownian motions on hyperbolic spaces

TL;DR

This paper investigates the long-term behavior of Brownian motions on hyperbolic spaces, deriving convergence results, central limit theorems, and explicit limit distributions using explicit representations.

Contribution

It provides a straightforward method to derive explicit limit distributions and Poisson kernels for Brownian motions on hyperbolic spaces, enhancing understanding of their asymptotic behavior.

Findings

Almost sure convergence of radial components

Central limit theorems for radial components

Explicit expressions for limit distributions and Poisson kernels

Abstract

Using the explicit representations of the Brownian motions on the hyperbolic spaces, we show that their almost sure convergence and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the limit distributions or the Poisson kernels.