Central limit theorem of Brownian motions in pinched negative curvature

TL;DR

This paper establishes a central limit theorem for distances related to Brownian motion and Green functions on negatively curved manifolds, with implications for ergodic properties and geodesic flow dynamics.

Contribution

It proves a new central limit theorem for Brownian motion on pinched negatively curved manifolds and explores related ergodic properties and applications to geodesic flows.

Findings

Central limit theorem for distances to Brownian paths and Green functions.

Ergodic properties of Brownian motions in negatively curved spaces.

Application to dynamics of geodesic flows.

Abstract

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.