Long Brownian bridges in hyperbolic spaces converge to Brownian trees

TL;DR

This paper proves that the scaled range of a long Brownian bridge in hyperbolic space converges to the Brownian continuum random tree, linking stochastic processes and geometric structures in hyperbolic geometry.

Contribution

It establishes a new convergence result connecting Brownian bridges in hyperbolic spaces to Brownian trees, using elementary probabilistic and geometric properties.

Findings

Convergence of Brownian bridge range to Brownian continuum random tree

Extension of results to infinite Brownian loops in hyperbolic space

Utilization of Bougerol-Jeulin theorem and hyperbolicity properties

Abstract

We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.