Long Brownian bridges in hyperbolic spaces converge to Brownian trees

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

• A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion,

• A property of invariance under re-rooting,

• 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.