Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Xinxin Chen (1) , Grégory Miermont (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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.

Domaines: Probabilités [math.PR]

Xinxin Chen, Grégory Miermont. Long Brownian bridges in hyperbolic spaces converge to Brownian trees. Electronic Journal of Probability, 2017, 22 (58), 15 pp. ⟨hal-01361103v2⟩