Long Brownian bridges in hyperbolic spaces converge to Brownian trees
Résumé
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]
Origine : Fichiers produits par l'(les) auteur(s)
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