Open Access
2017 Long Brownian bridges in hyperbolic spaces converge to Brownian trees
Xinxin Chen, Grégory Miermont
Electron. J. Probab. 22: 1-15 (2017). DOI: 10.1214/17-EJP68


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.


Download Citation

Xinxin Chen. Grégory Miermont. "Long Brownian bridges in hyperbolic spaces converge to Brownian trees." Electron. J. Probab. 22 1 - 15, 2017.


Received: 4 October 2016; Accepted: 8 May 2017; Published: 2017
First available in Project Euclid: 20 July 2017

zbMATH: 06797868
MathSciNet: MR3683367
Digital Object Identifier: 10.1214/17-EJP68

Primary: 58J65 , 60F17

Keywords: asymptotic cone , Brownian bridge in hyperbolic space , Brownian continuum random tree , Gromov-Hausdorff convergence , infinite Brownian loop

Vol.22 • 2017
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