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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

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.

Citation

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Xinxin Chen. Grégory Miermont. "Long Brownian bridges in hyperbolic spaces converge to Brownian trees." Electron. J. Probab. 22 1 - 15, 2017. https://doi.org/10.1214/17-EJP68

Information

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

Subjects:
Primary: 58J65, 60F17

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