Electronic Journal of Probability

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

Xinxin Chen and Grégory Miermont

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 58, 15 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

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

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Chen, Xinxin; Miermont, Grégory. Long Brownian bridges in hyperbolic spaces converge to Brownian trees. Electron. J. Probab. 22 (2017), paper no. 58, 15 pp. doi:10.1214/17-EJP68. https://projecteuclid.org/euclid.ejp/1500516020

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