Chen, Xinxin; Miermont, Grégory Long Brownian bridges in hyperbolic spaces converge to Brownian trees. (English) Zbl 1386.60121 Electron. J. Probab. 22, Paper No. 58, 15 p. (2017). Summary: 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 \(\bullet\) A theorem by P. Bougerol and T. Jeulin [Probab. Theory Relat. Fields 115, No. 1, 95–120 (1999; Zbl 0947.58032)], stating that the rescaled radial process converges to the normalized Brownian excursion, \(\bullet\) A property of invariance under re-rooting, \(\bullet\) 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. Cited in 2 Documents MSC: 60F17 Functional limit theorems; invariance principles 58J65 Diffusion processes and stochastic analysis on manifolds Keywords:Brownian bridge in hyperbolic space; Brownian continuum random tree; infinite Brownian loop; asymptotic cone; Gromov-Hausdorff convergence Citations:Zbl 0947.58032 PDFBibTeX XMLCite \textit{X. Chen} and \textit{G. Miermont}, Electron. J. Probab. 22, Paper No. 58, 15 p. (2017; Zbl 1386.60121) Full Text: DOI arXiv Euclid