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Long Brownian bridges in hyperbolic spaces converge to Brownian trees. (English) Zbl 1386.60121

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.

MSC:

60F17 Functional limit theorems; invariance principles
58J65 Diffusion processes and stochastic analysis on manifolds

Citations:

Zbl 0947.58032
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