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Brownian bridge on hyperbolic spaces and on homogeneous trees
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  • Published: August 1999

Brownian bridge on hyperbolic spaces and on homogeneous trees

  • Philippe Bougerol1 &
  • Thierry Jeulin2 

Probability Theory and Related Fields volume 115, pages 95–120 (1999)Cite this article

  • 221 Accesses

  • 16 Citations

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Abstract.

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H n , n > 2. For ν > 0, the Brownian bridge B (ν) of length ν on H is the process B t , 0 ≤t≤ν, conditioned by B 0 = B ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ3). The same result holds for the simple random walk on an homogeneous tree.

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Authors and Affiliations

  1. Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, F-75252 Paris, France. CNRS UMR 7599, e-mail: bougerol@ccr.jussieu.fr, , , , , , FR

    Philippe Bougerol

  2. UFR de Mathématiques, Université Paris 7, 2 Place Jussieu, F-75251 Paris, France. CNRS UPRESA 7055, e-mail: jeulin@math.jussieu.fr, , , , , , FR

    Thierry Jeulin

Authors
  1. Philippe Bougerol
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  2. Thierry Jeulin
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Received: 4 December 1998 / Revised version: 22 January 1999

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Bougerol, P., Jeulin, T. Brownian bridge on hyperbolic spaces and on homogeneous trees. Probab Theory Relat Fields 115, 95–120 (1999). https://doi.org/10.1007/s004400050237

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  • Issue Date: August 1999

  • DOI: https://doi.org/10.1007/s004400050237

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  • Mathematics Subject Classification (1991): 58G32, 60J15, 60J60, 60J65
  • Key words: Brownian bridge – Symmetric space – Tree – Excursion – Hyperbolic Space – Reflected random walk – Reflected Brownian motion
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