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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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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DOI: https://doi.org/10.1007/s004400050237
- Mathematics Subject Classification (1991): 58G32, 60J15, 60J60, 60J65
- Key words: Brownian bridge – Symmetric space – Tree – Excursion – Hyperbolic Space – Reflected random walk – Reflected Brownian motion