Brownian bridge on hyperbolic spaces and on homogeneous trees |
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| Authors: | Philippe Bougerol Thierry Jeulin |
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| Institution: | (1) Laboratoire de Probabilités, Université Paris 6, 4 Place Jussieu, F-75252 Paris, France. CNRS UMR 7599, e-mail: bougerol@ccr.jussieu.fr, FR;(2) UFR de Mathématiques, Université Paris 7, 2 Place Jussieu, F-75251 Paris, France. CNRS UPRESA 7055, e-mail: jeulin@math.jussieu.fr, FR |
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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.
Received: 4 December 1998 / Revised version: 22 January 1999 |
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| Keywords: | Mathematics Subject Classification (1991): 58G32 60J15 60J60 60J65 |
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