Brownian bridge on hyperbolic spaces and on homogeneous trees

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 greater than or equal to 2. For nu > 0, the Brownian bridge B-(nu) of length nu on H is the process B-t, 0 less than or equal to t less than or equal to nu, conditioned by B-0 = B-nu = 0, where 0 is an origin in H. It is proved that the process {1/root nu d(0, B-t nu((nu))), 0 less than or equal to t less than or equal to 1} converges weakly to the Brownian excursion when nu --> + infinity (the Brownian excursion is the radial part of the Brownian Bridge on IR3). The same result holds for the simple random walk on an homogeneous tree.