Brownian bridge on hyperbolic spaces and on homogeneous trees

Abstract

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space Hn, n > 2. For ν > 0, the Brownian bridge B(ν) of length ν on H is the process Bt, 0 ≤t≤ν, conditioned by B0 = Bν = 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.