Brownian bridge on hyperbolic spaces and on homogeneous trees (Q1124997)

Consider the Brownian bridge \(B^{(\nu)}\) of length \(\nu\) on a noncompact non Euclidean rank one symmetric space \(H\) (for instance a hyperbolic space), and with origin a fixed point \(o\) of \(H\). Let \(R_t^{(\nu)} = d(o,B_t^{(\nu)})\). It is proved that \((R_{t\nu}^{(\nu)}/\sqrt\nu ; 0\leq t\leq 1)\) converges in distribution to the real-valued Brownian excursion. The same result is proved for homogeneous trees. The introduction of the article explains the motivations for this study; in particular, the order of magnitude which is obtained for the Brownian bridge in this result has important physical applications.