Concentration of the Brownian bridge on the hyperbolic plane

Abstract

We consider a Brownian bridge on the hyperbolic plane with one extremity tending to infinity, in finite time. We show that the exact exponential rate according to which this process concentrates round the geodesical segment joining the origin o to the moving extremity z is $-\rho(o,z)$, where $\rho$ stands for the hyperbolic distance. This improves a result of A. Eberle.