Motion Among Random Obstacles on a Hyperbolic Space

Abstract

Abstract. We consider the motion of a particle along the geodesic lines of the Poincaré half-plane. The particle is specularly reflected when it hits randomlydistributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.