  We consider the motion of a particle along the geodesic lines of the
Poincar\`e half-plane. The particle is specularly reflected when it hits
randomly-distributed 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.
