  The kinetic Brownian motion on the sphere bundle of a Riemannian manifold $M$
is a stochastic process that models a random perturbation of the geodesic flow.
If $M$ is a orientable compact constantly curved surface, we show that in the
limit of infinitely large perturbation the $L^2$-spectrum of the infinitesimal
generator of a time rescaled version of the process converges to the Laplace
spectrum of the base manifold.
