  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 constant negatively 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. In addition, we give explicit error
estimates for the convergence to equilibrium. The proofs are based on
noncommutative harmonic analysis of $SL_2(\mathbb{R})$.
