Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces

Abstract

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 $S{L}_2(\mathbb{R})$.

Minor corrections. The main result of the spectral convergence is also covered with a simplified proof and in a more general setting in arXiv:2011.06434. This version still contains explicit error estimates not contained in arXiv:2011.06434