Spectral Asymptotics for Kinetic Brownian Motion on Hyperbolic Surfaces

TL;DR

This paper studies the spectral behavior of kinetic Brownian motion on hyperbolic surfaces, showing convergence of the generator's spectrum to the Laplace spectrum as perturbation grows large, with explicit error estimates.

Contribution

It establishes the spectral convergence of the kinetic Brownian motion generator to the Laplace operator spectrum on hyperbolic surfaces, using noncommutative harmonic analysis.

Findings

Spectrum of the generator converges to the Laplace spectrum

Explicit error estimates for convergence to equilibrium

Analysis based on noncommutative harmonic analysis of SL(2,R)

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 $SL_2(\mathbb{R})$.