Spectral Asymptotics for Kinetic Brownian Motion on Surfaces of Constant Curvature

TL;DR

This paper studies the spectral behavior of kinetic Brownian motion on surfaces of constant curvature, showing that as perturbations grow large, its generator's spectrum converges to the Laplace spectrum of the surface.

Contribution

It establishes a spectral asymptotic result connecting the generator of kinetic Brownian motion to the Laplace operator on constant curvature surfaces.

Findings

Spectrum of the generator converges to the Laplace spectrum as perturbation increases

Results apply specifically to orientable compact surfaces of constant curvature

Provides a link between stochastic dynamics and geometric spectral theory

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 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.