Spectral asymptotics for kinetic Brownian motion on Riemannian manifolds

TL;DR

This paper establishes the spectral convergence of kinetic Brownian motion generators to the Laplacian on closed Riemannian manifolds, extending previous results and confirming a conjecture on convergence rates.

Contribution

It generalizes spectral convergence results to all closed Riemannian manifolds and proves a conjecture on optimal convergence rates to equilibrium.

Findings

Spectral convergence of kinetic Brownian motion to the Laplacian.

Extension of previous results to general closed Riemannian manifolds.

Proof of a conjecture on optimal convergence rate to equilibrium.

Abstract

We prove the convergence of the spectrum of the generator of the kinetic Brownian motion to the spectrum of the base Laplacian for closed Riemannian manifolds. This generalizes recent work of Kolb--Weich--Wolf [arXiv:2011.06434] on constant curvature surfaces and of Ren--Tao [arXiv:2208.13111] on locally symmetric spaces. As an application, we prove a conjecture of Baudoin--Tardif [arXiv:1604.06813] on the optimal convergence rate to the equilibrium.