Small Mass Limit of a Langevin Equation on a Manifold

TL;DR

This paper investigates how a damped Langevin equation on a Riemannian manifold behaves as the particle's mass approaches zero, revealing a convergence to a limiting stochastic process with a noise-induced drift.

Contribution

It establishes the small mass limit of Langevin dynamics on manifolds, including the derivation of the noise-induced drift term in the limiting equation.

Findings

Solutions converge to a limiting equation as mass approaches zero

The limiting process includes a noise-induced drift term

Brownian motion emerges as a special case in the limit

Abstract

We study damped geodesic motion of a particle of mass $m$ on a Riemannian manifold, in the presence of an external force and noise. Lifting the resulting stochastic differential equation to the orthogonal frame bundle, we prove that, as $m \to 0$, its solutions converge to solutions of a limiting equation which includes a {\it noise-induced drift} term. A very special case of the main result presents Brownian motion on the manifold as a limit of inertial systems.