Small Mass Limit of a Langevin Equation on a Manifold

Jeremiah Birrell, Scott Hottovy, Giovanni Volpe, Jan Wehr

Published 2016-04-17Version 1

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 noise-induced drift term. A very special case of the main result presents the Brownian motion on the manifold as a limit of inertial systems.