Small mass asymptotic for the motion with vanishing friction

TL;DR

This paper studies the small mass limit of Langevin dynamics with variable, vanishing friction, introducing a regularization and analyzing the resulting Markov process in both one and multiple dimensions.

Contribution

It provides a new analysis of the small mass asymptotic for Langevin equations with vanishing friction, including the generator and boundary conditions of the limiting process.

Findings

Convergence of the regularized Langevin dynamics to a Markov process.

Explicit characterization of the generator and boundary conditions.

Validation of the approximation in both 1D and multidimensional cases.

Abstract

We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and boundary condition of this limiting Markov process and prove the convergence.