Langevin equations in the small-mass limit: Higher-order approximations

TL;DR

This paper develops a hierarchy of higher-order approximations for Langevin equations in the small-mass limit, improving accuracy beyond the standard order 1/2, applicable to systems with bounded and unbounded forces.

Contribution

It introduces a bootstrapping method to derive higher-order approximations for the overdamped limit of Langevin equations, extending prior results to arbitrary orders.

Findings

Achieves accuracy of order m^{ ext{ell}/2} for any positive integer ell.

Proves convergence in L^p norms for bounded forces.

Establishes convergence in probability for unbounded forces.

Abstract

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order $m^{\ell/2}$ over compact time intervals for any $\ell\in\mathbb{Z}^+$. This generalizes prior derivations of the homogenized equation for the position degrees of freedom in the $m\to 0$ limit, which result in order $m^{1/2}$ approximations. Our results cover bounded forces, for which we prove convergence in $L^p$ norms, and unbounded forces, in which case we prove convergence in probability.