A Weak Overdamped Limit Theorem for Langevin Processes

TL;DR

This paper proves that Langevin processes converge to their overdamped limit under certain conditions, using a classical mathematical method to establish tightness and identify the limit as a martingale problem.

Contribution

It provides a rigorous proof of the overdamped limit for Langevin processes using the perturbed test function method, under mild assumptions.

Findings

Convergence in distribution of Langevin processes to the overdamped limit.

The proof applies the perturbed test function method for tightness and limit identification.

The result holds with continuous potential gradients and mild initial energy control.

Abstract

In this paper, we prove convergence in distribution of Langevin processes in the overdamped asymptotics. The proof relies on the classical perturbed test function (or corrector) method, which is used both to show tightness in path space, and to identify the extracted limit with a martingale problem. The result holds assuming the continuity of the gradient of the potential energy, and a mild control of the initial kinetic energy.