Overdamped limit at stationarity for non-equilibrium Langevin diffusions

TL;DR

This paper proves that in the overdamped limit, the stationary distribution of non-equilibrium Langevin diffusions converges to a product of the overdamped stationary distribution and a Gaussian, clarifying the limiting behavior as damping increases.

Contribution

It establishes the convergence of stationary distributions of Langevin diffusions to a tensor product form in the high damping limit, extending understanding of non-equilibrium systems.

Findings

Stationary distribution converges to a tensor product in the overdamped limit.

The limiting distribution combines the overdamped stationary distribution and a Gaussian.

Results apply to possibly non-equilibrium Langevin diffusions.

Abstract

In this note, we establish that the stationary distribution of a possibly non-equilibrium Langevin diffusion converges, as the damping parameter goes to infinity (or equivalently in the Smoluchowski-Kramers vanishing mass limit), toward a tensor product of the stationary distribution of the corresponding overdamped process and of a Gaussian distribution.