# On the Langevin equation with variable friction

## TL;DR

This paper investigates the asymptotic behavior of the Langevin equation with variable friction, focusing on the small mass limit and the effects of vanishing friction regions, extending previous one-dimensional explicit results.

## Contribution

It provides new insights into the asymptotic limits of Langevin equations with variable friction in multi-dimensional settings, beyond previous one-dimensional explicit analyses.

## Key findings

- Analysis of the small mass asymptotic behavior with positive variable friction.
- Characterization of the solution's behavior when friction vanishes in certain regions.
- Extension of previous one-dimensional results to more general settings.

## Abstract

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.