Hypocoercive Langevin dynamics on the Lie group $\mathrm{SE}(2)$

TL;DR

This paper studies a Langevin diffusion on the Lie group SE(2), revealing how degenerate rotational noise leads to effective planar diffusion through geometric and averaging techniques.

Contribution

It provides an intrinsic geometric formulation of hypocoercivity on SE(2) and demonstrates the emergence of macroscopic diffusion via averaging over rotations.

Findings

Effective diffusion on  emerges from rotational averaging.

Intrinsic formulation using invariant vector fields clarifies hypocoercivity.

Highlights geometric mechanisms underlying diffusion on Lie groups.

Abstract

We consider a Langevin-type diffusion on the planar motion group $\mathrm{SE}(2)$, describing the coupled evolution of position and orientation with degenerate noise acting only in the rotational direction. Although hypocoercivity for related models on $\mathbb{R}^2 \times \mathbb{S}^1$ is well understood, our purpose is to present an intrinsic formulation on the Lie group $\mathrm{SE}(2)$, and to highlight the underlying geometric mechanism. By expressing the generator in terms of invariant vector fields and using the natural projection onto the kernel of the symmetric part, we show how an effective macroscopic diffusion on $\mathbb{R}^2$ emerges through averaging over the compact rotation subgroup.