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

We consider a Langevin-type diffusion on the planar motion group $\mathrm{S}\mathrm{E}(2)$, 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 $\mathrm{S}\mathrm{E}(2)$, 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.