Geometry and Hypocoercivity on the Special Euclidean Motion Groups SE(2) and SE(3)

16.10.2026, 11:00  –  Campus Golm, Building 09, Room 2.09.2.22
Freitag Seminar

Andrea V. Hurtado-Quiceno (Kaiserslautern-Landau)

The planar motion group SE(2) = R2 ⋊ S1 provides a natural geometric setting for stochastic
dynamics coupling position and orientation. In this talk, I will study a degenerate Kolmogorov
operator on SE(2) through its right-invariant vector fields and explain how the Lie group structure
enters the mechanism of hypocoercivity.
The diffusion acts only in the rotational direction, while the remaining spatial directions are re-
covered through transport and Lie brackets. In particular, the commutator of the rotational and
translational vector fields generates the missing spatial direction, reflecting Hörmander’s bracket-
generating condition. I will explain how this Lie algebra structure interacts with an abstract Hilbert
space hypocoercivity framework. Averaging over the compact rotational subgroup produces an
elliptic macroscopic operator on R2, connecting the geometry of the group with microscopic and
macroscopic coercivity and, ultimately, exponential convergence to equilibrium. Building on this
geometric viewpoint, we investigate the three-dimensional motion group SE(3), where the rota-
tional component is SO(3) and new geometric and analytic features arise.

This is joint work with Prof. Dr Martin Grothaus from RPTU University Kaiserslautern-Landau