Langevin-type dynamics and hypocoercivity on $\mathrm{S}\mathrm{E}(3)$

We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{S}\mathrm{E}(3)={\mathbb{R}}^3⋊\mathrm{S}\mathrm{O}(3)$ using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator $L=S-A$ decomposes into a symmetric part $S$, corresponding to a degenerate diffusion on the rotation component $\mathrm{S}\mathrm{O}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is ${\mu }_{\mathrm{\Phi }}={\gamma }_{\mathrm{\Phi }}\otimes {\nu }_3$, where ${\gamma }_{\mathrm{\Phi }}$ is a probability measure on ${\mathbb{R}}^3$ with density $Z(\mathrm{\Phi }{)}^{-1}{e}^{-\mathrm{\Phi }}$ with respect to the Lebesgue measure, and ${\nu }_3$ is the normalized Haar measure on $\mathrm{S}\mathrm{O}(3)$. As a consequence, if $\mathrm{\Phi }\in C^{\mathrm{\infty }}({\mathbb{R}}^3)$ is bounded from below, ${\gamma }_{\mathrm{\Phi }}$ satisfies a Poincaré inequality with constant $\mathrm{\Lambda }>0$, and there exists $c<\mathrm{\infty }$ such that \begin{equation*} |\nabla_ξ^2Φ(ξ)| \leq c\bigl(1+|\nabla_ξΦ(ξ)|\bigr), \quad ξ\in$\mathbb{R}$^3, \end{equation*} then the semigroup generated by the closure of $L$ converges exponentially fast to equilibrium in $L^2(\mathrm{S}\mathrm{E}(3),{\mu }_{\mathrm{\Phi }})$.