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

Abstract

We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{SE}(3)={\mathbb{R}}^3⋊\mathrm{SO}(3)$ 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 $\mathrm{SO}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is ${\mu }_{\Phi }={\gamma }_{\Phi }\otimes {\nu }_3$, where ${\gamma }_{\Phi }$ is a probability measure on ${\mathbb{R}}^3$ with density $Z(\Phi {)}^{-1}{e}^{-\Phi }$ with respect to the Lebesgue measure, and ${\nu }_3$ is the normalized Haar measure on $\mathrm{SO}(3)$. As a consequence, if $\Phi \in C^{\infty }({\mathbb{R}}^3)$ is bounded from below, ${\gamma }_{\Phi }$ satisfies a Poincaré inequality with constant $\Lambda >0$, and there exists $c<\infty$ 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 $L^2(\mathrm{SE}(3),{\mu }_{\Phi })$.

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