发表机构
RPTU（莱茵兰-普法尔茨科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在SE(3)群上构造亚压缩Kolmogorov型算子，利用抽象亚压缩性方法证明在势函数满足Poincaré不等式和梯度有界条件下，半群指数收敛到平衡态。
AI 中文摘要
我们利用ZXZ欧拉角在欧氏运动群$\mathrm{SE}(3)=\mathbb{R}^3\rtimes\mathrm{SO}(3)$上构造了亚压缩的Kolmogorov型算子，并使用抽象亚压缩性方法研究其长时间行为。算子$L=S-A$分解为对称部分$S$（对应于旋转分量$\mathrm{SO}(3)$上的退化扩散）和反对称部分$A$（捕捉由李群结构引起的输运和扭矩效应）。相关的不变概率测度为$\mu_\Phi=\gamma_\Phi\otimes\nu_3$，其中$\gamma_\Phi$是$\mathbb{R}^3$上的概率测度，其关于勒贝格测度的密度为$Z(\Phi)^{-1}e^{-\Phi}$，而$\nu_3$是$\mathrm{SO}(3)$上的归一化Haar测度。因此，若$\Phi\in C^\infty(\mathbb{R}^3)$下有界，$\gamma_\Phi$满足常数$\Lambda>0$的Poincaré不等式，并且存在$c<\infty$使得\begin{equation*} |\nabla_\xi^2\Phi(\xi)| \leq c\bigl(1+|\nabla_\xi\Phi(\xi)|\bigr), \quad \xi\in\mathbb{R}^3, \end{equation*} 则由$L$的闭包生成的半群在$L^2(\mathrm{SE}(3),\mu_\Phi)$中指数快速收敛到平衡态。
英文摘要
We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{SE}(3)=\mathbb{R}^3\rtimes\mathrm{SO}(3)$ 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 $μ_Φ=γ_Φ\otimesν_3$, where $γ_Φ$ is a probability measure on $\mathbb{R}^3$ with density $Z(Φ)^{-1}e^{-Φ}$ with respect to the Lebesgue measure, and $ν_3$ is the normalized Haar measure on $\mathrm{SO}(3)$. As a consequence, if $Φ\in C^\infty(\mathbb{R}^3)$ is bounded from below, $γ_Φ$ satisfies a Poincaré inequality with constant $Λ>0$, 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 $L^2(\mathrm{SE}(3),μ_Φ)$.
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