亚椭圆三阶朗之万扩散的弱平衡测度与容量-击中恒等式
Weak Equilibrium Measures and Capacity--Hitting Identities for the Hypoelliptic Third-Order Langevin Diffusion
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中文总结 AI 辅助
本文受加速采样算法启发，针对亚椭圆三阶朗之万扩散，通过证明椭圆正则化稳定性定理，结合特定策略与估计，构建弱平衡测度和弱容量，得到有界域及全空间的相关恒等式和正哈里斯常返性。
中文摘要 AI 辅助
受加速采样算法启发，我们为亚椭圆三阶朗之万扩散构建了弱平衡测度和弱容量。在此过程中，布朗噪声仅作用于最高阶辅助变量并通过三步霍尔曼德链到达物理变量，标准的一致椭圆边界通量理论在相空间球的特征点不直接适用。我们证明了相应击中律的椭圆正则化稳定性定理，进而定义了弱平衡测度和弱容量。证明结合了Lee--Ramil--Seo的边界击中稳定性策略与适用于三阶链的局部亚椭圆热核估计。我们得到了有界域弱容量-击中恒等式以及一个李雅普诺夫漂移论证，它产生正哈里斯常返性并将构造扩展到全空间弱平衡测度和全空间容量-击中恒等式。
英文摘要
We construct weak equilibrium measures and weak capacities for the hypoelliptic third-order Langevin diffusion motivated by an accelerated sampling algorithm (Mou et al. (2021) \textit{J. Mach. Learn. Res.}, \textbf{22}(42), 1--41). In this process, the Brownian noise acts only in the highest-order auxiliary variable and reaches the physical variables through a step-three Hörmander chain, so the standard uniformly elliptic boundary-flux theory is not directly applicable at characteristic points of phase-space balls. We prove an elliptic-regularization stability theorem for the corresponding hitting laws and then define the weak equilibrium measure and weak capacity. The proof combines the boundary-hitting stability strategy of Lee--Ramil--Seo (2026, \textit{arXiv:2503.12610v2}) with localized hypoelliptic heat-kernel estimates (Pigato (2022) \textit{Stoch. Process. Appl.}, \textbf{145}, 117--142) adapted to the third-order chain. We obtain the bounded-domain weak capacity--hitting identity and a Lyapunov drift argument in the spirit of Lee--Ramil--Seo that yields positive Harris recurrence and extends the construction to a whole-space weak equilibrium measure, and whole-space capacity--hitting identity.
