AI 中文总结
本文针对镜面反射的欠阻尼Langevin动力学，证明了在Poincaré不等式等条件下L^2指数收敛，速率按√m缩放，实现约束采样的平方根加速，并扩展到非凸域。
AI 中文摘要
我们研究了在边界处通过速度的镜面反射限制在有界域 $\Omega\subset\mathbb{R}^d$ 内的欠阻尼（动力学）Langevin 动力学。该过程是通常反射的过阻尼 Langevin 扩散的自然基于动量的类比，并且在实际中用于约束采样。仅假设位置边际 $\mu_x\propto e^{-U}$ 在 $\Omega$ 上满足 Poincaré 不等式，常数 $m>0$，$\nabla^2U\succeq-K\\,\mathrm{Id}$ 且 $\Omega$ 是凸域，我们证明该定律在 $L^2$ 中指数快速收敛到 Gibbs 测度，其显式速率按 $\sqrt m$ 缩放，当 $U$ 为凸时这是最优的。由于通常反射的过阻尼动力学恰好以速率 $m$ 收敛，这在小间隙机制下（即 $m$ 小时）建立了约束采样的平方根加速，与无约束情况下的已知加速相匹配。该显式速率与 Fan--Li--Lu 的无约束设置中的速率相同。证明改编了 Dolbeault--Mouhot--Schmeiser 的修正 $L^2$ hypoelliptic 方法，并使用了 Fan--Li--Lu 的间隙移位校正子。镜面对称性使得输运算子反对称，并且校正子自动选择过阻尼生成元的 Neumann 实现，这恰好是将每个辅助函数保持在镜面类内部的边界条件。全空间论证中使用的 Bochner 恒等式被加权 Reilly 公式替代，其边界贡献涉及 $\partial\Omega$ 的第二基本形式，并且对于凸 $\Omega$ 是非负的。最后，我们将结果扩展到域 $\Omega$ 非凸的设置。我们获得了依赖于域的显式收缩速率。
英文摘要
We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain $Ω\subset\mathbb{R}^d$ by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal $μ_x\propto e^{-U}$ satisfies a Poincaré inequality on $Ω$ with constant $m>0$, $\nabla^2U\succeq-K\,\mathrm{Id}$ and $Ω$ is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in $L^2$, with an explicit rate that scales like $\sqrt m$, which is optimal when $U$ is convex. Since the normally reflected overdamped dynamics converges exactly at rate $m$, this establishes a square-root acceleration for constrained sampling in the small-gap regime when $m$ is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan--Li--Lu. The proof adapts the modified $L^2$ hypocoercivity method of Dolbeault--Mouhot--Schmeiser with the gap-shifted corrector of Fan--Li--Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of $\partialΩ$ and is nonnegative for convex $Ω$. Finally, we extend our results to the setting where the domain $Ω$ is non-convex. We obtain an explicit contraction rate that depends on the domain.
Comments41 pages
