欠阻尼朗之万动力学的显式$L^2$收敛速率估计研究
On explicit $L^2$-convergence rate estimate for underdamped Langevin dynamics
AI总结：
该研究基于已有框架推导欠阻尼朗之万动力学$L^2$距离下的指数衰减显式估计，证明庞加莱型不等式，得到更简洁的速率表达式，且凸势下优化摩擦系数后速率快于过阻尼情形。
摘要：
我们基于文献[1]中提出的框架，给出了欠阻尼朗之万动力学在$L^2$距离下指数衰减速率的精细化显式估计。为实现这一结果，我们首先证明了一个空间上带吉布斯测度、动量上带高斯测度的庞加莱型不等式。我们的估计给出了更显式、更简洁的衰减速率表达式；此外，当势函数为凸且庞加莱常数$m \ll 1$时，经摩擦系数选择优化后，我们的估计显示衰减速率为$O(\sqrt{m})$，远快于过阻尼朗之万动力学的$m$阶速率。
英文摘要：
We provide a refined explicit estimate of exponential decay rate of underdamped Langevin dynamics in $L^2$ distance, based on a framework developed in [1]. To achieve this, we first prove a Poincaré-type inequality with Gibbs measure in space and Gaussian measure in momentum. Our estimate provides a more explicit and simpler expression of decay rate; moreover, when the potential is convex with Poincaré constant $m \ll 1$, our estimate shows the decay rate of $O(\sqrt{m})$ after optimizing the choice of friction coefficient, which is much faster than $m$ for the overdamped Langevi dynamics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
