We provide a framework to analyze the convergence of discretized kinetic Langevin dynamics for $M$-$\nabla$Lipschitz, $m$-convex potentials. Our approach gives convergence rates of $\mathcal{O}(m/M)$, with explicit stepsize restrictions, which are of the same order as the stability threshold for Gaussian targets and are valid for a large interval of the friction parameter. We apply this methodology to various integration schemes which are popular in the molecular dynamics and machine learning communities. Finally, we introduce the property "$\gamma$-limit convergent" (GLC) to characterize underdamped Langevin schemes that converge to overdamped dynamics in the high-friction limit and which have stepsize restrictions that are independent of the friction parameter; we show that this property is not generic by exhibiting methods from both the class and its complement. We further provide asymptotic bias estimates for the BAOAB scheme, which remain accurate in the high-friction limit by comparison to a modified stochastic dynamics which preserves the invariant measure.

翻译：我们提出一个分析离散化动力学朗之万动力学收敛性的框架，适用于$M$-$\nabla$Lipschitz、$m$-凸势函数。该方法给出$\mathcal{O}(m/M)$的收敛速率，并包含显式的步长限制条件——该阶次与高斯目标分布的稳定性阈值一致，且对摩擦参数的大范围区间均有效。我们将此方法论应用于分子动力学和机器学习领域流行的多种积分格式。最后，我们引入"$\gamma$-极限收敛"（GLC）性质，用以刻画在高摩擦极限下收敛至过阻尼动力学且步长限制条件独立于摩擦参数的欠阻尼朗之万格式；通过展示属于该类及其补类的方法，我们证明该性质并非通用。此外，我们提供BAOAB格式的渐近偏差估计，通过与保持不变测度的修正随机动力学对比，这些估计在高摩擦极限下仍保持准确性。