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. Further, 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. Finally, we 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：我们提出了一个分析框架，用于分析针对$M$-$\nabla$Lipschitz、$m$-凸势能的离散化动能朗之万动力学的收敛性。我们的方法给出了$\mathcal{O}(m/M)$量级的收敛速率，并附有明确的步长限制条件，这些条件与高斯目标下的稳定性阈值同阶，且适用于摩擦参数的大范围区间。我们将此方法应用于分子动力学和机器学习领域中流行的多种积分方案。此外，我们引入了“$\gamma$-极限收敛”（GLC）性质，用以刻画在强摩擦极限下收敛于过阻尼动力学、且步长限制与摩擦参数无关的欠阻尼朗之万方案；我们通过展示属于该类及其补集的方案，证明该性质并非普遍存在。最后，我们给出了BAOAB方案的渐近偏差估计，该估计通过与一个保持不变测度的修正随机动力学进行比较，在强摩擦极限下仍保持准确性。