We provide a framework to prove convergence rates for discretizations of kinetic Langevin dynamics for $M$-$\nabla$Lipschitz $m$-log-concave densities. Our approach provides 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 methods which are popular in the molecular dynamics and machine learning communities. Finally we introduce the property ``$\gamma$-limit convergent" (GLC) to characterise 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.

翻译：我们提出一个框架，用于证明$M$-$\nabla$Lipschitz $m$-对数凹密度下离散化动力朗之万动力学的收敛速率。该方法给出了与高斯目标稳定性阈值同阶的$\mathcal{O}(m/M)$收敛速率（含显式步长限制），且适用于宽摩擦参数区间。我们将此方法应用于分子动力学与机器学习界流行的多种积分方法。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到过阻尼动力学且步长限制独立于摩擦参数的欠阻尼朗之万格式；通过展示该性质类及其补集中的方法，我们证明此性质并非普遍成立。