We study the convergence in total variation and $V$-norm of discretization schemes of the underdamped Langevin dynamics. Such algorithms are very popular and commonly used in molecular dynamics and computational statistics to approximatively sample from a target distribution of interest. We show first that, for a very large class of schemes, a minorization condition uniform in the stepsize holds. This class encompasses popular methods such as the Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we provide mild conditions ensuring that the class of schemes that we consider satisfies a geometric Foster--Lyapunov drift condition, again uniform in the stepsize. This allows us to derive geometric convergence bounds, with a convergence rate scaling linearly with the stepsize. This kind of result is of prime interest to obtain estimates on norms of solutions to Poisson equations associated with a given numerical method.

翻译：我们研究了欠阻尼朗之万动力学离散格式在全变差范数和$V$-范数下的收敛性。这类算法在分子动力学和计算统计学中非常流行且广泛应用，用于对目标分布进行近似采样。首先，我们证明对于一大类格式，存在与步长均匀的小化条件。该类格式包括欧拉-丸山法和基于分裂策略的常用方法。其次，我们提供温和条件确保所考虑的格式类满足几何福斯特-利亚普诺夫漂移条件，且该条件同样与步长均匀。由此可推导几何收敛界，其收敛速率随步长线性缩放。这类结果对获取给定数值方法对应的泊松方程解的范数估计具有首要意义。