For the kinetic Langevin diffusion and its splitting discretizations, the hypoelliptic noise structure makes the relationship between couplings and total variation (TV) bounds more subtle than in the elliptic case. We establish that, for the kinetic Langevin equation with quadratic potential, no Markovian coupling (continuous or discrete) captures the asymptotic decay rate of the TV distance between two solutions with different initial values; the canonical iterated one-shot (or sticky) coupling, for which we derive an exact contraction formula, saturates this lower bound. On the constructive side, we show that the recent sharp TV bounds obtained by Chak and Monmarché admit a natural interpretation through an explicit non-Markovian coupling, built from an optimal coalescence trajectory characterized by a classical minimum-energy control problem. For the OBABO splitting scheme, this approach additionally eliminates the Hessian-Lipschitz, step-size, and final-time assumptions in the work of Chak and Monmarché.

翻译：针对动能朗之万扩散及其分裂离散化，亚椭圆噪声结构使得耦合与全变差（TV）界之间的关系比椭圆情形更为微妙。我们证明，对于具有二次势能的动能朗之万方程，任何马尔可夫耦合（连续或离散）都无法捕捉两个不同初值解之间TV距离的渐近衰减率；而标准迭代单次（或粘性）耦合（我们为其推导了精确的收缩公式）恰好达到该下界。在建设性方面，我们表明查克和蒙马谢最近获得的尖锐TV界可通过显式非马尔可夫耦合得到自然解释，该耦合基于由经典最小能量控制问题刻画的最优合并轨迹构建。对于OBABO分裂格式，该方法还消除了查克和蒙马谢工作中关于Hessian-Lipschitz条件、步长和最终时间假设的限制。