# On couplings for kinetic Langevin diffusions

arXiv: https://arxiv.org/abs/2605.31088 · HTML: https://mlsanity.com/paper/2605.31088/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 47 references that are in the map.
- First posted on arXiv 29 May 2026.

## Builds on (most cited first, 2 of 2)

- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Tail-Sensitive KL and R'enyi Convergence of Unadjusted Hamiltonian Monte Carlo via One-Shot Couplings](https://mlsanity.com/paper/2601.09019/) (2026)

## Built on by (most cited first, 1 of 1)

- [Provable Non-Acceleration of Standard Strang Splittings of Kinetic Langevin Dynamics](https://mlsanity.com/paper/2608.25279/) (2026)

## Abstract (arXiv)

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 Monmarch'e 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 Monmarch'e.