On couplings for kinetic Langevin diffusions

2026-05-29

AI Summary

This work investigates the impact of coupling strategies on the total variation (TV) distance decay rate in kinetic Langevin diffusion, particularly highlighting limitations under non-elliptic noise. Addressing the inadequacy of Markovian couplings in accurately capturing TV convergence rates, the authors construct an explicit non-Markovian coupling that achieves optimal contraction via optimal co-trajectory control and a minimal energy principle. By integrating the OBABO splitting scheme, sticky coupling, and refined TV analysis techniques, they derive an exact contraction formula. This result not only provides a natural explanation for the sharp TV bounds established by Chak and Monmarché but also eliminates dependencies on Hessian-Lipschitz regularity, step size restrictions, and final time horizons.

Abstract

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.

Problem

• kinetic Langevin diffusion
• coupling
• total variation distance
• hypoelliptic noise
• Markovian coupling

Innovation

• non-Markovian coupling
• kinetic Langevin diffusion
• total variation bounds
• optimal coalescence trajectory
• minimum-energy control