We introduce a new probabilistic approach to quantify convergence to
equilibrium for (kinetic) Langevin processes. In contrast to previous analytic
approaches that focus on the associated kinetic Fokker-Planck equation, our
approach is based on a specific combination of reflection and synchronous
coupling of two solutions of the Langevin equation. It yields contractions in a
particular Wasserstein distance, and it provides rather precise bounds for
convergence to equilibrium at the borderline between the overdamped and the
underdamped regime. In particular, we are able to recover kinetic behavior in
terms of explicit lower bounds for the contraction rate. For example, for a
rescaled double-well potential with local minima at distance $a$, we obtain a
lower bound for the contraction rate of order $\Omega (a^{-1})$ provided the
friction coefficient is of order $\Theta (a^{-1})$.

我们引入一种新的概率方法来量化（动力学）Langevin 过程的平衡收敛性，基于随机微分方程的两个解的特定组合，它产生了一种特定的 Wasserstein 距离缩小，并为过阻尼和欠阻尼之间的边界提供相当精确的收敛界限。

Langevin 动力学的耦合和定量收缩率

Couplings and quantitative contraction rates for Langevin dynamics

In this paper, we consider Langevin processes with mechanical constraints.
The latter are a fundamental tool in molecular dynamics simulation for sampling
purposes and for the computation of free energy differences. The results of
this paper can be divided into three parts. (i) We propose a simple
discretization of the constrained Langevin process based on a standard
splitting strategy. We show how to correct the scheme so that it samples {\em
exactly} the canonical measure restricted on a submanifold, using a Metropolis
rule in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm.
Moreover, we obtain, in some limiting regime, a consistent discretization of
the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling
exactly the correct canonical measure with constraints. The corresponding
numerical methods can be used to sample (without any bias) a probability
measure supported by a submanifold. (ii) For free energy computation using
thermodynamic integration, we rigorously prove that the longtime average of the
Lagrange multipliers of the constrained Langevin dynamics yields the gradient
of a rigid version of the free energy associated with the constraints. A second
order time discretization using the Lagrange multipliers is proposed. (iii) The
Jarzynski-Crooks fluctuation relation is proved for Langevin processes with
mechanical constraints evolving in time. An original numerical discretization
without time-step error is proposed. Numerical illustrations are provided for
(ii) and (iii).

本文研究了具有力学约束的 Langevin 过程，提出了离散化方案并在一定情况下证明了其正确性，同时还探讨了在热力学积分、波动关系等方面的应用。