Title: Couplings and quantitative contraction rates for Langevin dynamics,
Abstract: 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 $\Theta (a^{-1})$.