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 $\Theta (a^{-1})$.

References in corpus (2)

• Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation
• Hilbert space hypocoercivity for the Langevin dynamics revisited

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• Random Coordinate Underdamped Langevin Monte Carlo
• Well-posedness and long time behavior of singular Langevin stochastic differential equations
• Fast Convergence of Langevin Dynamics on Manifold: Geodesics meet Log-Sobolev