The Annals of Probability

Couplings and quantitative contraction rates for Langevin dynamics

Andreas Eberle, Arnaud Guillin, and Raphael Zimmer

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

Article information

Ann. Probab., Volume 47, Number 4 (2019), 1982-2010.

Received: March 2017
Revised: June 2018
First available in Project Euclid: 4 July 2019

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60H10: Stochastic ordinary differential equations [See also 34F05] 35Q84: Fokker-Planck equations 35B40: Asymptotic behavior of solutions

Langevin diffusion kinetic Fokker–Planck equation stochastic Hamiltonian dynamics reflection coupling convergence to equilibrium hypocoercivity quantitative bounds Wasserstein distance Lyapunov functions


Eberle, Andreas; Guillin, Arnaud; Zimmer, Raphael. Couplings and quantitative contraction rates for Langevin dynamics. Ann. Probab. 47 (2019), no. 4, 1982--2010. doi:10.1214/18-AOP1299.

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