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July 2019 Couplings and quantitative contraction rates for Langevin dynamics
Andreas Eberle, Arnaud Guillin, Raphael Zimmer
Ann. Probab. 47(4): 1982-2010 (July 2019). DOI: 10.1214/18-AOP1299

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

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Andreas Eberle. Arnaud Guillin. Raphael Zimmer. "Couplings and quantitative contraction rates for Langevin dynamics." Ann. Probab. 47 (4) 1982 - 2010, July 2019. https://doi.org/10.1214/18-AOP1299

Information

Received: 1 March 2017; Revised: 1 June 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114709
MathSciNet: MR3980913
Digital Object Identifier: 10.1214/18-AOP1299

Subjects:
Primary: 35B40, 35Q84, 60H10, 60J60

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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