COUPLINGS AND QUANTITATIVE CONTRACTION RATES FOR LANGEVIN DYNAMICS

• 成果类型: Article
• 署名作者: Eberle, Andreas; Guillin, Arnaud; Zimmer, Raphael
• 署名单位: University of Bonn; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1299
• 发表日期: 2019
• 页码: 1982-2010
• 关键词: exponential convergence; kinetic-theory; hypocoercivity; equilibrium; EQUATIONS; trend; degenerate; motion

摘要

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)).