COUPLINGS AND QUANTITATIVE CONTRACTION RATES FOR LANGEVIN DYNAMICS

被引:102
作者
Eberle, Andreas [1 ]
Guillin, Arnaud [2 ]
Zimmer, Raphael [1 ]
机构
[1] Univ Bonn, Inst Angew Math, Endenicher Allee 60, D-53115 Bonn, Germany
[2] Univ Clermont Auvergne, Lab Math Blaise Pascal, CNRS UMR 6620, Ave Landais, F-63177 Aubiere, France
关键词
Langevin diffusion; kinetic Fokker-Planck equation; stochastic Hamiltonian dynamics; reflection coupling; convergence to equilibrium; hypocoercivity; quantitative bounds; Wasserstein distance; Lyapunov functions; EXPONENTIAL CONVERGENCE; KINETIC-THEORY; HYPOCOERCIVITY; EQUILIBRIUM; EQUATIONS; TREND; DEGENERATE; MOTION;
D O I
10.1214/18-AOP1299
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
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)).
引用
收藏
页码:1982 / 2010
页数:29
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