Statistics

Physics

Statistical physics

Internal medicine

Channel (broadcasting)

Mathematical analysis

Contraction (grammar)

Maxima and minima

Langevin equation

Langevin dynamics

Applied mathematics

Computer network

Mathematics

Computer science

Probabilistic logic

Rate of convergence

Medicine

The Annals of Probability

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 (a -1 ) provided the friction coefficient is of order (a -1 ).

Couplings and quantitative contraction rates for Langevin dynamics