Coupling, concentration inequalities and stochastic dynamics

Jean René Chazottes, Pierre Collet, Frank Redig

Published 2007-08-16, updated 2015-06-29Version 4

In the context of interacting particle systems, we study the influence of the action of the semigroup on the concentration property of Lipschitz functions. As an application, this gives a new approach to estimate the relaxation speed to equilibrium of interacting particle systems. We illustrate our approach in a variety of examples for which we obtain several new results with short and non-technical proofs. These examples include the symmetric and asymmetric exclusion process and high-temperature spin-flip dynamics ("Glauber dynamics"). We also give a new proof of the Poincar\'e inequality, based on coupling, in the context of one-dimensional Gibbs measures. In particular, we cover the case of polynomially decaying potentials, where the log-Sobolev inequality does not hold.