A probabilistic approach to convex (ϕ)-entropy decay for Markov chains

We study the exponential dissipation of entropic functionals along the semigroup generated by a continuous time Markov chain and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry-Emery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. In particular, the validity of the method is not limited to the perturbative setting and we establish convex entropy decay for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we show that the exponential contraction of the Wasserstein distance implies MLSI. We also revisit classical examples often obtaining new inequalities and sometimes improving on the best known constants. In particular, we analyse the zero range dynamics, hardcore and Bernoulli-Laplace models and the Glauber dynamics for the Curie-Weiss and Ising model.