A probabilistic approach to convex $(\phi)$-entropy decay for Markov chains

Understand

We study the exponential dissipation of entropic functionals for continuous time Markov chains 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.
• The method is well suited to work in a non perturbative setting and we obtain new estimates for interacting random walks beyond the high temperature/weak interaction regime.
• In this framework, we also show that the exponential contraction of the Wasserstein distance along the semigroup implies MLSI.