  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. 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.
