  We develop a method, based on a Bochner-type identity, to obtain estimates on
the exponential rate of decay of the relative entropy from equilibrium of
Markov processes in discrete settings. When this method applies the relative
entropy decays in a convex way. The method is shown to be rather powerful when
applied to a class of birth and death processes. We then consider other
examples, including inhomogeneous zero-range processes and Bernoulli-Laplace
models. For these two models, known results were limited to the homogeneous
case, and obtained via the martingale approach, whose applicability to
inhomogeneous models is still unclear.
