On mathcal H¹ and entropic convergence for contractive PDMP

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincar\'e (and more generaly Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is developped.