Hypocoercive relaxation to equilibrium for some kinetic models via a third order differential inequality

This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincar\'e's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of Ft (the L2 norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that Ft satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.