  In his work about hypocercivity, Villani [18] considers in particular
convergence to equilibrium for the kinetic Langevin process. While his
convergence results in L 2 are given in a quite general setting, convergence in
entropy requires some boundedness condition on the Hessian of the Hamiltonian.
We will show here how to get rid of this assumption in the study of the
hypocoercive entropic relaxation to equilibrium for the Langevin diffusion. Our
method relies on a generalization to entropy of the multipliers method and an
adequate functional inequality. As a byproduct, we also give tractable
conditions for this functional inequality, which is a particular instance of a
weighted logarithmic Sobolev inequality, to hold.
