Entropic multipliers method for Langevin diffusion and weighted log Sobolev inequalities

Abstract

In his work about hypocoercivity, Villani [20] considers in particular convergence to equilibrium for the kinetic Langevin process. While his convergence results in L2 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.