Entropic multipliers method for langevin diffusion and weighted log sobolev inequalities

Patrick Cattiaux (1) , Arnaud Guillin (2) , Pierre Monmarché (3, 4) , Chaoen Zhang (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 UPEM - Université Paris-Est Marne-la-Vallée
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

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.

Mots clés: Langevin diffusion; entropic convergence; hypocercivity; weighted logarithmic Sobolev inequality; Lyapunov conditions

Domaines: Probabilités [math.PR]; Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Patrick Cattiaux, Arnaud Guillin, Pierre Monmarché, Chaoen Zhang. Entropic multipliers method for langevin diffusion and weighted log sobolev inequalities. Journal of Functional Analysis, 2019, 277 (11), ⟨10.1016/j.jfa.2019.108288⟩. ⟨hal-01571610⟩