An entropic interpolation proof of the HWI inequality

Ivan Gentil (1) , Christian Léonard (2, 3) , Luigia Ripani (1) , Luca Tamanini (4)

1 EDPA - Équations aux dérivées partielles, analyse
2 MODAL'X - Modélisation aléatoire de Paris X
3 FP2M - Fédération Parisienne de Modélisation Mathématique
4 Institut für Angewandte Mathematik

The HWI inequality is an "interpolation" inequality between the {\it Entropy} $H$, the {\it Fisher information} $I$ and the {\it Wasserstein distance} $W$. We present a pathwise proof of the HWI inequality which is obtained through a zero noise limit of the Schrödinger problem. Our approach consists in making rigorous the Otto-Villani heuristics in [23] taking advantage of the entropic interpolations, which are regular both in space and time, rather than the displacement ones.

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Ivan Gentil, Christian Léonard, Luigia Ripani, Luca Tamanini. An entropic interpolation proof of the HWI inequality. Stochastic Processes and their Applications, 2020, 130 (2), pp.907-923. ⟨10.1016/j.spa.2019.04.002⟩. ⟨hal-01840629v2⟩