About the analogy between optimal transport and minimal entropy

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

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

We describe some analogy between optimal transport and the Schrödinger problem where the transport cost is replaced by an entropic cost with a reference path measure. A dual Kantorovich type formulation and a Benamou-Brenier type representation formula of the entropic cost are derived, as well as contraction inequalities with respect to the entropic cost. This analogy is also illustrated with some numerical examples where the reference path measure is given by the Brownian or the Ornstein-Uhlenbeck process. Our point of view is measure theoretical and the relative entropy with respect to path measures plays a prominent role.

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

Ivan Gentil, Christian Léonard, Luigia Ripani. About the analogy between optimal transport and minimal entropy. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2017, 26, 3, pp.569-600. ⟨hal-01221325v2⟩