About the analogy between optimal transport and minimal entropy

Gentil, Ivan; Léonard, Christian; Ripani, Luigia

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 3, pp. 569-600.

Nous décrivons des analogies entre le transport optimal et le problème de Schrödinger lorsque le coût du transport est remplacé par un coût entropique avec une mesure de référence sur les trajectoires. Une formule duale de Kantorovich, une formulation de type Benamou–Brenier du coût entropique sont démontrées, ainsi que des inégalités de contraction par rapport au coût entropique. Cette analogie est aussi illustrée par des exemples numériques où la mesure de référence sur les trajectoires est donnée par le mouvement Brownien ou bien le processus d’Ornstein–Uhlenbeck.

Notre approche s’appuie sur la théorie de la mesure, plutôt que sur le contrôle optimal stochastique, et l’entropie relative joue un rôle fondamental.

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 motion or the Ornstein–Uhlenbeck process.

Our point of view is measure theoretical, rather than based on stochastic optimal control, and the relative entropy with respect to path measures plays a prominent role.

Mots clés : Schrödinger problem, entropic interpolation, Wasserstein distance, Kantorovich duality

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