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.Résumé. 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.

The defining equation ( * ) :ω t = −F (ω t ), of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation ( * ) into the family of slowed down gradient flow equations:ω ε t = −εF (ω ε t ), where ε > 0, and (ii) by considering the accelerationsω ε t . We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying Schrödinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero.We show heuristically that the solutions satisfy a Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequality under curvaturedimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.

We present a pathwise proof of the HWI inequality which is based on entropic interpolations rather than displacement ones. Unlike the latter, entropic interpolations are regular both in space and time. Consequently, our approach is closer to the Otto-Villani heuristics, presented in the first part of the article [23], than the original rigorous proof presented in the second part of [23].Ric V,N := Ric g − (N − n)Hess(1.4)Relative entropy. For any two probability measures p and r on a measurable space Z the relative entropy of p with respect to r is defined bywhere it is understood that this quantity is infinite when p is not absolutely continuous with respect to r. In our case, Z will be X, X × X or C([0, 1], X).