The Schrdinger problem is an entropic minimization problem and its a regular approximation of the Monge-Kantorovich problem, at the core of the Optimal Transport theory.

In this talk I will first introduce the two problems, then I will describe some analogy between optimal transport and the Schrdinger problem such as a dual Kantorovich type formulation, the dynamical Benamou-Brenier type representation formula, as well as a characterization formula and some properties of the respective solutions.

Finally I will mention, as an application of these analogies, some contraction inequalities with respect to the entropic cost, instead of the classical Wasserstein distance.