Mathematics

Aim of the talk is to present two a priori different problems: the optimal transport problem, which is probably better known, and the Schrödinger problem, a quite unknown topic which naturally arises in statistical mechanics. It is an entropy minimization problem and can be regarded as a sort of stochastic generalization/penalization of the former. Particular attention will be put on the strong relationship between the two problems.However, the framework we would like to work within is given by metric measure spaces with nice geometric properties, namely RCD*(K,N) spaces. For this reason, the presentation of the problems will be anticipated by a preliminary part, devoted to the development of first and second order calculus on metric measure spaces and the crucial CD and RCD conditions, due to Lott-Sturm-Villani (2006) and Ambrosio-Gigli-Savaré (2011) respectively.As a last step and as an example of application, we will sketch how to use the tools we will have seen so far and the connection between the two problems in order to get "entropic" approximations of Wasserstein geodesics that enable us to compute in a (suitable) weak sense the second order differentiation formula along Wasserstein geodesics.