Mathematics

After a brief introduction to optimal transport in a smooth setting, we shall present a quite unknown topic, which naturally arises in statistical mechanics: the Schrödinger problem. It is an entropy minimization problem and can be regarded as a sort of stochastic generalization of optimal transport; we will present its strong relationship with the Monge-Kantorovich problem. 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 second part of the talk will be devoted to the development of first 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, we will sketch how to apply the tools we will have seen so far in order to get "entropic" approximations of Wasserstein geodesics that enable us to compute in a (suitable) weak sense the second derivative of the density flow of a Wasserstein geodesic.