Séminaire des Doctorants et Doctorantes

From Otto to Schrödinger

by Gauthier Clerc




The optimal transport problem was born in 1781 in the "mémoire sur la théorie des déblais et remblais" of Gaspard Monge. From an heuristic point of view, this problem can be seen as follow : given a pile (déblai) and a hole (remblai), we want to find the optimal way to put the pile in the hole. During the twentieth century, this field received a new formalization in the measure theory framework. From this theory, a distance on the set of all probability measures of a given space was born: the Wasserstein distance. At the very beginning of the twenty first century, Felix Otto had the great idea of seeing the set of probability measures of a given space endowed with the Wasserstein distance as a "weak Riemannian manifold", on an article which gave birth to a theory named Otto calculus. We will see how the Otto calculus leads to the definition of some objects of the differential calculus in the Wasserstein space. At the end, we will see how the Otto calculus can lead us to have a better understanding of the Schrödinger problem, which is an entropic minimisation problem from quantum physics.