From the Schrödinger problem to the Monge-Kantorovich problem. An entropic approach to optimal transport.

In the early 30's, Schrödinger addressed and solved formally a statistical physics problem which is amazingly analogous to quantum mechanics. It is a large deviation problem which is similar to the Monge-Kantorovich optimal transport problem. This similarity is not incidental. Indeed, it will be shown that the optimal transport problem is the limit of a sequence of well-chosen Schrödinger problems. Analytically, this amounts to describe the optimal transport problem as a Gamma-limit of relative entropy minimization problems under prescribed marginal constraints. The minimizers of these problems might be interpreted as some kind of geodesics in the space of probability measures.