A Survey on dynamical transport distances

In this paper, we review some transport models based on the continuity equation, starting with the so-called Benamou − Brenier formula, which is nothing but a fluid mechanics reformulation of the Monge − Kantorovich problem with cost c(x, y) = |x − y|². We discuss some of its applications (gradient flows, sharp functional inequalities, etc.), as well as some variants and generalizations to dynamical transport problems, where interaction effects among mass particles are considered. Bibliography: 43 titles.