Geodesics for a class of distances in the space of probability measures
  In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savar e. We first prove the existence of a potential function and then give necessary and suffi cient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves. 
Pierre Cardaliaguet (CEREMADE), Guillaume Carlier (CEREMADE), Bruno   Nazaret (CEREMADE)
math.OC math.AP