Geodesics for a class of distances in the space of probability measures

Pierre Cardaliaguet (1) , Guillaume Carlier (1) , Bruno Nazaret (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: dynamical transport distances; power mobility; geodesics in the space of probability measures; optimality conditions

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Pierre Cardaliaguet, Guillaume Carlier, Bruno Nazaret. Geodesics for a class of distances in the space of probability measures. Calculus of Variations and Partial Differential Equations, 2013, 48 (3), pp.395-420. ⟨hal-00686908⟩