Minimal Action on the Space of Measures and Hamilton-Jacobi Equations

We study the minimal action induced by a convex Lagrangian on the torus in the space of prob-ability measures, in connection with the viscosity solutions of a pair of corresponding conjugate time-dependent Hamilton Jacobi equations. To this scope, we prove a version of Kantorovich dual-ity theorem in the aforementioned framework through an argument that, as far as we can judge, is simple and new. It is based on the construction of approximate linear optimization problems posed in a space of matrices and a passage at the limit exploiting the density property of the convex hull of Dirac measures in the space of the probability measures on the torus with the narrow topology. In the proof it is solely used the finite dimensional version of Hahn-Banach theorem.