Minimal Measures, one dimensional currents and the Monge-Kantorovich Problem

In recent works L.C. Evans has noticed a strong analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the theory devel- oped in the last years for the optimal mass transportation (or Monge–Kantorovich) problem. In this paper we start to investigate this analogy by proving that to each minimal measure it is possible to associate, in a natural way, a family of curves on the space of probability measures. These curves are absolutely continuous with respect to the metric structure related to the optimal mass transportation problem. Some minimality properties of such curves are also addressed.