Schrödinger dynamics and optimal transport of measures on the torus

Abstract

The aim of this paper is to recover displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by semiclassical measures associated with solutions of Schrödinger's equations defined on the flat torus. Under an additional assumption, we show the completing viewpoint by proving that a family of displacement interpolations can always be viewed as these time dependent semiclassical measures.