Schr\"odinger dynamics and optimal transport of measures on the torus

TL;DR

This paper explores how Schr"odinger equations on the torus can be used to recover and interpret displacement interpolations of probability measures through semiclassical measures, linking quantum dynamics with optimal transport.

Contribution

It establishes a novel connection between Schr"odinger dynamics and optimal transport measures on the torus, showing that displacement interpolations can be represented as semiclassical measures.

Findings

Displacement interpolations can be recovered via semiclassical measures.

Under certain assumptions, all displacement interpolations correspond to these measures.

The work bridges quantum dynamics and optimal transport theory.

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 Schr\"odinger 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.