Three superposition principles: currents, continuity equations and curves of measures

TL;DR

This paper develops a general superposition principle for measure curves solving continuity equations in metric spaces, linking them to ODE solutions and extending probabilistic representations in optimal transport.

Contribution

It introduces a broad superposition principle in metric spaces without smooth structures, connecting measure curves, currents, and ODE solutions, and extends existing probabilistic representations.

Findings

Established a superposition principle for measure curves in metric spaces.

Connected measure curves with solutions of ODEs via observables.

Extended probabilistic representations for curves in Wasserstein spaces.

Abstract

We establish a general superposition principle for curves of measures solving a continuity equation on metric spaces without any smooth structure nor underlying measure, representing them as marginals of measures concentrated on the solutions of the associated ODE defined by some algebra of observables. We relate this result with decomposition of acyclic normal currents in metric spaces. As an application, a slightly extended version of a probabilistic representation for absolutely continuous curves in Kantorovich-Wasserstein spaces, originally due to S. Lisini, is provided in the metric framework. This gives a hierarchy of implications between superposition principles for curves of measures and for metric currents.