A continuous model of transportation revisited

TL;DR

This paper compares and unifies two continuous models of optimal transport with congestion effects, demonstrating their equivalence through advanced mathematical tools and extending their frameworks.

Contribution

It provides a comprehensive analysis showing the equivalence of Beckmann's and Carlier et al.'s models in a functional analytic setting, using Smirnov's decomposition theorem.

Findings

Models are equivalent in a functional analytic framework

Extended models to natural functional spaces

Used Smirnov decomposition to establish equivalence

Abstract

We review two models of optimal transport, where congestion effects during the transport can be possibly taken into account. The first model is Beckmann's one, where the transport activities are modeled by vector fields with given divergence. The second one is the model by Carlier et al. (SIAM J Control Optim 47: 1330-1350, 2008), which in turn is the continuous reformulation of Wardrop's model on graphs. We discuss the extensions of these models to their natural functional analytic setting and show that they are indeed equivalent, by using Smirnov decomposition theorem for normal 1-currents.