Time-periodic branched transport

TL;DR

This paper introduces a novel framework for modeling time-varying branched transport problems, capturing cyclic changes in source and target distributions with a focus on transportation networks like roads or circulatory systems.

Contribution

It develops a new concept of time-dependent transport paths, defines associated energies and distances, and establishes the existence of optimal paths and a metric structure on measure spaces.

Findings

Existence of energy-minimizing transport paths over time

Definition of a new metric structure for time-varying measures

Application to cyclic transportation networks

Abstract

We develop a new framework for branched transport between probability measures which are allowed to vary in time. This framework can be used to model problems where the underlying transportation network displays a branched structure, but the source and target mass distributions can change cyclically over time, such as road networks or circulatory systems. We introduce the notion of time-dependent transport paths along with associated energies and distances, and prove existence of transport paths whose energy achieves the distance. We also show the time-dependent transport yields a metric structure on subsets of appropriately defined measure-valued Sobolev spaces.