# Dynamic Models of Wasserstein-1-Type Unbalanced Transport

## TL;DR

This paper introduces a dynamic formulation for Wasserstein-1-type unbalanced transport problems, providing an equivalent static model, analyzing optimal solutions, and comparing with existing models in the literature.

## Contribution

It derives a computationally efficient static formulation from dynamic models with transport costs proportional to distance, and analyzes the relationship between dynamic and static models.

## Key findings

- Derived an equivalent static formulation for dynamic unbalanced transport models.
- Analyzed the structure of optimal mass transport and change.
- Compared the proposed models with existing formulations in the literature.

## Abstract

We consider a class of convex optimization problems modelling temporal mass transport and mass change between two given mass distributions (the so-called dynamic formulation of unbalanced transport), where we focus on those models for which transport costs are proportional to transport distance. For those models we derive an equivalent, computationally more efficient static formulation, we perform a detailed analysis of the model optimizers and the associated optimal mass change and transport, and we examine which static models are generated by a corresponding equivalent dynamic one. Alongside we discuss thoroughly how the employed model formulations relate to other formulations found in the literature.