Talk by Martin Burger
On July 21st, 2021, Prof. Dr. Martin Burger (Friedrich-Alexander-Universität Erlangen-Nürnberg) gave a talk about "Transport distances and gradient flows on networks" as part of the research seminar Analysis of the FernUniversität in Hagen.
In this talk we study a dynamical optimal transport problem on a network that allows for transport of mass between different edges if some penalty is paid. We show existence of minimisers using duality and discuss the relationships of the transport distance to other metrics such as the Fisher-Rao and the classical Wasserstein metric and analyse the resulting distance functional in limiting cases of the penalty amount. This leads e.g. to certain distances on the graph only. Moreover, we discuss gradient flow structures arising from those distances, which naturally yield action functionals. Finally we provide some examples of interacting systems on networks that can be formulated as such gradient flows by defining suitable energy functionals.