Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph

Matthias Erbar; Dominik Forkert; Jan Maas; Delio Mugnolo; Matthias Erbar; Dominik Forkert; Jan Maas; Delio Mugnolo

doi:10.3934/nhm.2022023

Networks and Heterogeneous Media

2022, Volume 17, Issue 5: 687-717. doi: 10.3934/nhm.2022023

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Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph

1.
Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501 Bielefeld, Germany
2.
Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria
3.
FernUniversität in Hagen, Lehrgebiet Analysis, Fakultät Mathematik und Informatik, D-58084 Hagen, Germany

Received: 01 May 2021 Revised: 01 April 2022 Published: 16 June 2022
Primary: 35R02; Secondary: 49Q22, 60B05

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.
- Metric graph,
- optimal transport,
- gradient flow,
- entropy,
- McKean–Vlasov equation
Citation: Matthias Erbar, Dominik Forkert, Jan Maas, Delio Mugnolo. Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph[J]. Networks and Heterogeneous Media, 2022, 17(5): 687-717. doi: 10.3934/nhm.2022023

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Abstract

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.

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