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

Matthias Erbar; Dominik Forkert; Jan Maas; Delio Mugnolo

doi:10.3934/nhm.2022023

Article Contents

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

This issue Previous Article On the complete aggregation of the Wigner-Lohe model for identical potentials Next Article Martingale solutions of stochastic nonlocal cross-diffusion systems

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

^*Corresponding author: Jan Maas
^*Corresponding author: Jan Maas

Received: May 2021

Revised: April 2022

Early access: June 2022

Published: October 2022

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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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Mathematics Subject Classification: Primary: 35R02; Secondary: 49Q22, 60B05.

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Figure 1. The supergraph $ {{{\mathfrak G}}_{{\operatorname{ext}}}} $ is constructed by adjoining an additional leaf at every node in $ {V} $

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Figure 2. The support of probability measures $ {\mu} $ and $ {\nu} $ on a metric graph induced by a oriented star with 3 leaves

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Figure 3. Plot of the entropy along the geodesic interpolation

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