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

@article{Erbar2022GradientFF,
  title={Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph},
  author={Matthias Erbar and Dominik Forkert and Jan Maas and Delio Mugnolo},
  journal={Networks and Heterogeneous Media},
  year={2022}
}
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… 
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