Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph
Erbar M, Forkert D, Maas J, Mugnolo D (2022)
Networks and Heterogeneous Media: An Applied Mathematics Journal 17(5): 687-717.
Zeitschriftenaufsatz
| Veröffentlicht | Englisch
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Autor*in
Erbar, MatthiasUniBi;
Forkert, Dominik;
Maas, Jan;
Mugnolo, Delio
Einrichtung
Abstract / Bemerkung
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.
Stichworte
Metric graph;
optimal transport;
gradient flow;
entropy;
McKean- Vlasov;
equation
Erscheinungsjahr
2022
Zeitschriftentitel
Networks and Heterogeneous Media: An Applied Mathematics Journal
Band
17
Ausgabe
5
Seite(n)
687-717
ISSN
1556-1801
eISSN
1556-181X
Page URI
https://pub.uni-bielefeld.de/record/2980855
Zitieren
Erbar M, Forkert D, Maas J, Mugnolo D. Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media: An Applied Mathematics Journal. 2022;17(5):687-717.
Erbar, M., Forkert, D., Maas, J., & Mugnolo, D. (2022). Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media: An Applied Mathematics Journal, 17(5), 687-717. https://doi.org/10.3934/nhm.2022023
Erbar, Matthias, Forkert, Dominik, Maas, Jan, and Mugnolo, Delio. 2022. “Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph”. Networks and Heterogeneous Media: An Applied Mathematics Journal 17 (5): 687-717.
Erbar, M., Forkert, D., Maas, J., and Mugnolo, D. (2022). Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media: An Applied Mathematics Journal 17, 687-717.
Erbar, M., et al., 2022. Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media: An Applied Mathematics Journal, 17(5), p 687-717.
M. Erbar, et al., “Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph”, Networks and Heterogeneous Media: An Applied Mathematics Journal, vol. 17, 2022, pp. 687-717.
Erbar, M., Forkert, D., Maas, J., Mugnolo, D.: Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph. Networks and Heterogeneous Media: An Applied Mathematics Journal. 17, 687-717 (2022).
Erbar, Matthias, Forkert, Dominik, Maas, Jan, and Mugnolo, Delio. “Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph”. Networks and Heterogeneous Media: An Applied Mathematics Journal 17.5 (2022): 687-717.
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