October  2022, 17(5): 687-717. doi: 10.3934/nhm.2022023

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

Received  May 2021 Revised  April 2022 Published  October 2022 Early access  June 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 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.

Citation: Matthias Erbar, Dominik Forkert, Jan Maas, Delio Mugnolo. Gradient flow formulation of diffusion equations in the Wasserstein space over a Metric graph. Networks and Heterogeneous Media, 2022, 17 (5) : 687-717. doi: 10.3934/nhm.2022023
References:
[1]

L. Ambrosio and N. Gigli, A user's guide to optimal transport, Modelling and Optimisation of Flows on Networks, 2062 (2013), 1-155.  doi: 10.1007/978-3-642-32160-3_1.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Springer Science & Business Media, 2008.

[3]

L. AmbrosioN. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391.  doi: 10.1007/s00222-013-0456-1.

[4]

J.-D. Benamou and Y. Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, Monge Ampére Equation: Applications to Geometry and Optimization, 226 (1999), 1-12.  doi: 10.1090/conm/226/03232.

[5]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[6]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013.

[7]

M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks, Models and theory. Lecture Notes in Mathematics, 1955. Springer-Verlag, Berlin, 2009.

[8]

V. I. Bogachev, Measure Theory, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[9]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.

[10]

M. Burger, I. Humpert and J.-F. Pietschmann, Dynamic optimal transport on networks, arXiv: 2101.03415, 2021.

[11]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[12]

D. Cordero-ErausquinR. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257.  doi: 10.1007/s002220100160.

[13]

N. Gigli and B.-X. Han, The continuity equation on metric measure spaces, Calc. Var. Partial Differential Equations, 53 (2015), 149-177.  doi: 10.1007/s00526-014-0744-7.

[14]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[15]

M. Kramar FijavžD. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.  doi: 10.1007/s00245-006-0887-9.

[16]

P. Kuchment, Quantum graphs: An introduction and a brief survey, Analysis on Graphs and Its Applications, 77 (2008), 291-312.  doi: 10.1090/pspum/077/2459876.

[17]

M. LieroA. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Invent. Math., 211 (2018), 969-1117.  doi: 10.1007/s00222-017-0759-8.

[18]

G. Lumer, Connecting of local operators and evolution equations on networks, Potential Theory (Proc. Copenhagen 1979), 234 (1980), 230-243. 

[19]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292.  doi: 10.1016/j.jfa.2011.06.009.

[20]

J. M. MazónJ. D. Rossi and J. Toledo, Optimal mass transport on metric graphs, SIAM J. Optim., 25 (2015), 1609-1632.  doi: 10.1137/140995611.

[21]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.

[22]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Networks Het. Media, 2 (2007), 55-79.  doi: 10.3934/nhm.2007.2.55.

[23]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems. Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[24]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.  doi: 10.1006/jfan.1999.3557.

[25]

M.-K. v. Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[26]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[27]

C. Villani, Optimal Transport: Old and New, volume 338., Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[28]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 2003,251–279. doi: 10.1142/S021919970300094X.

show all references

References:
[1]

L. Ambrosio and N. Gigli, A user's guide to optimal transport, Modelling and Optimisation of Flows on Networks, 2062 (2013), 1-155.  doi: 10.1007/978-3-642-32160-3_1.

[2]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Springer Science & Business Media, 2008.

[3]

L. AmbrosioN. Gigli and G. Savaré, Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., 195 (2014), 289-391.  doi: 10.1007/s00222-013-0456-1.

[4]

J.-D. Benamou and Y. Brenier, A numerical method for the optimal time-continuous mass transport problem and related problems, Monge Ampére Equation: Applications to Geometry and Optimization, 226 (1999), 1-12.  doi: 10.1090/conm/226/03232.

[5]

J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393.  doi: 10.1007/s002110050002.

[6]

G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, 186. American Mathematical Society, Providence, RI, 2013.

[7]

M. Bernot, V. Caselles and J.-M. Morel, Optimal Transportation Networks, Models and theory. Lecture Notes in Mathematics, 1955. Springer-Verlag, Berlin, 2009.

[8]

V. I. Bogachev, Measure Theory, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.

[9]

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry, Studies in Mathematics, 33. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/033.

[10]

M. Burger, I. Humpert and J.-F. Pietschmann, Dynamic optimal transport on networks, arXiv: 2101.03415, 2021.

[11]

S.-N. ChowW. HuangY. Li and H. Zhou, Fokker-Planck equations for a free energy functional or Markov process on a graph, Arch. Ration. Mech. Anal., 203 (2012), 969-1008.  doi: 10.1007/s00205-011-0471-6.

[12]

D. Cordero-ErausquinR. J. McCann and M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., 146 (2001), 219-257.  doi: 10.1007/s002220100160.

[13]

N. Gigli and B.-X. Han, The continuity equation on metric measure spaces, Calc. Var. Partial Differential Equations, 53 (2015), 149-177.  doi: 10.1007/s00526-014-0744-7.

[14]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker–Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.

[15]

M. Kramar FijavžD. Mugnolo and E. Sikolya, Variational and semigroup methods for waves and diffusion in networks, Appl. Math. Optim., 55 (2007), 219-240.  doi: 10.1007/s00245-006-0887-9.

[16]

P. Kuchment, Quantum graphs: An introduction and a brief survey, Analysis on Graphs and Its Applications, 77 (2008), 291-312.  doi: 10.1090/pspum/077/2459876.

[17]

M. LieroA. Mielke and G. Savaré, Optimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures, Invent. Math., 211 (2018), 969-1117.  doi: 10.1007/s00222-017-0759-8.

[18]

G. Lumer, Connecting of local operators and evolution equations on networks, Potential Theory (Proc. Copenhagen 1979), 234 (1980), 230-243. 

[19]

J. Maas, Gradient flows of the entropy for finite Markov chains, J. Funct. Anal., 261 (2011), 2250-2292.  doi: 10.1016/j.jfa.2011.06.009.

[20]

J. M. MazónJ. D. Rossi and J. Toledo, Optimal mass transport on metric graphs, SIAM J. Optim., 25 (2015), 1609-1632.  doi: 10.1137/140995611.

[21]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.

[22]

D. Mugnolo, Gaussian estimates for a heat equation on a network, Networks Het. Media, 2 (2007), 55-79.  doi: 10.3934/nhm.2007.2.55.

[23]

D. Mugnolo, Semigroup Methods for Evolution Equations on Networks, Understanding Complex Systems. Springer, Cham, 2014. doi: 10.1007/978-3-319-04621-1.

[24]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400.  doi: 10.1006/jfan.1999.3557.

[25]

M.-K. v. Renesse and K.-T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl. Math., 58 (2005), 923-940.  doi: 10.1002/cpa.20060.

[26]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2.

[27]

C. Villani, Optimal Transport: Old and New, volume 338., Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[28]

Q. Xia, Optimal paths related to transport problems, Commun. Contemp. Math., 5 2003,251–279. doi: 10.1142/S021919970300094X.

Figure 1.  The supergraph $ {{{\mathfrak G}}_{{\operatorname{ext}}}} $ is constructed by adjoining an additional leaf at every node in $ {V} $
Figure 2.  The support of probability measures $ {\mu} $ and $ {\nu} $ on a metric graph induced by a oriented star with 3 leaves
Figure 3.  Plot of the entropy along the geodesic interpolation
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