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May  2016, 36(5): 2653-2671. doi: 10.3934/dcds.2016.36.2653

Invariance properties of the Monge-Kantorovich mass transport problem

1. 

School of Mathematics and Statistics, Carleton University, Ottawa, K1S5B6, Canada

Received  April 2015 Revised  July 2015 Published  October 2015

We consider the multimarginal Monge-Kantorovich transport problem in an abstract setting. Our main results state that if a cost function and marginal measures are invariant by a family of transformations, then a solution of the Kantorovich relaxation problem and a solution of its dual can be chosen so that they are invariant under the same family of transformations. This provides a new tool to study and analyze the support of optimal transport plans and consequently to scrutinize the Monge problem. Birkhoff's Ergodic theorem is an essential tool in our analysis.
Citation: Abbas Moameni. Invariance properties of the Monge-Kantorovich mass transport problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2653-2671. doi: 10.3934/dcds.2016.36.2653
References:
[1]

M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions,, Trans. Amer. Math. Soc., 363 (2011), 4203. doi: 10.1090/S0002-9947-2011-05174-3.

[2]

G. Buttazzo, L. De Pascale and P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory,, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.062502.

[3]

G. Carlier, On a class of multidimensional optimal transportation problems,, J. Convex Anal., 10 (2003), 517.

[4]

G. Carlier and B. Nazaret, Optimal transportation for the determinant,, ESAIM Control Optim. Calc., 14 (2008), 678. doi: 10.1051/cocv:2008006.

[5]

P.-A. Chiappori, R. J. McCann and L. P. Nesheim, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness,, Econom. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z.

[6]

M. Colombo, L. De Pascale and S. Di Marino, Mutlimarginal optimal transport maps for 1-dimensional repulsive costs,, Canad. J. Math., 67 (2015), 350. doi: 10.4153/CJM-2014-011-x.

[7]

W. Gangbo and R. J. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113.

[8]

W. Gangbo and A. Swiech, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.

[9]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields,, Comm. Pure Appl. Math., 66 (2013), 905. doi: 10.1002/cpa.21430.

[10]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields,, Geom. Funct. Anal., 24 (2014), 1129. doi: 10.1007/s00039-014-0287-2.

[11]

N. E. Gretsky, J. M. Ostroy and W. R. Zame, Perfect competition in the continuous assignment model,, J. Econ. Theory, 88 (1999), 60. doi: 10.1006/jeth.1999.2540.

[12]

H. Heinich, Probleme de Monge pour n probabilities,, C.R. Math. Acad. Sci. Paris, 334 (2002), 793. doi: 10.1016/S1631-073X(02)02341-5.

[13]

Y.-H. Kim and B. Pass, A general condition for Monge solutions in the multi-marginal optimal transport problem,, SIAM J. Math. Anal., 46 (2014), 1538. doi: 10.1137/130930443.

[14]

V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,, Set-Valued Analysis, 7 (1999), 7. doi: 10.1023/A:1008753021652.

[15]

G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna,, Arch. Ration. Mech. Anal., 199 (2011), 269. doi: 10.1007/s00205-010-0330-x.

[16]

B. Pass, Uniqueness and monge solutions in the multimarginal optimal transportation problem,, SIAM J. Math. Anal., 43 (2011), 2758. doi: 10.1137/100804917.

[17]

B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional,, Nonlinearity, 26 (2013), 2731. doi: 10.1088/0951-7715/26/9/2731.

[18]

S. T. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications,, Theory Probab. Appl., 29 (1985), 647.

[19]

S. T. Rachev and L. Rüchendorf, Mass Transportation Problems. Vol. I. Theory,, Probability and its Applications (New York), (1998).

[20]

C. Villani, Optimal Transport, Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9.

show all references

References:
[1]

M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions,, Trans. Amer. Math. Soc., 363 (2011), 4203. doi: 10.1090/S0002-9947-2011-05174-3.

[2]

G. Buttazzo, L. De Pascale and P. Gori-Giorgi, Optimal-transport formulation of electronic density-functional theory,, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.062502.

[3]

G. Carlier, On a class of multidimensional optimal transportation problems,, J. Convex Anal., 10 (2003), 517.

[4]

G. Carlier and B. Nazaret, Optimal transportation for the determinant,, ESAIM Control Optim. Calc., 14 (2008), 678. doi: 10.1051/cocv:2008006.

[5]

P.-A. Chiappori, R. J. McCann and L. P. Nesheim, Hedonic price equilibria, stable matching, and optimal transport: Equivalence, topology, and uniqueness,, Econom. Theory, 42 (2010), 317. doi: 10.1007/s00199-009-0455-z.

[6]

M. Colombo, L. De Pascale and S. Di Marino, Mutlimarginal optimal transport maps for 1-dimensional repulsive costs,, Canad. J. Math., 67 (2015), 350. doi: 10.4153/CJM-2014-011-x.

[7]

W. Gangbo and R. J. McCann, The geometry of optimal transportation,, Acta Math., 177 (1996), 113.

[8]

W. Gangbo and A. Swiech, Optimal maps for the multidimensional Monge-Kantorovich problem,, Comm. Pure Appl. Math., 51 (1998), 23. doi: 10.1002/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H.

[9]

N. Ghoussoub and A. Moameni, A self-dual polar factorization for vector fields,, Comm. Pure Appl. Math., 66 (2013), 905. doi: 10.1002/cpa.21430.

[10]

N. Ghoussoub and A. Moameni, Symmetric Monge-Kantorovich problems and polar decompositions of vector fields,, Geom. Funct. Anal., 24 (2014), 1129. doi: 10.1007/s00039-014-0287-2.

[11]

N. E. Gretsky, J. M. Ostroy and W. R. Zame, Perfect competition in the continuous assignment model,, J. Econ. Theory, 88 (1999), 60. doi: 10.1006/jeth.1999.2540.

[12]

H. Heinich, Probleme de Monge pour n probabilities,, C.R. Math. Acad. Sci. Paris, 334 (2002), 793. doi: 10.1016/S1631-073X(02)02341-5.

[13]

Y.-H. Kim and B. Pass, A general condition for Monge solutions in the multi-marginal optimal transport problem,, SIAM J. Math. Anal., 46 (2014), 1538. doi: 10.1137/130930443.

[14]

V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem,, Set-Valued Analysis, 7 (1999), 7. doi: 10.1023/A:1008753021652.

[15]

G. Loeper, Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna,, Arch. Ration. Mech. Anal., 199 (2011), 269. doi: 10.1007/s00205-010-0330-x.

[16]

B. Pass, Uniqueness and monge solutions in the multimarginal optimal transportation problem,, SIAM J. Math. Anal., 43 (2011), 2758. doi: 10.1137/100804917.

[17]

B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional,, Nonlinearity, 26 (2013), 2731. doi: 10.1088/0951-7715/26/9/2731.

[18]

S. T. Rachev, The Monge-Kantorovich mass transference problem and its stochastic applications,, Theory Probab. Appl., 29 (1985), 647.

[19]

S. T. Rachev and L. Rüchendorf, Mass Transportation Problems. Vol. I. Theory,, Probability and its Applications (New York), (1998).

[20]

C. Villani, Optimal Transport, Old and New,, Grundlehren der Mathematischen Wissenschaften, (2009). doi: 10.1007/978-3-540-71050-9.

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