Invariance properties of the Monge-Kantorovich mass transport problem

Abbas  Moameni

doi:10.3934/dcds.2016.36.2653

Article Contents

2016, Volume 36, Issue 5: 2653-2671. Doi: 10.3934/dcds.2016.36.2653

This issue Previous Article Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces Next Article The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions

Invariance properties of the Monge-Kantorovich mass transport problem

Abbas Moameni^1,

1.
School of Mathematics and Statistics, Carleton University, Ottawa, K1S5B6

Received: March 31, 2015

Revised: June 30, 2015

Published: May 2017

Abstract Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 49Q20, 49K30; Secondary: 91B24.

Citation:

Related Papers

Cited by

References

[1]	M. Beiglböck and W. Schachermayer, Duality for Borel measurable cost functions, Trans. Amer. Math. Soc., 363 (2011), 4203-4224.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), 062502.doi: 10.1103/PhysRevA.85.062502.
[3]	G. Carlier, On a class of multidimensional optimal transportation problems, J. Convex Anal., 10 (2003), 517-529.
[4]	G. Carlier and B. Nazaret, Optimal transportation for the determinant, ESAIM Control Optim. Calc., 14 (2008), 678-698.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), 317-354.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), 350-368.doi: 10.4153/CJM-2014-011-x.
[7]	W. Gangbo and R. J. McCann, The geometry of optimal transportation, Acta Math., 177 (1996), 113-161.
[8]	W. Gangbo and A. Swiech, Optimal maps for the multidimensional Monge-Kantorovich problem, Comm. Pure Appl. Math., 51 (1998), 23-45.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), 905-933.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), 1129-1166.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), 60-118.doi: 10.1006/jeth.1999.2540.
[12]	H. Heinich, Probleme de Monge pour n probabilities, C.R. Math. Acad. Sci. Paris, 334 (2002), 793-795.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), 1538-1550.doi: 10.1137/130930443.
[14]	V. Levin, Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem, Set-Valued Analysis, 7 (1999), 7-32.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), 269-289.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), 2758-2775.doi: 10.1137/100804917.
[17]	B. Pass, Remarks on the semi-classical Hohenberg-Kohn functional, Nonlinearity, 26 (2013), 2731-2744.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), 647-676.
[19]	S. T. Rachev and L. Rüchendorf, Mass Transportation Problems. Vol. I. Theory, Probability and its Applications (New York), Springer-Verlag, New York, 1998.
[20]	C. Villani, Optimal Transport, Old and New, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2009.doi: 10.1007/978-3-540-71050-9.