Transfunctions applied to plans, Markov operators and optimal transport
Abstract
A transfunction is a function which maps between sets of finite measures on measurable spaces. In this paper we characterize transfunctions that correspond to Markov operators and to plans; such a transfunction will contain the “instructions” common to several Markov operators and plans. We also define the adjoint of transfunctions in two settings and provide conditions for existence of adjoints. Finally, we develop approximations of identity in each setting and use them to approximate weakly-continuous transfunctions with simple transfunctions; one of these results can be applied to some optimal transport problems to approximate the optimal cost with simple Markov transfunctions.
Keywords
Markov operator; plan; Radon adjoint; transfunction; transport
References
L. Ambrosio, Optimal transport maps in Monge–Kantorovich problem, in: T. Li (ed.), Proceedings of the International Congress of Mathematicians. Vol. III, Higher Education Press, Beijing, 2002, pp. 131–140.
J. Bentley, Construction of regular non-atomic strictly-positive measures in secondcountable non-atomic locally compact Hausdorff spaces, Ann. Math. Sil. 36 (2022), no. 1, 15–25.
J. Bentley and P. Mikusiński, Localized transfunctions, Int. J. Appl. Math. 31 (2018), no. 6, 689–707.
J.R. Brown, Approximation theorems for Markov operators, Pacific J. Math. 16 (1966), no. 1, 13–23.
S. Campi, P. Gronchi, and P. Salani, A proof of a Loomis–Whitney type inequality via optimal transport, J. Math. Anal. Appl. 471 (2019), no. 1–2, 489–495.
P. Jiménez Guerra and B. Rodríguez-Salinas, A general solution of the Monge–Kantorovich mass-transfer problem, J. Math. Anal. Appl. 202 (1996), no. 2, 492–510.
H.W. Kuhn, The Hungarian method for the assignment problem, Naval Res. Logist. Quart. 2 (1955), 83–97.
P. Mikusiński, Transfunctions, arXiv preprint, 2015. Available at arXiv: 1507.03441.
P. Mikusiński and M.D. Taylor, Markov operators and n-copulas, Ann. Polon. Math. 96 (2009), no. 1, 75–95.
C. Villani, Optimal Transport: Old and New, Springer-Verlag, Berlin-Heidelberg, 2009.
Department of Mathematics, University of Central Florida United States
https://orcid.org/0000-0002-0858-2178
Department of Mathematics, University of Central Florida United States
https://orcid.org/0000-0002-1890-8039
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
