Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Simultaneous optimal transport
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by Ruodu Wang and Zhenyuan Zhang;
Trans. Amer. Math. Soc.
DOI: https://doi.org/10.1090/tran/9393
Published electronically: May 8, 2025

Abstract:

We propose a general framework of mass transport between non-negative vector-valued measures, which will be called simultaneous optimal transport (SOT). The new framework is motivated by the need to transport resources of different types simultaneously, i.e., in single trips, from specified origins to destinations; similarly, in economic matching, one needs to couple two groups, e.g., buyers and sellers, by equating supplies and demands of different goods at the same time. The mathematical structure of simultaneous transport is very different from the classic setting of optimal transport, leading to many new challenges. The Monge and Kantorovich formulations are contrasted and connected. Existence conditions and duality formulas are established. More interestingly, by connecting SOT to a natural relaxation of martingale optimal transport (MOT), we introduce the MOT-SOT parity, which allows for explicit solutions of SOT in many interesting cases.
References
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Bibliographic Information
  • Ruodu Wang
  • Affiliation: Department of Statistics and Actuarial Science, University of Waterloo, Canada
  • MR Author ID: 880329
  • ORCID: 0000-0003-3849-4555
  • Email: wang@uwaterloo.ca
  • Zhenyuan Zhang
  • Affiliation: Department of Mathematics, Stanford University
  • MR Author ID: 1404846
  • Email: zzy@stanford.edu
  • Received by editor(s): June 20, 2023
  • Received by editor(s) in revised form: October 30, 2024, and December 20, 2024
  • Published electronically: May 8, 2025
  • Additional Notes: The first author was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, RGPIN-2018-03823, RGPAS-2018-522590).
  • © Copyright 2025 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc.
  • MSC (2020): Primary 49Q22, 60E15, 46G10
  • DOI: https://doi.org/10.1090/tran/9393