---
OA_place: repository
OA_type: green
_id: '20569'
abstract:
- lang: eng
  text: 'This is the first part of a general description in terms of mass transport
    for time-evolving interacting particles systems, at a mesoscopic level. Beyond
    kinetic theory, our framework naturally applies in biology, computer vision, and
    engineering. The central object of our study is a new discrepancy d between two
    probability distributions in position and velocity states, which is reminiscent
    of the 2-Wasserstein distance, but of second-order nature. We construct d in two
    steps. First, we optimise over transport plans. The cost function is given by
    the minimal acceleration between two coupled states on a fixed time horizon T.
    Second, we further optimise over the time horizon T > 0. We prove the existence
    of optimal transport plans and maps, and study two time-continuous characterisations
    of d. One is given in terms of dynamical transport plans. The other one -- in
    the spirit of the Benamou--Brenier formula -- is formulated as the minimisation
    of an action of the acceleration field, constrained by Vlasov''s equations. Equivalence
    of static and dynamical formulations of d holds true. While part of this result
    can be derived from recent, parallel developments in optimal control between measures,
    we give an original proof relying on two new ingredients: Galilean regularisation
    of Vlasov''s equations and a kinetic Monge--Mather shortening principle. Finally,
    we establish a first-order differential calculus in the geometry induced by d,
    and identify solutions to Vlasov''s equations with curves of measures satisfying
    a certain d-absolute continuity condition. One consequence is an explicit formula
    for the d-derivative of such curves.'
acknowledgement: "This work was partially inspired by an unpublished note from 2014
  by Guillaume Carlier,\r\nJean Dolbeault, and Bruno Nazaret. GB deeply thanks Jean
  Dolbeault for proposing\r\nthis problem to him, guiding him into the subject, and
  sharing the aforementioned note.\r\nWe are grateful to Karthik Elamvazhuthi for
  making us aware of the work [20].\r\nThe work of GB has received funding from the
  European Union’s Horizon 2020 research and innovation programme under the Marie
  Sklodowska-Curie grant agreement\r\nNo 101034413.\r\nJM and FQ gratefully acknowledge
  support from the Austrian Science Fund (FWF)\r\nproject 10.55776/F65."
article_number: '2502.15665'
article_processing_charge: No
arxiv: 1
author:
- first_name: Giovanni
  full_name: Brigati, Giovanni
  id: 63ff57e8-1fbb-11ee-88f2-f558ffc59cf1
  last_name: Brigati
- first_name: Jan
  full_name: Maas, Jan
  id: 4C5696CE-F248-11E8-B48F-1D18A9856A87
  last_name: Maas
  orcid: 0000-0002-0845-1338
- first_name: Filippo
  full_name: Quattrocchi, Filippo
  id: 3ebd6ba8-edfb-11eb-afb5-91a9745ba308
  last_name: Quattrocchi
  orcid: 0009-0000-9773-1931
citation:
  ama: 'Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part
    1: Second-order discrepancies between probability measures. <i>arXiv</i>. doi:<a
    href="https://doi.org/10.48550/arXiv.2502.15665">10.48550/arXiv.2502.15665</a>'
  apa: 'Brigati, G., Maas, J., &#38; Quattrocchi, F. (n.d.). Kinetic Optimal Transport
    (OTIKIN) -- Part 1: Second-order discrepancies between probability measures. <i>arXiv</i>.
    <a href="https://doi.org/10.48550/arXiv.2502.15665">https://doi.org/10.48550/arXiv.2502.15665</a>'
  chicago: 'Brigati, Giovanni, Jan Maas, and Filippo Quattrocchi. “Kinetic Optimal
    Transport (OTIKIN) -- Part 1: Second-Order Discrepancies between Probability Measures.”
    <i>ArXiv</i>, n.d. <a href="https://doi.org/10.48550/arXiv.2502.15665">https://doi.org/10.48550/arXiv.2502.15665</a>.'
  ieee: 'G. Brigati, J. Maas, and F. Quattrocchi, “Kinetic Optimal Transport (OTIKIN)
    -- Part 1: Second-order discrepancies between probability measures,” <i>arXiv</i>.
    .'
  ista: 'Brigati G, Maas J, Quattrocchi F. Kinetic Optimal Transport (OTIKIN) -- Part
    1: Second-order discrepancies between probability measures. arXiv, 2502.15665.'
  mla: 'Brigati, Giovanni, et al. “Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order
    Discrepancies between Probability Measures.” <i>ArXiv</i>, 2502.15665, doi:<a
    href="https://doi.org/10.48550/arXiv.2502.15665">10.48550/arXiv.2502.15665</a>.'
  short: G. Brigati, J. Maas, F. Quattrocchi, ArXiv (n.d.).
corr_author: '1'
date_created: 2025-10-28T13:12:08Z
date_published: 2025-08-10T00:00:00Z
date_updated: 2026-10-13T22:31:24Z
day: '10'
department:
- _id: GradSch
- _id: JaMa
doi: 10.48550/arXiv.2502.15665
ec_funded: 1
external_id:
  arxiv:
  - '2502.15665'
fulldoi: https://doi.org/10.48550/arXiv.2502.15665
keyword:
- optimal transport
- kinetic theory
- second-order discrepancy
- Vlasov equation
- Wasserstein splines.
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2502.15665
month: '08'
oa: 1
oa_version: Preprint
project:
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: arXiv
publication_status: draft
related_material:
  record:
  - id: '20563'
    relation: dissertation_contains
    status: public
status: public
title: 'Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-order discrepancies between
  probability measures'
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...