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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.15665 |
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| _version_ | 1866913981594075136 |
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| author | Brigati, Giovanni Maas, Jan Quattrocchi, Filippo |
| author_facet | Brigati, Giovanni Maas, Jan Quattrocchi, Filippo |
| contents | 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 $\mathsf d$ between two probability distributions in position and velocity states, which is reminiscent of the $2$-Wasserstein distance, but of second-order nature. We construct $\mathsf d$ 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 $\mathsf d$. 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 $\mathsf d$ 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 $\mathsf d$, and identify solutions to Vlasov's equations with curves of measures satisfying a certain $\mathsf d$-absolute continuity condition. One consequence is an explicit formula for the $\mathsf d$-derivative of such curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15665 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies Between Probability Measures Brigati, Giovanni Maas, Jan Quattrocchi, Filippo Analysis of PDEs Functional Analysis Metric Geometry Probability 49Q22 (Primary), 82C40, 35Q83 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 $\mathsf d$ between two probability distributions in position and velocity states, which is reminiscent of the $2$-Wasserstein distance, but of second-order nature. We construct $\mathsf d$ 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 $\mathsf d$. 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 $\mathsf d$ 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 $\mathsf d$, and identify solutions to Vlasov's equations with curves of measures satisfying a certain $\mathsf d$-absolute continuity condition. One consequence is an explicit formula for the $\mathsf d$-derivative of such curves. |
| title | Kinetic Optimal Transport (OTIKIN) -- Part 1: Second-Order Discrepancies Between Probability Measures |
| topic | Analysis of PDEs Functional Analysis Metric Geometry Probability 49Q22 (Primary), 82C40, 35Q83 |
| url | https://arxiv.org/abs/2502.15665 |