Optimal transport of measures via autonomous vector fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913718959341568 |
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| author | De Nitti, Nicola Fernández-Real, Xavier |
| author_facet | De Nitti, Nicola Fernández-Real, Xavier |
| contents | We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06503 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal transport of measures via autonomous vector fields De Nitti, Nicola Fernández-Real, Xavier Optimization and Control Analysis of PDEs Classical Analysis and ODEs 49Q22, 93B05, 28A50 We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems. |
| title | Optimal transport of measures via autonomous vector fields |
| topic | Optimization and Control Analysis of PDEs Classical Analysis and ODEs 49Q22, 93B05, 28A50 |
| url | https://arxiv.org/abs/2405.06503 |