Optimal transport of signed measures: existence, uniqueness and fractal structure
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915999766282240 |
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| author | Barthelemy, Bwo'Nyahre Baidi Yannick, Kouakep Tchaptche Elvies, Houpa Danga Duplex |
| author_facet | Barthelemy, Bwo'Nyahre Baidi Yannick, Kouakep Tchaptche Elvies, Houpa Danga Duplex |
| contents | This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable regularity and structural assumptions (H1-H5), we prove existence and uniqueness of an optimal transport map T for a cost that distinguishes same-sign and opposite sign transports via a positional penalty lambda. We derive coupled Monge-Ampere equations and a double Legendre transform system characterizing the solution. Moreover, we show that the optimal transport preserves Hausdorff dimension and Ahlfors regularity of fractal sets. The proof relies on a novel adaptive regularization technique that respects the signed and fractal nature of the measures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01523 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal transport of signed measures: existence, uniqueness and fractal structure Barthelemy, Bwo'Nyahre Baidi Yannick, Kouakep Tchaptche Elvies, Houpa Danga Duplex Analysis of PDEs 49Q22, 35J96, 28A80, 28A75, 28A78 This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable regularity and structural assumptions (H1-H5), we prove existence and uniqueness of an optimal transport map T for a cost that distinguishes same-sign and opposite sign transports via a positional penalty lambda. We derive coupled Monge-Ampere equations and a double Legendre transform system characterizing the solution. Moreover, we show that the optimal transport preserves Hausdorff dimension and Ahlfors regularity of fractal sets. The proof relies on a novel adaptive regularization technique that respects the signed and fractal nature of the measures. |
| title | Optimal transport of signed measures: existence, uniqueness and fractal structure |
| topic | Analysis of PDEs 49Q22, 35J96, 28A80, 28A75, 28A78 |
| url | https://arxiv.org/abs/2605.01523 |