A quantitative stability result for regularity of optimal transport on compact manifolds
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915385698156544 |
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| author | Warren, Micah |
| author_facet | Warren, Micah |
| contents | We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. \ We do this in spite of unhelpful MTW\ curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded. \ Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_09035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A quantitative stability result for regularity of optimal transport on compact manifolds Warren, Micah Analysis of PDEs 35J60, 49Q22 We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. \ We do this in spite of unhelpful MTW\ curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded. \ Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded. |
| title | A quantitative stability result for regularity of optimal transport on compact manifolds |
| topic | Analysis of PDEs 35J60, 49Q22 |
| url | https://arxiv.org/abs/2507.09035 |