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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2411.12109 |
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| _version_ | 1866911285524824064 |
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| author | De Philippis, Guido Shenfeld, Yair |
| author_facet | De Philippis, Guido Shenfeld, Yair |
| contents | Caffarelli's contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: The trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim-Milman transport map |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12109 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal transport maps, majorization, and log-subharmonic measures De Philippis, Guido Shenfeld, Yair Analysis of PDEs Probability 49Q22, 39B62 Caffarelli's contraction theorem bounds the derivative of the optimal transport map between a log-convex measure and a strongly log-concave measure. We show that an analogous phenomenon holds on the level of the trace: The trace of the derivative of the optimal transport map between a log-subharmonic measure and a strongly log-concave measure is bounded. We show that this trace bound has a number of consequences pertaining to volume-contracting transport maps, majorization and its monotonicity along Wasserstein geodesics, growth estimates of log-subharmonic functions, the Wehrl conjecture for Glauber states, and two-dimensional Coulomb gases. We also discuss volume-contraction properties for the Kim-Milman transport map |
| title | Optimal transport maps, majorization, and log-subharmonic measures |
| topic | Analysis of PDEs Probability 49Q22, 39B62 |
| url | https://arxiv.org/abs/2411.12109 |