Optimal transport maps, majorization, and log-subharmonic measures
Analysis of PDEs
2025-11-26 v3 Probability
Abstract
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
Cite
@article{arxiv.2411.12109,
title = {Optimal transport maps, majorization, and log-subharmonic measures},
author = {Guido De Philippis and Yair Shenfeld},
journal= {arXiv preprint arXiv:2411.12109},
year = {2025}
}