Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal transport maps, majorization, and log-subharmonic measures

Guido De Philippis, Yair Shenfeld

TL;DR

This work extends the contraction framework of optimal transport to the trace level by proving a sharp bound on the Laplacian of the Brenier map between log-subharmonic and strongly log-concave measures: $\| riangle \Phi\|_{L^{\infty}} \le n\sqrt{\alpha/\kappa}$ under $\triangle V \le \alpha n$ and $\nabla^{2}W \succeq \kappa I$. The authors develop an $L^p$-based argument, introduce the $\Delta_{\epsilon}$ operator, and apply smoothing via Ornstein–Uhlenbeck and heat semigroups to derive a robust trace bound, which then yields majorization, entropy monotonicity along Wasserstein geodesics, and stability results. These trace bounds underpin applications to growth estimates in Fock spaces, the Wehrl conjecture for Glauber states, two-dimensional Coulomb gases, and the Kim–Milman transport map. The paper advances a unified, transport-based approach to several problems in probability, analysis, and mathematical physics by connecting trace-level regularity, majorization, and entropy with concrete models and conjectures.

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

Optimal transport maps, majorization, and log-subharmonic measures

TL;DR

This work extends the contraction framework of optimal transport to the trace level by proving a sharp bound on the Laplacian of the Brenier map between log-subharmonic and strongly log-concave measures: under and . The authors develop an -based argument, introduce the operator, and apply smoothing via Ornstein–Uhlenbeck and heat semigroups to derive a robust trace bound, which then yields majorization, entropy monotonicity along Wasserstein geodesics, and stability results. These trace bounds underpin applications to growth estimates in Fock spaces, the Wehrl conjecture for Glauber states, two-dimensional Coulomb gases, and the Kim–Milman transport map. The paper advances a unified, transport-based approach to several problems in probability, analysis, and mathematical physics by connecting trace-level regularity, majorization, and entropy with concrete models and conjectures.

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

Paper Structure

This paper contains 13 sections, 29 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. The divergence of the Brenier map
  3. Volume contracting maps and majorization
  4. Volume contraction of the Kim-Milman map
  5. Growth estimates in Fock spaces and log-subharmonic functions
  6. The Wehrl conjecture for Glauber states
  7. Two-dimensional Coulomb gases
  8. The divergence of the Brenier map
  9. Majorization
  10. The Kim-Milman heat flow transport map
  11. Appendix
  12. The operator $\Delta_{\epsilon}$
  13. Smoothing under Ornstein-Uhlenbeck and heat semigroups

Key Result

Theorem 1.2

Let $\mathop{}\!\mathrm{d}\mu=e^{-V}\mathop{}\!\mathrm{d} x$ and $\mathop{}\!\mathrm{d}\nu=e^{-W}\mathop{}\!\mathrm{d} x$ be probability measures on $\mathbb R^{n}$, with $\mu$ supported on all of $\mathbb R^{n}$, such that there exist $\alpha>0,\kappa>0$ with Let $\nabla\Phi:\mathbb R^{n}\to\mathbb R^{n}$ be the Brenier map transporting $\mu$ to $\nu$. Then, where $\|\cdot\|_{L^{\infty}(\mathop

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2: Caffarelli MR1800860, Kolesnikov kolesnikov2011mass
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5: Lipschitz bounds
  • Definition 1.6
  • Definition 1.7
  • Theorem 1.8: Melbourne-Roberto MR2023
  • Theorem 1.9: Volume contraction of the Brenier map
  • Theorem 1.10: Majorization
  • ...and 46 more