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Stability of optimal transport on metric measure spaces

Bang-Xian Han, Zhuo-Nan Zhu

Abstract

We prove a quantitative stability of Kantorovich potentials on metric measure spaces with lower Ricci curvature bound, thereby confirming a recent conjecture of Kitagawa, Letrouit and Mérigot. Our proof, which employs the heat kernel-regularized $c$-transform, does not rely on linear structure or sectional curvature bounds. As a corollary, we get a quantitative stability of optimal transport maps on Alexandrov spaces with lower curvature bound.

Stability of optimal transport on metric measure spaces

Abstract

We prove a quantitative stability of Kantorovich potentials on metric measure spaces with lower Ricci curvature bound, thereby confirming a recent conjecture of Kitagawa, Letrouit and Mérigot. Our proof, which employs the heat kernel-regularized -transform, does not rely on linear structure or sectional curvature bounds. As a corollary, we get a quantitative stability of optimal transport maps on Alexandrov spaces with lower curvature bound.
Paper Structure (21 sections, 22 theorems, 124 equations)

This paper contains 21 sections, 22 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Motivation and setting
  3. Optimal transport.
  4. General setting.
  5. Main results
  6. Strategy: heat kernel-regularized c-transform
  7. Stability of Kantorovich potentials
  8. Heat kernel estimate
  9. Heat kernel regularization
  10. Gradient estimate
  11. Global concavity estimate
  12. Stability estimate for Kantorovich functionals
  13. Passing to the limit
  14. Lower bound:
  15. Upper bound:
  16. ...and 6 more sections

Key Result

Theorem 1.2

Let $(X,{\mathrm d},\mathfrak m)$ be an ${\rm RCD}(K,N)$ metric measure space. Let $S\subseteq X$ be a John domain and $Y\subseteq X$ be compact with $\mathfrak m(Y)>0$. Let $\rho\in \mathcal{P}(S)$ be with $a_1\mathfrak m\hbox{$|_{S}$} \leq \rho\leq a_2\mathfrak m\hbox{$|_{S}$}$ for some positive c where $\phi_\mu$ and $\phi_\nu$ are the Kantorovich potentials from $\rho$ to $\mu$ and $\rho$ to $

Theorems & Definitions (43)

  • Definition 1.1: John domain
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 33 more