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On the Lipschitz properties of transportation along heat flows

Dan Mikulincer, Yair Shenfeld

TL;DR

The paper develops dimension-free Lipschitz bounds for transport maps φ^{flow} that push the standard Gaussian γ_d forward to broad target measures μ by transporting along the heat flow, extending Caffarelli-type results beyond strong log-concavity. Using the Ornstein–Uhlenbeck semigroup and a flow of diffeomorphisms, the authors obtain explicit operator-norm bounds on ∥∇φ^{flow}∥ under κ-log-concavity with bounded support and under Gaussian-mixture structure, as well as a bound on the inverse map for β-semi-log-convex measures. These bounds yield eigenvalue comparison inequalities, dimensional Gaussian functional inequalities, and majorization-type consequences, helping to transfer Gaussian regularity to wider classes of measures. The results connect to and contrast with Brownian transport maps, offering a finite-dimensional, Lipschitz-control framework with potential broad applicability to concentration and spectral theory in high dimensions.

Abstract

We prove new Lipschitz properties for transport maps along heat flows, constructed by Kim and Milman. For (semi)-log-concave measures and Gaussian mixtures, our bounds have several applications: eigenvalues comparisons, dimensional functional inequalities, and domination of distribution functions.

On the Lipschitz properties of transportation along heat flows

TL;DR

The paper develops dimension-free Lipschitz bounds for transport maps φ^{flow} that push the standard Gaussian γ_d forward to broad target measures μ by transporting along the heat flow, extending Caffarelli-type results beyond strong log-concavity. Using the Ornstein–Uhlenbeck semigroup and a flow of diffeomorphisms, the authors obtain explicit operator-norm bounds on ∥∇φ^{flow}∥ under κ-log-concavity with bounded support and under Gaussian-mixture structure, as well as a bound on the inverse map for β-semi-log-convex measures. These bounds yield eigenvalue comparison inequalities, dimensional Gaussian functional inequalities, and majorization-type consequences, helping to transfer Gaussian regularity to wider classes of measures. The results connect to and contrast with Brownian transport maps, offering a finite-dimensional, Lipschitz-control framework with potential broad applicability to concentration and spectral theory in high dimensions.

Abstract

We prove new Lipschitz properties for transport maps along heat flows, constructed by Kim and Milman. For (semi)-log-concave measures and Gaussian mixtures, our bounds have several applications: eigenvalues comparisons, dimensional functional inequalities, and domination of distribution functions.
Paper Structure (7 sections, 12 theorems, 59 equations)

This paper contains 7 sections, 12 theorems, 59 equations.

Table of Contents

  1. Introduction and main results
  2. Transport along heat flows and the Brownian transport map.
  3. Applications
  4. Acknowledgments.
  5. Proofs
  6. Preliminaries
  7. Lipschitz properties of transportation along heat flows

Key Result

Theorem 1

Let $\mu$ be a $\kappa$-log-concave probability measure on $\mathbb R^d$, and set $D := \mathrm{diam}(\mathrm{supp}(\mu))$. Then, for the map $\varphi^{\mathrm{flow}}:\mathbb R^d \to \mathbb R^d$, which satisfies $\varphi^{\mathrm{flow}}_*\gamma_d = \mu$, the following holds:

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • proof
  • Corollary 7
  • proof
  • Remark 8
  • ...and 13 more