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Optimal transport of measures via autonomous vector fields

Nicola De Nitti, Xavier Fernández-Real

TL;DR

The paper addresses whether one can transport a source probability measure \\mu_0 to a target \\mu_1 using an autonomous velocity field, linking exact Lagrangian controllability (flow map) with the Eulerian continuity equation. The authors construct autonomous vector fields in one space dimension by solving a linear homogeneous functional equation so that the time-1 flow realizes the monotone (optimal) transport map, typically achieving Lipschitz regularity; they further extend to multiple dimensions via Sudakov's disintegration, gluing one-dimensional flow constructions along transport rays. The main contributions are: (i) a rigorous 1D realization of the monotone Monge map as a time-1 flow of an autonomous velocity, with detailed regularity and flow-uniqueness properties; (ii) a generalized multi-D approach reducing to 1D problems along transport rays and producing a corresponding autonomous velocity field; and (iii) a suite of examples illustrating the construction and regularity phenomena, including cases with fixed points and unbounded supports. This work broadens the toolbox for exact controllability of transport dynamics and provides explicit mechanisms to embed optimal transport maps into autonomous flows, with implications for controllability and evolution of measures.

Abstract

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time-$1$ map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems.

Optimal transport of measures via autonomous vector fields

TL;DR

The paper addresses whether one can transport a source probability measure \\mu_0 to a target \\mu_1 using an autonomous velocity field, linking exact Lagrangian controllability (flow map) with the Eulerian continuity equation. The authors construct autonomous vector fields in one space dimension by solving a linear homogeneous functional equation so that the time-1 flow realizes the monotone (optimal) transport map, typically achieving Lipschitz regularity; they further extend to multiple dimensions via Sudakov's disintegration, gluing one-dimensional flow constructions along transport rays. The main contributions are: (i) a rigorous 1D realization of the monotone Monge map as a time-1 flow of an autonomous velocity, with detailed regularity and flow-uniqueness properties; (ii) a generalized multi-D approach reducing to 1D problems along transport rays and producing a corresponding autonomous velocity field; and (iii) a suite of examples illustrating the construction and regularity phenomena, including cases with fixed points and unbounded supports. This work broadens the toolbox for exact controllability of transport dynamics and provides explicit mechanisms to embed optimal transport maps into autonomous flows, with implications for controllability and evolution of measures.

Abstract

We study the problem of transporting one probability measure to another via an autonomous velocity field. We rely on tools from the theory of optimal transport. In one space-dimension, we solve a linear homogeneous functional equation to construct a suitable autonomous vector field that realizes the (unique) monotone transport map as the time- map of its flow. Generically, this vector field can be chosen to be Lipschitz continuous. We then use Sudakov's disintegration approach to deal with the multi-dimensional case by reducing it to a family of one-dimensional problems.
Paper Structure (13 sections, 17 theorems, 89 equations, 2 figures)

This paper contains 13 sections, 17 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The problems
  3. Our results and structure of the paper
  4. Some related results
  5. Construction in the one-dimensional case
  6. Measures with bounded supports
  7. Measures with unbounded supports
  8. Proofs of the main results
  9. On the assumptions of Theorem 2.2
  10. Construction in the multi-dimensional case
  11. Preliminaries on Sudakov's theorem
  12. Proof of Theorem 3.1 (and Theorem C)
  13. Examples

Key Result

Theorem A

Let $\mu_0, \mu_1 \in \mathcal{P}_{\mathrm{a.c.}}(\mathbb{R})$ be two probability measures with convex supportWe recall that, given a (non-negative) measure $\mu$ on a measurable topological space $(X,\Sigma = {\rm Borel}(X))$, $\mathop{\mathrm{supp}}\nolimits \mu \coloneqq \overline{\{A\in \Sigma:\

Figures (2)

  • Figure 1: On the left, the vector field transporting a Gaussian $\bar{\mu}_0 (x) = e^{-x^2}$ into a translated and rescaled Gaussian $\bar{\mu}_1(x) = 2e^{-4(x-1)^2}$ is given by the linear function $v$ here depicted (as explained in \ref{['ex:gaussian']} and \ref{['ex:affine']}). On the right, the densities $\bar{\mu}_0$ (in red) and $\bar{\mu}_1$ (in green) from \ref{['ex:infty2']}. The velocity field $v$ (in blue) can be constructed arbitrarily in the interval $[2, 3]$, and this fixes the values uniquely in $[0, 2]$ as well (in this case, we are not trying to match higher derivatives as in \ref{['cor:Ck']}, so $v$ is not necessarily $C^1$). Plot created with MATLAB MATLAB.
  • Figure 2: The densities $\bar{\mu}_0$ (in red) and $\bar{\mu}_1$ (in green) from \ref{['ex:infty-2']} for the $C^1$ map $\mathrm{T}$ (left) and for the $C^\infty$ map $\mathrm{T}_\infty$ (right). Plot created with MATLAB MATLAB.

Theorems & Definitions (43)

  • Theorem A: Exact controllability, $d=1$
  • Remark 1.1: Non-uniqueness of the velocity field
  • Remark 1.2: On the solution to \ref{['prob:1bis']}
  • Remark 1.3: On the positivity assumption
  • Theorem B: Approximate controllability, $d=1$
  • Theorem C: Exact controllability, $d \ge 1$
  • Theorem 2.1: One-dimensional Monge's problem
  • Theorem 2.2: Exact controllability, $d=1$
  • Remark 2.3: Continuity equation
  • Remark 2.4: Higher regularity
  • ...and 33 more