Optimal transport of measures via autonomous vector fields

TL;DR

This paper develops a method for transporting probability measures using autonomous vector fields, leveraging optimal transport theory and reducing multi-dimensional problems to one-dimensional cases.

Contribution

It introduces a novel approach to construct autonomous vector fields for measure transport, including solutions in higher dimensions via disintegration techniques.

Findings

Constructed Lipschitz continuous vector fields in 1D for optimal transport.

Reduced multi-dimensional transport problems to 1D cases using Sudakov's disintegration.

Provided explicit solutions for the flow-based transport maps.

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.