Optimal transport of signed measures: existence, uniqueness and fractal structure

TL;DR

This paper establishes a comprehensive theory for optimal transport of signed measures, including existence, uniqueness, and fractal structure preservation, extending classical results with new regularization techniques.

Contribution

It extends optimal transport theory to signed measures with fractal parts, proving existence, uniqueness, and structural preservation under new regularity assumptions.

Findings

Existence and uniqueness of optimal transport maps for signed measures.

Coupled Monge-Ampere equations characterize solutions.

Optimal transport preserves fractal dimensions and regularity.

Abstract

This paper develops a comprehensive theory of optimal transport for signed (real) measures on Rd. Extending the classical Brenier theorem, we consider Jordan decompositions of measures with possibly fractal singular parts. Under suitable regularity and structural assumptions (H1-H5), we prove existence and uniqueness of an optimal transport map T for a cost that distinguishes same-sign and opposite sign transports via a positional penalty lambda. We derive coupled Monge-Ampere equations and a double Legendre transform system characterizing the solution. Moreover, we show that the optimal transport preserves Hausdorff dimension and Ahlfors regularity of fractal sets. The proof relies on a novel adaptive regularization technique that respects the signed and fractal nature of the measures.