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

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.