Optimal transport of signed measures: existence, uniqueness and fractal structure part I: the separated-support case research announcement, with a numerical validation report

We announce a rigorous theory of optimal transport for signed (real) measures on Rd in the separated-support regime: the supports of the positive and negative parts of measures are assumed spatially disjoint.

Under this structural hypothesis, together with regularity and non-degeneracy conditions on the absolutely continuous, discrete, and fractal singular components of the underlying Jordan and Lebesgue decompositions, we obtain existence and uniqueness of an optimal transport map for a cost that distinguishes same-sign and opposite-sign transports via a positional penalty, a coupled Monge-Ampere system together with a double Legendre transform characterising the solution, and exact preservation of Hausdorff dimension on the fractal singular components, the latter via a novel adaptive regularisation kernel and a direct measure-theoretic argument requiring no continuity of the transport map.

This note states the results and outlines the two main proof techniques, full proofs, uniform estimates, and technical lemmas appear in the complete version of this work.

We further report, in full and exhaustive summary, a complete numerical validation of this theory in ambient dimension one, two, and three: an entropic Sinkhorn solver for the single combined transport problem underlying the existence proof, applied to an explicit test case in which inter-sign transport is forced by mass imbalance and calibrated to occur with a large, robust cost margin, recovers exactly the predicted four-region structure with zero measured crossing of fractal mass between signs, and an exact-assignment refinement confirms pointwise-exact preservation of Hausdorff dimension under transport, with box-counting estimates converging to the theoretical dimension as resolution increases.

The case of overlapping supports and the fully general unbalanced setting are the subject of Part II, in preparation.