Optimal transport of signed fractal measures with dimensional distortion: a variational characterization

Abstract

We extend the optimal transport theory for signed measures supported on Ahlfors-regular fractal sets (Bwo'Nyahre et al., 2026) to allow a controlled dimensional distortion between source and target. A penalization term $\epsilon \Phi (d_s(x)-d_t(y))$ -- where $\Phi$ is a fixed smooth strictly convex function and $d_s,d_t$ are the local Hausdorff dimensions of the fractal supports -- is added to the transport cost on inter-sign regions, with~$\epsilon \ge 0$ controlling the tolerance for distortion. Under hypotheses H1--H7, we prove: the existence and uniqueness of an optimal transport map~$T^{\epsilon }$ for every~$\epsilon >0$; coupled Monge--Ampère equations with a distortion correction term, generalizing the classical Brenier--Caffarelli equation; a double Legendre--Fenchel characterization of the optimal potentials, giving a complete variational description of the transport in each of the four sign regimes. The double Legendre--Fenchel system (Theorem~4.2) is the central contribution: it shows that the optimal potentials are the unique fixed points of a system of conjugacy equations, one per transport regime, and it provides the foundation for numerical algorithms and asymptotic analysis.

We extended optimal transport for signed fractal measures to controlled dimensional distortion, establishing a well-posed penalized problem with a unique map~$T^{\epsilon }$ for~$\epsilon \in (0,{\epsilon }_{\max })$ and coupled Monge--Ampère equations. Future work will optimize~$\epsilon$, assess window-size consistency, and test sensitivity to gradual versus abrupt dimension shifts