Instability of the ray-monotone selector for $W_1$-optimal transport

TL;DR

This paper demonstrates the instability of the ray-monotone selector for $W_1$-optimal transport plans under weak convergence, providing a counterexample and analyzing the limits of optimal plan sets.

Contribution

It constructs a counterexample to the stability of the secondary variational selector in $W_1$-optimal transport and analyzes the limits of optimal plan sets under weak convergence.

Findings

Counterexample shows instability of the selector under weak convergence.

Identifies the Kuratowski limit of optimal plan sets.

Derives a non-commutation result for additive perturbations.

Abstract

For the distance cost $c(x,y)=|x-y|$, the set $O(\mu,\nu)$ of $W_1$-optimal plans is generally not a singleton. Under the classical absolute-continuity hypotheses in the Euclidean case, secondary variational selection by the quadratic energy $C_2$ yields the ray-monotone $W_1$-optimal plan. We provide a counterexample to an open problem posed by Santambrogio that concerns the stability of this selector under weak convergence of the marginals. More precisely, we construct a fixed absolutely continuous source $\mu$ and absolutely continuous targets $\nu_n\rightharpoonup\nu$ such that $\gamma^{\mathrm{sel}}(\mu,\nu_n)\rightharpoonup\gamma^{\mathrm{hom}}$, where $\gamma^{\mathrm{hom}}\in O(\mu,\nu)$ but $\gamma^{\mathrm{hom}}\neq\gamma^{\mathrm{sel}}(\mu,\nu)$. We also identify the narrow Kuratowski limit of the optimal-plan sets $O(\mu,\nu_n)$, derive the constrained $\Gamma$-limit for secondary energies of the form $\int \Phi(|x-y|)\,d\gamma$ with $\Phi\in C([0,2])$, and deduce a non-commutation result for the additive perturbation $c_\varepsilon(x,y)=|x-y|+\varepsilon|x-y|^2$.