Characterization of measures on the real line that are critically unstable under small shifts

TL;DR

This paper characterizes measures on the real line that are critically unstable under small shifts by analyzing their mass distribution on porous sets and identifying those with maximal initial speed in the Wasserstein tangent cone.

Contribution

It provides a characterization of measures with critical stability properties under small perturbations in the Wasserstein metric, linking stability to mass distribution on porous sets.

Findings

Measures with critical stability concentrate most of their mass on porous sets.

The critical rate of Wasserstein distance under shifts is achieved by specific measures.

Identification of measures with maximal initial speed in the Wasserstein tangent cone.

Abstract

We study the perturbation of a measure $\mu \in \mathscr{P}(\mathbb{R})$ consisting in superposing two copies of $\mu$, each slightly shifted by a small distance $\pm h$. The difference between $\mu$ and its perturbation is measured with a Wasserstein distance. For any $\mu$, this distance is bounded from above by $h$. We show that measures for which this critical rate is achieved when $h$ goes to 0 are characterized as the ones giving most of their mass to some particular porous sets. This is used to identify which measures $\mu$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed.