Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Averil Aussedat

Abstract

We study the perturbation of a measure $μ\in \mathscr{P}(\mathbb{R})$ consisting in superposing two copies of $μ$, each slightly shifted by a small distance $\pm h$. The difference between $μ$ and its perturbation is measured with a Wasserstein distance. For any $μ$, 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 $μ$ on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed.

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

Abstract

We study the perturbation of a measure consisting in superposing two copies of , each slightly shifted by a small distance . The difference between and its perturbation is measured with a Wasserstein distance. For any , this distance is bounded from above by . We show that measures for which this critical rate is achieved when 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 on the real line have a 2-Wasserstein tangent cone equal to the set of directions inducing curves with maximal initial speed.
Paper Structure (18 sections, 11 theorems, 75 equations)

This paper contains 18 sections, 11 theorems, 75 equations.

Table of Contents

  1. Preliminaries
  2. Wasserstein distances
  3. Monotone plan
  4. Dual formulation
  5. Tangent cones in the case $p=2$
  6. Submeasure estimate
  7. Proof of \ref{['res:charLS1']}
  8. Link with the geometric tangent cone to $\mathop{\mathrm{\mathscr{P}}}\nolimits_2(\mathbb{R})$
  9. Reduction to centred measure fields
  10. Reduction to the unit symmetric measure field
  11. Step 0: restriction to a suitable set
  12. Step 1: reduction to a symmetric measure field
  13. Step 2: reduction to a unit symmetric measure field
  14. Proof of \ref{['res:answerQ1']}
  15. Measures for which the limit sup vanishes
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mu \in \mathop{\mathrm{\mathscr{P}}}\nolimits(\mathbb{R})$. The following conditions are equivalent:

Theorems & Definitions (28)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 2: Large-mass submeasures of a given measure are close
  • proof
  • Lemma 3: Pointwise bound on the monotone plan
  • proof
  • Remark 4
  • Lemma 5
  • proof
  • ...and 18 more