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A Benamou-Brenier formula for transport distances between stationary random measures

Martin Huesmann, Bastian Müller

Abstract

We derive a Benamou-Brenier type dynamical formulation for the Wasserstein metric $\mathsf W_p$ between stationary random measures recently introduced in [EHJM23]. A key step is a reformulation of the metric $\mathsf W_p$ using Palm probabilities.

A Benamou-Brenier formula for transport distances between stationary random measures

Abstract

We derive a Benamou-Brenier type dynamical formulation for the Wasserstein metric between stationary random measures recently introduced in [EHJM23]. A key step is a reformulation of the metric using Palm probabilities.
Paper Structure (6 sections, 13 theorems, 140 equations)

This paper contains 6 sections, 13 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Connection to the literature.
  3. Structure of the article.
  4. Setup
  5. Metric structure of $\Omega\times\mathop{\mathrm{\mathbb{R}}}\nolimits^d$
  6. Continuity Equation

Key Result

Theorem 1.1

For two stationary random measures $\mathsf{P}_i$, $i=0,1$, with unit intensity, we have for $p>1$ where the infimum runs over all admissible probability spaces $(\Omega,\mathcal{F}, \mathsf{Q})$ and solutions to the continuity equation $((\mathop{\mathrm{\mathbb{P}}}\nolimits_t)_{t\in [0,1]},(V_t)_{t\in [0,1]})$ such that

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 21 more