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Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space

Alessandro Pinzi, Giuseppe Savaré

TL;DR

The paper develops a comprehensive geometric framework for the space of random measures endowed with the Wasserstein-on-Wasserstein metric. It introduces a nested lifting (from curves of random measures to laws on curves and then to laws of particle trajectories) and proves both metric and differential superposition principles, yielding a Benamou–Brenier-type formula in this iterated setting. The authors characterize absolutely continuous curves, define tangent/cotangent structures, and connect these to non-local vector fields and derivations, establishing a robust CE-based description and a universal measurable selection mechanism. These results illuminate the geometry of the Wasserstein on Wasserstein space, provide a principled way to describe geodesics via laws of random geodesics, and link the theory to interacting particle systems with potential applications to gradient-flow-type analyses in higher-order transport spaces.

Abstract

We study the geometric structure of the space of random measures $\mathcal{P}_p(\mathcal{P}_p(X))$, endowed with the Wasserstein on Wasserstein metric, where $(X, d)$ is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When $X$ is $\mathbb{R}^d$, we study the differential structure of $\mathcal{P}_p(\mathcal{P}_p(\mathbb{R}^d))$ in analogy with the simpler Wasserstein space $\mathcal{P}_p(\mathbb{R}^d)$. We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields $b:[0,T]\times \mathbb{R}^d \times \mathcal{P}(\mathbb{R}^d) \to \mathbb{R}^d$. In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems.

Nested superposition principle for random measures and the geometry of the Wasserstein on Wasserstein space

TL;DR

The paper develops a comprehensive geometric framework for the space of random measures endowed with the Wasserstein-on-Wasserstein metric. It introduces a nested lifting (from curves of random measures to laws on curves and then to laws of particle trajectories) and proves both metric and differential superposition principles, yielding a Benamou–Brenier-type formula in this iterated setting. The authors characterize absolutely continuous curves, define tangent/cotangent structures, and connect these to non-local vector fields and derivations, establishing a robust CE-based description and a universal measurable selection mechanism. These results illuminate the geometry of the Wasserstein on Wasserstein space, provide a principled way to describe geodesics via laws of random geodesics, and link the theory to interacting particle systems with potential applications to gradient-flow-type analyses in higher-order transport spaces.

Abstract

We study the geometric structure of the space of random measures , endowed with the Wasserstein on Wasserstein metric, where is a complete separable metric space. In this setting, we prove a metric superposition principle, in the spirit of the result by S. Lisini, that will allow us to recover important geometric features of the space. When is , we study the differential structure of in analogy with the simpler Wasserstein space . We show that continuity equations for random measures involving the abstract concept of derivation acting on cylinder functions can be more conveniently described by suitable non-local vector fields . In this way, we can: 1) characterize the absolutely continuous curves on the Wasserstein on Wasserstein space; 2) define and characterize its tangent bundle; 3) prove a superposition principle for the solutions to the standard non-local continuity equation in terms of solutions of interacting particle systems.

Paper Structure

This paper contains 44 sections, 78 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces of curves
  4. Space of continuous curves.
  5. Space of absolutely continuous curves in metric spaces
  6. Geodesics
  7. Spaces of measures
  8. Narrow topology over the spaces of measures
  9. Wasserstein metric
  10. Laws of random probability measures
  11. Curves of measures
  12. Continuous curves
  13. Topology over $\mathcal{P}(C_T(Y))$
  14. Absolutely continuous curves
  15. Continuity equation over
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $(X,d)$ be a complete and separable metric space and $p>1$. Let $\boldsymbol{M}=(M_t)_{t\in[0,T]} \in AC_T^p(\mathcal{P}_p(\mathcal{P}_p(X)))$. Then, there exist $\Lambda \in \mathcal{P}(C_T(\mathcal{P}(X)))$ and $\mathfrak{L}\in \mathcal{P}(\mathcal{P}(C_T(X)))$ satisfying:

Theorems & Definitions (169)

  • Theorem 1.1: Nested metric superposition and minimal energy liftings
  • Theorem 1.2: Nested superposition principle for the random continuity equation in $\mathbb{R}^d$
  • Theorem 1.3: Benamou-Brenier formula
  • Definition 2.1: Absolutely continuous curves
  • Proposition 2.2
  • Definition 2.3: $p$-action of a curve
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 159 more