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Viability Theory in the $1$-Wasserstein Space

Benoît Bonnet-Weill, Alberto Domínguez Corella, Hélène Frankowska

TL;DR

This work advances viability theory for dynamics on the $1$-Wasserstein space ${\rm P}_1(\mathbb{R}^d)$ by addressing two regularity regimes for admissible velocities. In the Lipschitz case, viability is characterized by the intersection $D\mathcal{Q}(\tau|\nu) \cap V(\tau,\nu) \neq \emptyset$ for ${\mathcal L}^1$-a.e. $\tau$ and all $\nu \in \mathcal{Q}(\tau)$, extending previous results to the less-smooth $W_1$ geometry. When velocities are only upper semicontinuous, the paper proves a simpler, yet robust, sufficient condition based on the infinitesimal behavior of the Aumann integral, namely $D\mathcal{Q}(\tau|\nu) \cap \liminf_{h\to0^+} \frac{1}{h} \int_{\tau}^{\tau+h} V(s, \mathbb{B}(\nu,r)) ds \neq \emptyset$ for a.e. $\tau$, $\nu$, and $r>0$, with a constructive proof using maximal families of approximate solutions and compactness. The analysis hinges on a careful treatment of $W_1$-geometry (notably the lack of strong superdifferentiability), measurable selection arguments, and a Wasserstein-version of the Castaing–Valadier closure, with the potential for extension to general ${\rm P}_p(\mathbb{R}^d)$ spaces. Overall, the results lay foundational viability criteria for measure-valued dynamics under time-dependent constraints, with implications for mean-field control and transport-based HJB theory.

Abstract

In this article, we establish necessary and sufficient viability conditions for continuity inclusions over the 1-Wasserstein space. Depending on the regularity properties of the dynamics, we derive two results which are based on fairly different proof strategies. When the admissible velocities are Lipschitz in the measure variable, we show that it is necessary and sufficient for viable solutions to exist that the latter intersect the graphical derivative of the constraints. On the other hand, when the admissible velocities are merely upper semicontinuous in the measure variable, we provide a sufficient condition for viability involving the infinitesimal behaviour of their Aumann integral over a neighbouring set of measures.

Viability Theory in the $1$-Wasserstein Space

TL;DR

This work advances viability theory for dynamics on the -Wasserstein space by addressing two regularity regimes for admissible velocities. In the Lipschitz case, viability is characterized by the intersection for -a.e. and all , extending previous results to the less-smooth geometry. When velocities are only upper semicontinuous, the paper proves a simpler, yet robust, sufficient condition based on the infinitesimal behavior of the Aumann integral, namely for a.e. , , and , with a constructive proof using maximal families of approximate solutions and compactness. The analysis hinges on a careful treatment of -geometry (notably the lack of strong superdifferentiability), measurable selection arguments, and a Wasserstein-version of the Castaing–Valadier closure, with the potential for extension to general spaces. Overall, the results lay foundational viability criteria for measure-valued dynamics under time-dependent constraints, with implications for mean-field control and transport-based HJB theory.

Abstract

In this article, we establish necessary and sufficient viability conditions for continuity inclusions over the 1-Wasserstein space. Depending on the regularity properties of the dynamics, we derive two results which are based on fairly different proof strategies. When the admissible velocities are Lipschitz in the measure variable, we show that it is necessary and sufficient for viable solutions to exist that the latter intersect the graphical derivative of the constraints. On the other hand, when the admissible velocities are merely upper semicontinuous in the measure variable, we provide a sufficient condition for viability involving the infinitesimal behaviour of their Aumann integral over a neighbouring set of measures.

Paper Structure

This paper contains 13 sections, 24 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces and integration
  4. Optimal transport and the $1$-Wasserstein space
  5. Set-valued analysis
  6. Continuity equations and inclusions in Wasserstein spaces
  7. Viability in the Lipschitz case
  8. Viability in the upper semicontinuous case
  9. A family of approximate solutions
  10. Proof of Theorem \ref{['Thm4']}
  11. On the (lack of) superdifferentiability of the 1-Wasserstein distance
  12. On the necessity of absolute left or right continuity for the viability of time-dependent constraints
  13. Heuristics subtending Proposition \ref{['prop:CastaingValadier']}

Key Result

Theorem 2.2

Let $\{f_n\}_{n\in\mathbb N} \subseteq L^1(I,X)$, and suppose that there exist $m(\cdot)\in L^1(I, \mathbb{R}_+)$ and a family $\{K_{t}\}_{t \in I}$ of compact subsets of $X$ such that for $\mathcal{L}^1$-almost every $t \in I$. Then, there exists a subsequence $\{f_{n_k}\}_{k\in\mathbb N} \subseteq L^1(I,X)$ that converges weakly to some $f \in L^1(I,X)$, and in particular for every scalarly-$^

Theorems & Definitions (54)

  • Definition 2.1: Space of Bochner integrable functions
  • Theorem 2.2: Weak compactness criterion in Bochner spaces
  • Definition 2.3: Bochner integral of $C^0(\mathbb{R}^d,\mathbb{R}^d)$-valued maps
  • Lemma 2.4: Weak $L^1$-compactness criterion for $C^0(\mathbb{R}^d,\mathbb{R}^d)$-valued maps
  • proof
  • Definition 2.5: Basic continuity notions for set-valued maps
  • Definition 2.6: Lipschitz continuity for set-valued maps
  • Definition 2.7: Measurability of set-valued maps and measurable selections
  • Definition 2.8: Absolute continuity of set-valued maps
  • Proposition 2.9: Absolute continuity of the metric distance
  • ...and 44 more