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A Lagrangian approach to totally dissipative evolutions in Wasserstein spaces

Giulia Cavagnari, Giuseppe Savaré, Giacomo Enrico Sodini

TL;DR

This work develops a Lagrangian-Hilbertian framework for totally dissipative multivalued probability vector fields on the Wasserstein space, revealing a precise correspondence with law-invariant dissipative operators in a suitable Hilbert space. By lifting MPVFs to a Lagrangian operator in $\mathcal{X}=L^2(\Omega;\mathsf X)$, the authors import maximal-dissipative operator theory to obtain well-posedness, stability, and Eulerian-Lagrangian equivalence for flows in $\mathcal{P}_2(\mathsf X)$, including a robust resolvent/Yosida-semigroup machinery that preserves law invariance. They show that total dissipativity strengthens metric dissipativity and that a dense discrete core suffices to reconstruct a unique maximal totally dissipative MPVF, whose flow provides a mean-field description of particle systems and yields structural insights for gradient flows of geodesically convex/displacement-convex functionals. The results unify discrete-to-continuum limits, offer a solid backbone for numerical schemes (implicit Euler/JKO) in Wasserstein spaces, and reveal new convexity and regularity properties for functionals along generalized geodesics. Overall, the paper extends dissipative evolution theory beyond gradient flows to a broad class of multivalued, law-invariant evolutions in measure spaces, with clear mean-field and stability implications.

Abstract

We introduce and study the class of totally dissipative multivalued probability vector fields (MPVF) $\boldsymbol{\mathrm F}$ on the Wasserstein space $(\mathcal{P}_2(\mathsf{X}),W_2)$ of Euclidean or Hilbertian probability measures. We show that such class of MPVFs is in one to one correspondence with law-invariant dissipative operators in a Hilbert space $L^2(Ω,\mathcal{B},\mathbb{P};\mathsf{X})$ of random variables, preserving a natural maximality property. This allows us to import in the Wasserstein framework many of the powerful tools from the theory of maximal dissipative operators in Hilbert spaces, deriving existence, uniqueness, stability, and approximation results for the flow generated by a maximal totally dissipative MPVF and the equivalence of its Eulerian and Lagrangian characterizations. We will show that demicontinuous single-valued probability vector fields satisfying a metric dissipativity condition are in fact totally dissipative. Starting from a sufficiently rich set of discrete measures, we will also show how to recover a unique maximal totally dissipative version of a MPVF, proving that its flow provides a general mean field characterization of the asymptotic limits of the corresponding family of discrete particle systems.Such an approach also reveals new interesting structural properties for gradient flows of displacement convex functionals with a core of discrete measures dense in energy.

A Lagrangian approach to totally dissipative evolutions in Wasserstein spaces

TL;DR

This work develops a Lagrangian-Hilbertian framework for totally dissipative multivalued probability vector fields on the Wasserstein space, revealing a precise correspondence with law-invariant dissipative operators in a suitable Hilbert space. By lifting MPVFs to a Lagrangian operator in , the authors import maximal-dissipative operator theory to obtain well-posedness, stability, and Eulerian-Lagrangian equivalence for flows in , including a robust resolvent/Yosida-semigroup machinery that preserves law invariance. They show that total dissipativity strengthens metric dissipativity and that a dense discrete core suffices to reconstruct a unique maximal totally dissipative MPVF, whose flow provides a mean-field description of particle systems and yields structural insights for gradient flows of geodesically convex/displacement-convex functionals. The results unify discrete-to-continuum limits, offer a solid backbone for numerical schemes (implicit Euler/JKO) in Wasserstein spaces, and reveal new convexity and regularity properties for functionals along generalized geodesics. Overall, the paper extends dissipative evolution theory beyond gradient flows to a broad class of multivalued, law-invariant evolutions in measure spaces, with clear mean-field and stability implications.

Abstract

We introduce and study the class of totally dissipative multivalued probability vector fields (MPVF) on the Wasserstein space of Euclidean or Hilbertian probability measures. We show that such class of MPVFs is in one to one correspondence with law-invariant dissipative operators in a Hilbert space of random variables, preserving a natural maximality property. This allows us to import in the Wasserstein framework many of the powerful tools from the theory of maximal dissipative operators in Hilbert spaces, deriving existence, uniqueness, stability, and approximation results for the flow generated by a maximal totally dissipative MPVF and the equivalence of its Eulerian and Lagrangian characterizations. We will show that demicontinuous single-valued probability vector fields satisfying a metric dissipativity condition are in fact totally dissipative. Starting from a sufficiently rich set of discrete measures, we will also show how to recover a unique maximal totally dissipative version of a MPVF, proving that its flow provides a general mean field characterization of the asymptotic limits of the corresponding family of discrete particle systems.Such an approach also reveals new interesting structural properties for gradient flows of displacement convex functionals with a core of discrete measures dense in energy.
Paper Structure (25 sections, 83 theorems, 425 equations)

This paper contains 25 sections, 83 theorems, 425 equations.

Table of Contents

  1. Introduction
  2. Total dissipativity
  3. Preliminaries
  4. Wasserstein distance in Hilbert spaces and strong-weak topology
  5. Duality pairings
  6. Multivalued probability vector fields, metric dissipativity and EVI solutions
  7. Invariant dissipative operators in Hilbert spaces and totally dissipative MPVFs
  8. Law invariant dissipative operators
  9. Totally dissipative MPVFs
  10. Totally dissipative PVFs concentrated on maps
  11. Lagrangian and Eulerian flow generated by a totally dissipative MPVF
  12. Totally convex functionals in P
  13. Metric dissipativity, total dissipativity and maximal extensions
  14. Local optimality and injectivity of couplings
  15. Local optimality of couplings between discrete measures
  16. ...and 10 more sections

Key Result

Theorem 2.1

Let $\mathbb X, X$ be Lusin completely regular topological spaces, $\boldsymbol\mu\in\mathcal{P}(\mathbb{X})$ and $r:\mathbb{X}\to X$ a Borel map. Denote with $\mu=r_{\sharp}\boldsymbol\mu\in\mathcal{P}(X)$. Then there exists a $\mu$-a.e. uniquely determined Borel family of probability measures $\{\ for every bounded Borel map $\varphi:\mathbb{X}\to\mathbb R$.

Theorems & Definitions (196)

  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4: Strong-weak topology in $\mathcal{P}_2(\mathsf {T X})$
  • Proposition 2.5
  • Definition 2.6: Geodesics
  • Definition 2.7: Geodesic and total convexity
  • Remark 2.8
  • Theorem 2.9: Properties of geodesics
  • Definition 2.10: The space $\mathop{\mathrm{Cyl}}\nolimits(\mathsf X)$ of cylindrical functions
  • ...and 186 more