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The first Fundamental Theorem of Calculus for functions defined on Wasserstein space

Xavier Erny

TL;DR

This work extends calculus to the Wasserstein space $\\mathcal{P}_1(\\mathbb{R})$ by developing a first fundamental theorem of calculus for measure-variable functions. It defines and compares Fréchet- and Gateaux-type differentiability via the linear functional derivative $\\delta F(m,x)$ and the Dawson derivative $\\Delta F(m,x)$, introducing regularity classes $C^{0,1}$ and $C^{1,1}$. Central contributions include a constructive criterion that upgrades Dawson-differentiability to Fréchet-differentiability (Proposition \ref{prop:criteria}) and a precise integral representation yielding an explicit antiderivative $F$ with $\\delta F = H$ (Theorem \ref{thm:TFA}), under symmetry and regularity assumptions on $H$. The paper further develops a density- and approximation-based framework using atomic measures to bridge finite-atom and general measure spaces, culminating in a robust, general theorem that generalizes the fundamental theorem of calculus to infinite-dimensional measure spaces. This advances analytical tools for mean-field type problems and optimal transport by enabling rigorous antiderivative constructions for measure-dependent functionals.

Abstract

We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the sense of the linear functional derivative, then its integral is differentiable and the derivative coincides with the integrand. Our approach relies on a general differentiability criterion that connects the linear functional derivative, viewed as a Fréchet-derivative, and Dawson's weaker notion, which corresponds to a Gateaux-derivative. Under suitable regularity assumptions, it is possible to upgrade Gateaux-differentiability to Fréchet-differentiability in the infinite-dimensional setting of Wasserstein space.

The first Fundamental Theorem of Calculus for functions defined on Wasserstein space

TL;DR

This work extends calculus to the Wasserstein space by developing a first fundamental theorem of calculus for measure-variable functions. It defines and compares Fréchet- and Gateaux-type differentiability via the linear functional derivative and the Dawson derivative , introducing regularity classes and . Central contributions include a constructive criterion that upgrades Dawson-differentiability to Fréchet-differentiability (Proposition \ref{prop:criteria}) and a precise integral representation yielding an explicit antiderivative with (Theorem \ref{thm:TFA}), under symmetry and regularity assumptions on . The paper further develops a density- and approximation-based framework using atomic measures to bridge finite-atom and general measure spaces, culminating in a robust, general theorem that generalizes the fundamental theorem of calculus to infinite-dimensional measure spaces. This advances analytical tools for mean-field type problems and optimal transport by enabling rigorous antiderivative constructions for measure-dependent functionals.

Abstract

We establish an analogue of the first fundamental theorem of calculus for functions defined on the Wasserstein space of probability measures. Precisely, we show that if a function on the Wasserstein space is sufficiently regular in the sense of the linear functional derivative, then its integral is differentiable and the derivative coincides with the integrand. Our approach relies on a general differentiability criterion that connects the linear functional derivative, viewed as a Fréchet-derivative, and Dawson's weaker notion, which corresponds to a Gateaux-derivative. Under suitable regularity assumptions, it is possible to upgrade Gateaux-differentiability to Fréchet-differentiability in the infinite-dimensional setting of Wasserstein space.
Paper Structure (11 sections, 12 theorems, 156 equations)

This paper contains 11 sections, 12 theorems, 156 equations.

Table of Contents

  1. Definitions and main results
  2. Differentiability of measure-variable functions
  3. Main result
  4. Proof of Theorem \ref{['prop:criteria']}
  5. Atomic measures approximation
  6. "Differentiability" of $F$ on a set of atomic measures
  7. End of the proof of Theorem \ref{['prop:criteria']}
  8. Proof of Theorem \ref{['thm:TFA']}
  9. Counter-example for Theorem \ref{['thm:TFA']}
  10. Symmetrical property of the second-order derivative
  11. Proof of Proposition \ref{['prop:mncv']}

Key Result

Lemma 1.3

Let $G:\mathcal{P}_1(\mathbb{R})\times\mathbb{R}\to\mathbb{R}$ be $C^{0,1}$ and $K>0$. Then $G$ is uniformly continuous w.r.t. the measure-variable $m$ uniformly w.r.t. the real-variable $x$ on the set $\mathcal{P}([-K,K])\times[-K,K]$.

Theorems & Definitions (28)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Definition 1.7
  • ...and 18 more