DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The coupling definition of the 1-Wasserstein distance

Definition

Let $d \geq 1$ . For Borel probability measures $μ, ν$ on $R^{d}$ with finite first moments $\int ∣ x ∣ d μ (x) < \infty$ and $\int ∣ y ∣ d ν (y) < \infty$ , let $Π (μ, ν)$ be the set of Borel probability measures on $R^{d} \times R^{d}$ whose first and second marginals are $μ$ and $ν$ . Define $W_{1} (μ, ν) : = \inf_{π \in Π (μ, ν)} \int_{R^{d} \times R^{d}} ∣ x - y ∣ d π (x, y) .$ This definition makes no assertion that the infimum is attained. It applies only to probability measures with finite first moment; it is not a distance assigned here to a general signed measure.

Here $∣ \cdot ∣$ is the Euclidean norm from The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ , and probability measure has the meaning of Probability measures and probability spaces.

Depends on

Used by

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

François Golse, Mean-Field Limits in Statistical Dynamics, Lecture 1 (standard reference, not scraped)