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.

Globally Lipschitz mean-field interaction kernel

Definition

Let $d \geq 1$ , with the Euclidean norm from The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . An interaction kernel is a continuous map $K : R^{d} \times R^{d} \to R^{d}$ . It is globally Lipschitz with constant $L \geq 0$ in the convention used on this page if $∣ K (x, y) - K (x^{'}, y^{'}) ∣ \leq L (∣ x - x^{'} ∣ + ∣ y - y^{'} ∣)$ for all $x, y, x^{'}, y^{'} \in R^{d}$ . This implies the linear-growth estimate $∣ K (x, y) ∣ \leq ∣ K (0, 0) ∣ + L (∣ x ∣ + ∣ y ∣) .$ The notation $L$ always refers to this two-variable convention; the stability factor below is therefore $2 L$ .

Depends on

The Euclidean inner product $\langle x,y\rangle = \sum_{k<n} x_k y_k$ on $\mathbb{R}^n$

Used by

Dependency tree · two levels

14 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)
Simon Mayer, The Vlasov Equation and Mean Field Limit of a Particle System (standard reference, not scraped)