Alphabeta Math
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.

Characteristic solution of a probability mean-field equation

Definition

Fix T>0, a globally Lipschitz kernel K as in Globally Lipschitz mean-field interaction kernel, and a Borel probability measure μ0 on Rd with finite first moment as in The coupling definition of the 1-Wasserstein distance. A characteristic solution on [0,T] consists of a jointly Borel map X:[0,T]×Rd→Rd, defined up to a μ0-null set of labels, such that t↦Xt(x) is absolutely continuous for μ0-almost every x, X0(x)=x, and Xt(x)=x+∫0t∫RdK(Xs(x),Xs(y)) dμ0(y) ds for μ0-almost every x. It transports μ0 to μt=(Xt)#μ0, where (Xt)#μ0(A):=μ0(Xt−1(A)). The weighted path integrability convention is ∫Rdsup⁡0≤t≤T∣Xt(x)∣dμ0(x)<∞; in particular every time slice has finite first moment.

Depends on

Used by

Dependency tree · two levels

12 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