LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generated

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.

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Mass-normalized fragmentation is bounded in weighted total variation

Statement

Assume the daughter-count bound $B_{x} (E) \leq D$ of Continuous-size coagulation-fragmentation weak model with $D \geq 1$ . The fragmentation operator there satisfies

$∥ F μ ∥_{w} \leq β_{*} (D + 1) ∥ μ ∥_{w} (μ \in M_{w} (E)) .$

Facts & Assumptions

Given: $E = [a, \infty)$ , $w (x) = 1 + x^{2}$ , $0 \leq β \leq β_{*}$ , daughter support in $[a, x]$ , daughter number at most $D$ , and daughter first moment $x$ .

[F1]

Continuous-size coagulation-fragmentation weak model: $F μ = \int β (x) (B_{x} - δ_{x}) μ (d x)$ , $B_{x} (E) \leq D$ , and $\int z B_{x} (d z) = x$ .

[F2]

Weighted total-variation space for population measures: $∥ σ ∥_{w} = \int (1 + x^{2}) d ∣ σ ∣$ .

Proof

1.1givenF1algebra

Since $0 \leq z \leq x$ on the support of $B_{x}$ , $z^{2} \leq x z$ and hence $\int z^{2} B_{x} (d z) \leq x^{2}$ . Therefore $\int w (z) B_{x} (d z) \leq D + x^{2}$ .

2.1givenF1F2step 1.1algebra∎

The weighted variation of the signed measure $B_{x} - δ_{x}$ is at most $\int w d B_{x} + w (x) \leq D + 1 + 2 x^{2} \leq (D + 1) (1 + x^{2})$ because $D \geq 1$ . Integrating against $β (x) ∣ μ ∣ (d x)$ and using [F2] gives the claim.

Depends on

Used by

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Sources

Philippe Laurençot and Stéphane Mischler, From the discrete to the continuous coagulation-fragmentation equations, §§1–2 (standard reference, not scraped)
Jacek Banasiak, Analytic fragmentation semigroups and classical solutions to coagulation-fragmentation equations, §§1–2 (standard reference, not scraped)