Vlasov equations on directed hypergraph measures

The paper proves well-posedness of the Vlasov fixed-point equation via characteristics and a Banach contraction on the weighted path metric d_α, and shows (a) propagation of x-continuity of measures and (b) stability/approximation under discretizations; see Proposition 4.2–4.3 and Theorem 5.6, together with Appendix C–D for the contraction/continuity details . The model’s solution reproduces the same structure: a flow-based fixed point T(ν·) = (S^ν(t,0,·))#ν0, contraction in d_α (Definition 2.26/Prop. 2.27 in the paper), and stability via uniform velocity-field mismatch, leading to d0-convergence under the paper’s discretizations (5.1) . A minor caveat: the paper’s proof of continuity in x uses continuity of x↦η^x (assumption (A4)′) in Appendix D, while (A4)′ is not explicitly listed in Proposition 4.3’s assumptions; the model explicitly assumes (A4)′ in that step, which matches the proof ingredients .