VLASOV EQUATIONS ON DIGRAPH MEASURES

Core well-posedness and continuity properties in the paper are correct (e.g., continuity in t and x for the solution map, and exactness of the particle ODE discretization) as shown via the fixed-point/characteristics method and explicit bounds on d∞ with constants built from BL(gi), BL(h), and ‖η‖ (see Proposition 4.4 and its proof, including the continuity-in-time estimate and mass conservation, as well as the continuity-in-x argument) . Step I in the approximation proof also correctly identifies the finite ODE scheme as an exact weak solution for the discretized data (equation (8.5)) . However, the main discretization theorems (Theorem 5.11 and its summary Theorem C) state the single-limit claim lim_{n→∞} d∞(ν^{m,n}_t, ν_t) = 0 for fixed m, which is not supported by the proof: Steps II–III/IV of Section 8.3 use continuity in η, h, and ν0 to obtain only a two-step limit lim_{m→∞} lim_{n→∞} d∞(·,·) = 0 (or a diagonal limit), since h_m remains piecewise-constant in x for fixed m and cannot approach h as n→∞ alone . The theorem statements therefore overclaim; the correct conclusion is the two-step (or diagonal) convergence. In addition, there is a minor typo in the data construction: the atoms y_{ℓ,m,n(…)} used to build the DGM approximants must be points of X, not Y, while the text writes {y_{ℓ,m,n…}} ⊆ Y in the summary (equations (2.5b)/(5.1)) . The model’s solution identifies both issues and gives the standard corrected statement (two-step/diagonal convergence) along with a clean Dobrushin-type stability estimate and the needed uniform-in-x continuity assumption on h (which the paper tacitly uses in Appendix B as “(A8)”) .