Global Dynamics of McKean-Vlasov SDEs via Stochastic Order

The paper proves (under Assumption 1) that the McKean–Vlasov semigroup is order-preserving (Theorem 4.4), establishes a compact attracting set via coming-down-from-infinity (Theorem 4.8), characterizes and totally orders invariant measures (Lemma 4.10), and shows finiteness using analyticity of the self-consistency function together with boundedness of its zero set (proof of Theorem 2.1; see (4.24)–(4.26)), leading to global convergence to an order interval and one-sided attraction to the extremal invariant measures (Theorem 2.1) . By contrast, the candidate solution’s main ideas align with the paper’s results (e.g., the explicit stationary family µm in (1.4) and the fixed-point characterization), but two key steps are incorrect or insufficient: (i) the maximum-principle proof of order preservation for CDF differences uses an invalid identity relating the mean difference to an integral of H := Fν − Fµ over ℝ and ignores boundary/integrability issues; the paper instead proves order preservation via an extrinsic-law comparison principle and a two-case argument around the time when means coincide (Theorem 4.3 → Theorem 4.4) ; (ii) the bracketing of P*_tB by sup/inf of stochastic order for an arbitrary W2-bounded set B is asserted without the needed order-boundedness argument; the paper supplies this through a general attractor framework and the existence of a globally attracting, order-bounded set (Theorem 4.2 + Theorem 4.8) . Hence the results are correct in the paper, but the model’s proof contains material gaps.