UNIFORM MEASURE ATTRACTORS OF MCKEAN-VLASOV STOCHASTIC REACTION-DIFFUSION EQUATIONS ON UNBOUNDED THIN DOMAIN

The paper establishes (i) existence/uniqueness of uniform measure attractors for the McKean–Vlasov stochastic reaction–diffusion equations on unbounded thin domains and for the limit problem, and (ii) upper semicontinuity of these attractors as the thin domain collapses, by transforming to a fixed strip, proving uniform (in ε and g) moment/tail estimates, obtaining a uniform absorbing set, asymptotic compactness via tightness, and using a joint-continuity argument based on Vitali’s theorem together with a kernel-section representation of the attractor. These steps and results are explicit in Lemma/Corollary 4.x, Lemma 5.2, Lemma 5.3, Theorems 5.1–5.2, and Theorem 6.1 of the paper, including the use of the averaging/embedding operators M and I and assumptions (6.1)–(6.2) for the thin-domain limit . By contrast, the model’s Phase-2 outline asserts properties not proved under the paper’s hypotheses: (a) a Lipschitz/Feller contraction on the law space and exponential contraction in W2, whereas the paper explicitly avoids a duality/Feller route for McKean–Vlasov equations and instead establishes joint continuity via Vitali’s theorem (because P*g is not the dual of Pg) ; and (b) O(ε) rates for the thin-domain convergence of one-step operators and attractors, while the paper proves convergence without any rate (Theorem 6.1 and (6.18)–(6.22)) . Hence, while the model’s high-level structure broadly mirrors the paper, it over-claims key estimates under weaker assumptions, so its proof is not correct as stated.