Long-time Behaviour of the Non-autonomous Stochastic FitzHugh-Nagumo Systems on Thin Domains

The paper establishes existence and upper semicontinuity of pullback measure attractors for the non-autonomous stochastic FitzHugh–Nagumo system on thin domains via: (i) uniform estimates and pullback absorbing families, (ii) a decomposition of the partially dissipative component v, (iii) an averaging operator M on the fixed cylinder, (iv) convergence of the transition operators (Theorem 7.2), and (v) passage to ω-limits (Theorem 4.6). These elements are clearly stated and coherent in the manuscript (existence: Theorems 4.4–4.5; upper semicontinuity: Theorem 4.6; Section 7 using M; and the thin-domain fixed formulation with anisotropic diffusion, Lemma 7.1, and Lemmas in Section 5–6) . The candidate’s solution proves the same limiting result but via a different route: a fast vertical alignment argument leveraging the ε−2 anisotropy, an explicit averaging operator P (identical in role to M), a Grönwall step for the averaged dynamics, and a coupling argument on measures. This alternative proof strategy is compatible with the paper’s assumptions and core lemmas (e.g., the static estimate ||w−Mw|| ≤ c ε ||w||_{H^1_ε}, Lemma 7.1), though it asserts an explicit O(ε) rate the paper itself does not claim. No contradiction arises; the model’s proof is a plausible, differently organized argument toward the same conclusion.