Non-autonomous stochastic lattice systems with Markovian switching

The paper’s Section 5 defines λ := max_j λ(j) (see the setup around (4.10)–(4.11) and (4.6)–(4.9), where λ is introduced as the regime-dependent coefficient and then λ = max_j λ(j) is fixed) , but the subsequent L^2 energy and contractivity estimates (e.g., Lemma 5.1 and Lemma 5.3) use λ as if it were a uniform lower bound on λ(r(t)) (compare (5.3)–(5.6) and the choice ϖ1 = 2λ − 2 − 2β0^2 − 4||β||^2, and γ = 7/4 λ − ||L||^2 − 4 λ^{-1} L_f^2 in (5.7)–(5.8)) . This is a structural gap: to make those inequalities valid uniformly in time and regimes, one needs λ_* := inf_j λ(j), not the maximum. The candidate solution explicitly corrects this and consistently works with λ_* throughout, yielding a complete, coherent proof; it also supplies a constructive Wasserstein-based pullback construction and uniqueness that align with the paper’s main theorem (Theorem 6.1) under the corrected hypothesis . Minor constant factors in the candidate’s L^2 diffusion bound can be tightened (a factor of 2), but this does not affect the overall argument once (5.1)–(5.2) are read with λ = λ_*.