WELL-POSEDNESS OF A STOCHASTIC PARAMETRICALLY-FORCED NONLINEAR SCHRÖDINGER EQUATION

The paper proves global well-posedness for the 1D stochastic PFNLS in both L2 and H1 via a truncated fixed-point scheme (on X_T^s := C([0,T];Hs) ∩ L^r(0,T;W^{s,p})) and a blow-up criterion, then excludes blow-up by deriving an energy evolution identity (Proposition 4.4) and estimates that control E0 and E1; in particular, it carefully handles the Itô correction and stochastic terms and shows the L2 a priori estimate E sup_{t≤T0} ||z(t)||_2^2 ≤ E||z0||_2^2 e^{2ε(γ+μ)T0} (Theorem 2.6) . The candidate solution correctly identifies the mild formulation, Strichartz and BDG ingredients, the composition z↦M_z∘Φ for the noise, and the L2 Itô cancellation. However, it makes two critical missteps: (i) it asserts F_Φ ∈ W^{s,∞} from Φ ∈ γ(L^2;W^{s,2+δ}), which is stronger than what the paper proves and generally false under (H6); the paper only uses the weaker but sufficient F_Φ ∈ W^{s,1+δ/2} to control zF_Φ terms , and (ii) it claims a global extension for s=1 from an L2 a priori bound alone, omitting the H1 energy evolution and BDG estimates that are essential in the paper’s proof of non-blow-up in H1 (Section 4) . It also asserts stronger stochastic Strichartz bounds and membership in “all admissible” Strichartz pairs without justification or alignment with the paper’s fixed pair (r,p). Therefore the paper’s argument is correct and complete for its claims, while the model’s proof is incomplete/incorrect for s=1 and overstates regularity of the Itô correction.