Persistence of Invariant Tori for Stochastic Nonlinear Schrödinger in the Sense of Most Probable Paths

The paper states three main claims for the stochastic NLS with additive, diagonal, time‑dependent noise: (1) an Onsager–Machlup functional whose minimizers coincide with deterministic Hamiltonian trajectories, (2) a Freidlin–Wentzell large deviation principle (LDP) with explicit rate J, and (3) “persistence” of invariant KAM tori in the sense that the most probable paths remain on the deterministic invariant tori. These are asserted for general infinite‑lattice Hamiltonian systems under (C1)–(C2) and then specialized to NLS, with the rate spelled out in (31) and in the NLS form with Σ−1 defined modewise (see Theorem 1.1, Theorem 1.2, and Theorem 1.3 in the file, e.g., the OM functional formula and hypotheses (C1)–(C2) , the general LDP statement and rate function derivation via tube estimates and upper/lower bounds , and the NLS specialization including the explicit Σ−1 and the KAM torus statement ). However, the proofs are not fully rigorous as written: the “Onsager–Machlup functional” is essentially the 1/ε² LDP action and is used with a heuristic tube‑probability ansatz (Definition 2.10), without addressing the well‑known subtleties of OM functionals in infinite dimensions and possible divergence terms; the LDP proof via tube coverings does not establish exponential tightness or goodness in the stated infinite‑dimensional path space, and there are notational inconsistencies (e.g., A ⊂ R^d in Theorem 1.2 for an infinite‑dimensional setting and an apparent superfluous spatial integral in the NLS rate J) . Moreover, the paper assumes a globally Lipschitz Hamiltonian vector field (C1) and then claims that the NLS Hamiltonian satisfies it, which is generally false for polynomial NLS nonlinearities on the full Hilbert space; at best one has local Lipschitz on bounded sets. By contrast, the candidate solution invokes the standard weak‑convergence/variational representation method (Budhiraja–Dupuis–Maroulas) on a Hilbert space SDE du = b(u)dt + εΣ dW to derive a good LDP with rate J(ψ) = 1/2 ∫ ||Σ^{-1}(ψ̇ − b(ψ))||² dt and correctly identifies minimizers with deterministic trajectories; it then uses classical PDE KAM results to describe the invariant tori for the deterministic flow. This approach is standard and correct at the level of principle, assuming appropriate well‑posedness and continuity of solution maps (Da Prato–Zabczyk) and invertible Σ. Thus, while the paper’s claims are directionally aligned with the model, the paper’s arguments require major tightening; the model’s solution provides a correct and more orthodox path to the result.