{"id":"baf5dabb-69d4-49d6-9468-087878e6c0a3","arxiv_id":"2502.02162","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"The authors construct invariant probability measures supported on level sets of the renormalized mass for the defocusing cubic nonlinear Schrodinger equation on the one- and two-dimensional torus.","lead":"This paper builds new probability measures that the cubic Schrodinger flow leaves unchanged on the torus, with the total mass pinned to a fixed value. It works in one and two dimensions and handles very irregular solutions using Malliavin calculus and surface measures.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The recursive L^p estimate for the renormalized nonlinearity in Section 3 (Props. 3.1 and 3.2) rests on an admitted but unsupplied independence step; without a replacement, the W^∞ regularity supporting Theorem 5.1 is unproven.","rationale":"The strongest claim requires three ingredients: a surface measure σ_r from the Airault-Malliavin theorem, W^∞ regularity of the renormalized vector field :B: so that it has a good redefinition on V_r, and a compactness/flow argument. The reader identifies the second ingredient as the weakest, and I agree. The W^∞ estimates in Propositions 3.1-3.4 are exactly the bridge between the Gaussian Wiener-space estimates and the σ_r-level-set dynamics. The independence device appears twice (Props. 3.1 and 3.2), is explicitly labeled erroneous, and is not backed by a replacement; without it, the higher L^p integrability of :B: and its gradients is not established. I do not regard this as fatal: hypercontractivity for bounded-degree Gaussian chaos is standard and would likely close the gap, converting the uniform L^2 estimate into a uniform L^p estimate. But because the paper neither states nor applies that tool, a conditional acceptance with the missing argument as a condition is the appropriate verdict. I also considered two secondary issues but chose not to weight them equally: the theorem asserts σ_r for every r>0 while the proof only uses ρ(r)>0, though positivity is plausible for a nondegenerate Gaussian image; and the Duhamel formula in Theorem 5.1 omits the +u linear term present in Eq. (4), though adding a constant phase rotation would likely repair the statement. Neither undermines the core construction as directly as the unsupplied L^p argument.","tokens_in":19556,"tokens_out":11811,"duration_ms":113598,"concrete_test":"Check whether Prop. 3.1 can be proved by substituting Gaussian hypercontractivity for the independence step: show that for the degree-at-most-3 polynomial P_{N,k}=:B:_{N,k} one has E|P_{N,k}|^{2p} ≤ C_p (E|P_{N,k}|^2)^p with C_p independent of N and k, then use Minkowski's inequality to sum over k with weights |k|^{2β}, obtaining sup_N E||:B:_N||^p_{H^β} < ∞ from the uniform L^2 bound already displayed in Section 3. Repeat the same derivation for ∇^s :B: (degree at most 3-s) to verify Prop. 3.2/3.4. If this yields the stated W^∞ regularity, the gap is a missing proof detail and the paper's conclusion stands; if the L^p constants grow with N or the derivative sums diverge, the d=2 surface-measure construction lacks proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Prop. 3.1 (Section 3, after Eq. (8)) the displayed chain replaces the random factor |B_k-C_N φ_k|^2 by the deterministic value I_{N,1}+I_{N,2} under the asserted independence of X=|B_k-C_N φ_k|^{2p} and |B_k|^2-C_N^2 |φ_k|^2. The text explicitly says this independence is used erroneously and that the argument can be made rigorous, but no rigorous replacement is given. Moreover, the intermediate bracket with |B_k|^2-C_N^2 |φ_k|^2 is not algebraically equivalent to |B_k-C_N φ_k|^2, so the displayed derivation is not valid as written. Prop. 3.2 repeats the same recursive independence device for the gradient, and Propositions 3.3-3.4 rely on the same pattern. These estimates are the only route to the W^∞ regularity of :B: on the Wiener space, which is what permits the redefinition :B:^* on the level set V_r and the Ambrosio-Figalli flow argument in Theorem 5.1. Thus the central claim genuinely depends on closing this gap. A plausible repair exists: for fixed N,k, :B:_{N,k} is a polynomial of degree at most 3 in the Gaussian variables, so Gaussian hypercontractivity would convert the already-proved uniform L^2 bounds into uniform L^p bounds, and similarly for derivatives. But the paper itself does not supply this argument.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs probability measures σ_r supported on level sets V_r of a renormalized mass E for the defocusing cubic NLS on the torus in dimensions d=1,2, and claims existence and uniqueness of a flow u_t solving the (renormalized) NLS, with σ_r invariant under the flow. The construction uses the Airault-Malliavin coarea formula for surface measures on a Gaussian (abstract Wiener) space, Sobolev estimates for the Wick-ordered nonlinearity :B:, and the Ambrosio-Figalli theory of flows for Sobolev vector fields. The main theorem is stated as Theorem 1.1 and Theorem 5.1.","tokens_in":19966,"tokens_out":24015,"duration_ms":229825,"significance":"If the main theorem is correct, this would be the first construction of invariant measures supported on mass level sets for the 2D cubic NLS, going beyond Bourgain's Gibbs-measure constructions and the 1D mass-conditioned results of Oh-Quastel and Brereton. The paper's approach is original: it combines Malliavin calculus, the Airault-Malliavin coarea formula, and Ambrosio-Figalli flows. A notable strength is that the renormalization constants Z_k and the subtraction term C_N are derived from Gaussian variances, not fitted to force the conclusion, and the L^2 estimates for the Wick-ordered nonlinearity are largely credible. However, several load-bearing steps in the proof are not rigorously supplied, and at least one displayed equation appears inconsistent with the NLS dynamics.","major_comments":[{"comment":"The recursive L^p estimates use an admitted erroneous independence assumption. After Eq. (8), the displayed chain replaces the random factor |B_k - C_N φ_k|^2 by the deterministic value I_{N,1}+I_{N,2}, but |B_k|^2 - C_N^2|φ_k|^2 is not equal to I_{N,1}+I_{N,2}; these are expectations, not identities. The text states that independence is used erroneously and that the argument can be made rigorous, yet no replacement is supplied. Proposition 3.2 repeats the same device for the gradient, and Propositions 3.3-3.4 inherit the pattern. Since the W^∞ regularity of :B: is the basis for the redefinition :B:^* and for the flow argument in Theorem 5.1, this gap is load-bearing. A repair via Gaussian hypercontractivity for polynomial :B:_{N,k} seems plausible but is not in the manuscript.","section":"Section 3, Propositions 3.1 and 3.2"},{"comment":"The integral equation displayed in Eq. (11) and in Theorem 5.1(i) does not appear to be the Duhamel formula for the cubic NLS. With A=Δ as defined in Section 2.3, the equation i∂_t u = -Δu + |u|^2 u is equivalent to ∂_t φ_k = -i|k|^2 φ_k - i B_k(φ), whose integral form is u_t = e^{itA}u_0 - i∫_0^t e^{i(t-s)A} B(u_s) ds. The manuscript instead writes u_t = e^{itA}u_0 + ∫_0^t e^{-i(t-s)A} :B:^*(u_s) ds; differentiating this expression does not yield the NLS evolution. This is a central statement of the theorem, and the subsequent estimates are built on this formula. Please correct the sign/conjugation and re-derive the relevant bounds.","section":"Section 5, Eq. (11) and Theorem 5.1(i)"},{"comment":"The Gaussian measure μ_2 and the renormalized mass E are defined through sums over k≠0, so the phase space is effectively the zero-mean subspace of L^2. However, the cubic NLS (1) does not preserve the zero-mean subspace: the zero Fourier mode evolves according to dφ_0/dt = -i B_0(φ), which is generally nonzero even when φ_0(0)=0. The flow constructed in Theorem 5.1 uses only the modes k≠0 and therefore appears to solve a projected, zero-mean system rather than the full equation (1) or (4). The paper must either include the zero mode in the construction (for instance via a massive Gaussian weight) or explicitly prove invariance for the projected system and state the theorem accordingly. Without this, the claim that σ_r is invariant for the NLS flow is not established.","section":"Sections 2.3, 4, and Theorem 5.1"},{"comment":"The proof of E∈W^8 is incomplete. The displayed expansion of ∫|E|^p dμ_a is not the correct multinomial expansion of (∑_k X_k)^p: it keeps only the diagonal term and the term with all indices distinct, omitting terms such as X_k^2 X_l with repeated indices. Consequently the displayed equalities are not identities. The conclusion may still be true (for instance by Rosenthal's inequality or hypercontractivity for the independent centered variables X_k=|φ_k|^2-Z_k), but the proof as written does not establish it. Since E∈W^8 is required by Theorem 2.1 for the surface measure construction, a rigorous L^p bound for E is needed.","section":"Section 4, Proposition 4.1"},{"comment":"The uniqueness of the flow is delegated to [2, Theorem 4.7] after only a brief indication. To apply the Ambrosio-Figalli theorem, one must verify that the limit vector field :B:^* has Sobolev regularity with respect to the surface measure σ_r (not only with respect to μ_2) and that its divergence with respect to σ_r is zero. The paper states these properties but does not prove them; the divergence-free condition is shown only for the truncated fields :B:^n, and the passage to the limit is not justified. Please spell out the verification or provide a reference with the precise conditions met here.","section":"Section 5, uniqueness step"}],"minor_comments":[{"comment":"The abstract contains a LaTeX artifact \"Schr\\\"odinger\"; the final PDF should be checked for this and similar encoding issues.","section":"Abstract and title"},{"comment":"The product measure Π_{k∈Z^d} dμ_k^a is not a probability measure for k=0 because |k|^a=0 makes the density zero; the paper should explicitly state whether the zero mode is excluded from the Gaussian measure and how the level sets V_r are defined in that case.","section":"Section 2.1"},{"comment":"The definition E(φ)=∫|φ|^2 dx - 1_{d=2}∑_{k≠0} 2/|k|^2 mixes an integral over all frequencies with a renormalization over nonzero frequencies; the zero-mode contribution should be made explicit.","section":"Section 1, Eq. (5)"},{"comment":"In the estimates for the second tightness condition, the passage from convergence of :B:^n in L^p_μ to convergence in L^p_σr is not demonstrated; a short proof using the coarea formula and Proposition 3.1 would clarify the step.","section":"Section 5, tightness proof"},{"comment":"Reference [10] has a typo in the journal name (\"J. Funct. AnaL\"); it should read \"J. Funct. Anal.\".","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper has a promising construction and the L^2 core estimates are plausible, but the admitted independence gap in Section 3 and the apparent zero-mode issue are serious. The zero-mode problem is not mentioned in the reader's report; it may require either a substantial reworking of the theorem statement (to the projected zero-mean system) or an extension of the Gaussian framework. The Duhamel sign error is likely a fixable local mistake, but it should be corrected carefully. I recommend major revision rather than rejection because the methods are valuable and the gaps, while load-bearing, may be addressable within the same technical framework."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe headline: this paper does something genuinely new—2D invariant measures for cubic NLS conditioned on renormalized mass level sets—and the L^2 part of the analysis is mostly solid. But the proof as written has a load-bearing gap in the L^p estimates for the renormalized nonlinearity, and the authors know it. They explicitly write that they use independence erroneously and that the argument can be made rigorous, but no rigorous replacement is supplied. I'd take the paper for peer review, with the request to close that gap.\n\nWhat's new and good: The construction via Airault-Malliavin surface measures is a real extension of the Oh-Quastel and Brereton results to d=2 with Wick-ordered mass and nonlinearity. The renormalization constants are derived from Gaussian variances, not chosen to make the conclusion true—no circularity. The L^2 bounds on :B: and the convergence of truncated vector fields appear carefully done, and the appendix's series estimates seem fine. The paper is honest about what it borrows and what it adds.\n\nThe soft spot: Propositions 3.1 and 3.2 (and 3.3–3.4, which lean on the same device) prove the W^∞ regularity of :B: by a recursive step that assumes independence of |B_k−C_N φ_k|^{2p} and |B_k|^2−C_N^2|φ_k|^2. The text says the independence is used erroneously and a rigorous argument exists but isn't given. This is not a cosmetic gap: W^∞ regularity is exactly what lets the authors redefine :B: on the level set V_r and invoke the Ambrosio–Figalli flow machinery in Theorem 5.1. Without it, the main theorem is unsupported. The good news is the fix is likely standard—for fixed N,k, :B:_{N,k} is a degree-3 polynomial in Gaussian variables, so Gaussian hypercontractivity should turn the proved uniform L^2 bounds into the needed L^p bounds. But the paper doesn't supply that argument, so a referee can't verify the central claim as written.\n\nMinor point: Theorem 1.1 says \"for all r>0\", while Theorem 5.1 restricts to r with ρ(r)>0. That's probably just a statement mismatch, but worth clarifying; ρ(r)>0 for every r>0 isn't proved.\n\nBottom line: I'd send this to a good stochastic-analysis referee. The construction is interesting and the missing step is plausibly repairable, but the version in front of us is conditional on that repair. For my own work I'd cite it cautiously, after the fix appears.","headline":"Genuinely new 2D mass-constrained invariant measures for NLS, but the paper's central W^∞ regularity claim rests on an admitted, unsupplied independence argument; deserve peer review with a request to close that gap.","tokens_in":20479,"tokens_out":3245,"would_cite":true,"duration_ms":30880,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q55","28C20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For the defocusing cubic NLS on the 1- and 2-dimensional torus, the paper constructs, for every r>0, a probability measure on the mass level set {E=r} and a unique flow, defined almost surely, that preserves it.","keywords":["invariant measure","surface measure","cubic nonlinear Schrödinger equation","mass level set","renormalization","torus","Wiener space","quasi-sure analysis"],"falsifier":"Take $d=2$, a frequency $k\\ne 0$, a truncation $N$, and $p=2$. Expand $E[|B_k-C_N\\varphi_k|^6]$ directly using the exact Gaussian moments $E[|\\varphi_j|^{2m}]=2^m m! |j|^{-2m}$, keeping all cross-contractions between the two factors $|B_k-C_N\\varphi_k|^4$ and $|B_k|^2-C_N^2|\\varphi_k|^2$, which share $\\varphi_k$. If the ratio to $E[|B_k-C_N\\varphi_k|^4]$ is unbounded as $N\\to\\infty$, Proposition 3.1 fails; if it is bounded, the recursive step can likely be made rigorous.","tokens_in":19362,"feed_emoji":"🌊","tokens_out":17601,"duration_ms":153135,"temperature":0.7,"pith_summary":"For the defocusing cubic Schrödinger equation on the torus $\\mathbb{T}^d$ with $d=1,2$, the paper constructs, for every $r>0$, a probability measure $\\sigma_r$ supported on the level set $V_r=\\{\\varphi: E(\\varphi)=r\\}$ of the renormalized mass, together with a unique flow $u_t$, defined $\\sigma_r$-almost everywhere for all real times, that solves the cubic NLS equation (after renormalization in dimension 2) and leaves $\\sigma_r$ invariant. The interest is that the measure is concentrated on a thin mass shell rather than spread over the whole support of the Gibbs measure, and the paper states this is the first such result in dimension 2, where both the nonlinearity and the mass must be renormalized. The one-dimensional case is treated as a simpler companion that needs no renormalization. If the construction is correct, rough initial data drawn from $\\sigma_r$ still evolve uniquely for all times and the mass shell is a genuinely invariant structure.","feed_headline":"Invariant measures pinned to mass shells for cubic NLS","feed_subtitle":"In dimensions 1 and 2 a flow exists almost surely on each renormalized mass surface and preserves the surface measure.","key_machinery":"The mechanism is the disintegration of a weighted Gaussian measure on a Wiener space into surface measures on level sets of a smooth functional. The two objects that carry the argument are the renormalized mass $E(\\varphi)=\\sum_k(|\\varphi_k|^2-Z_k)$, with $Z_k=c_1|k|^{-2}$ in dimension 2, and the renormalized cubic nonlinearity $:B:$, defined as the limit of $:B:^N_k=B_k-4\\sum_{1\\le|m|\\le N}|m|^{-2}\\varphi_k$. The paper proves that $E$, the inverse of the norm of its gradient, and all stochastic derivatives of $:B:$ belong to the intersection of all Sobolev classes $W^\\infty$ with respect to the Gaussian measure; this makes the surface-measure construction applicable and lets the vector field $A-:B:$ be redefined on the level sets with zero divergence, so that the surface measures are invariant under the truncated flows.","core_discovery":"The central claim is Theorem 1.1, sharpened as Theorem 5.1: for every $r>0$ there is a Borel probability measure $\\sigma_r$ on $L^2$ with support in $V_r=\\{E=r\\}$, and there is a flow $u\\in C(\\mathbb{R}\\times V_r)$ such that $u_t$ solves equation (1) when $d=1$ and the renormalized equation (4) when $d=2$, for $\\sigma_r$-almost every initial condition, and $(u_t)_*\\sigma_r=\\sigma_r$ for every $t\\in\\mathbb{R}$. The flow is obtained as a limit of finite-dimensional truncations: the truncated vector fields are divergence-free with respect to the surface measure on each level set, tightness gives a weak limit of their laws on path space, a standard representation theorem for weak limits gives an almost-sure representation, and the invariance passes to the limit. Uniqueness follows from the continuity-equation framework for Sobolev vector fields on Wiener spaces.","pith_inferences":["The paper does not discuss momentum level sets, but the same surface-measure machinery would plausibly apply to $\\{P=b\\}$ in dimension 2 once the momentum functional and the inverse of its gradient are shown to have the same $W^\\infty$ regularity.","A numerical check of finite-dimensional truncations on $V_r$ could test the invariance before the analysis is fixed: if the empirical law of $E(v^n_t(\\varphi))$ drifts away from $r$ as $n$ grows, the claimed limiting invariance would be called into question.","If the missing recursive estimate cannot be repaired by direct Gaussian moment expansions, the theorem might still be true by another route, such as bounding the mixed moments with Wick-contraction combinatorics; the paper's own admitted independence error points to that combinatorics as the natural repair."],"forward_implications":["Initial data drawn from $\\sigma_r$ are almost surely in $H^\\beta$ for every $\\beta<0$, so the flow is defined on rough data concentrated on a mass shell rather than on smooth data.","The invariance $(u_t)_*\\sigma_r=\\sigma_r$ holds for all real times, so the mass shell $\\{E=r\\}$ is preserved exactly in measure, not just approximately.","In dimension 1 the same construction yields the result without renormalization, giving a simpler family of mass-conditioned invariant measures for the cubic NLS.","The paper notes the argument extends to dispersion $(-\\Delta)^s$ for $s>0$ and odd-power nonlinearities, with the renormalization constant adjusted from $a=2$ to $a=2s$.","By the paper's account this is the first two-dimensional construction of invariant measures conditioned on a mass level; earlier mass- and momentum-conditioned Gibbs measures were one-dimensional."],"supporting_citations":[{"why":"constructs surface measures on level sets of non-degenerate functionals on Wiener spaces and proves the coarea formula used to define sigma_r.","marker":"[1]"},{"why":"supplies the continuity-equation framework for Sobolev vector fields on Wiener spaces used to prove existence and uniqueness of the flow u_t.","marker":"[2]"},{"why":"renormalizes the 2D defocusing cubic NLS and builds the weighted Gibbs measure mu, providing the starting measure and the renormalization pattern for :B:.","marker":"[5]"},{"why":"introduces the surface-measure construction on level sets of the enstrophy for the Euler equation, the template the paper adapts to mass level sets.","marker":"[9]"},{"why":"provides the quasi-sure analysis and redefinition technology that allows E, grad E, and :B: to be evaluated on level sets.","marker":"[22]"},{"why":"establishes earlier invariant Gibbs measures conditioned on mass and momentum in one dimension, the comparison point that this two-dimensional result extends.","marker":"[23]"}],"fun_headline_variants":["Mass-shell measures made invariant for cubic NLS","Cubic NLS keeps measures pinned to mass levels","Invariant flows on each mass shell for cubic NLS","Surface measures survive cubic NLS flow","Cubic NLS: new invariant measures on mass sets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on being able to repair a gap in Section 3: the proof of the key norm bounds for the renormalized nonlinearity uses an independence of two random variables that the text itself calls erroneous, and no rigorous replacement is supplied.","fun_headline_variants_meta":{"raw":{"variants":["Mass-shell measures made invariant for cubic NLS","Cubic NLS keeps measures pinned to mass levels","Invariant flows on each mass shell for cubic NLS","Surface measures survive cubic NLS flow","Cubic NLS: new invariant measures on mass sets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000856,"raw_usage":{"total_tokens":3627,"prompt_tokens":764,"completion_tokens":2863,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":380,"completion_tokens_details":{"reasoning_tokens":2789}},"tokens_in":380,"tokens_out":2863,"duration_ms":17688,"temperature":1.0,"reasoning_tokens":2789,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T13:07:01.034227+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $d=2$, a frequency $k\\ne 0$, a truncation $N$, and $p=2$. Expand $E[|B_k-C_N\\varphi_k|^6]$ directly using the exact Gaussian moments $E[|\\varphi_j|^{2m}]=2^m m! |j|^{-2m}$, keeping all cross-contractions between the two factors $|B_k-C_N\\varphi_k|^4$ and $|B_k|^2-C_N^2|\\varphi_k|^2$, which share $\\varphi_k$. If the ratio to $E[|B_k-C_N\\varphi_k|^4]$ is unbounded as $N\\to\\infty$, Proposition 3.1 fails; if it is bounded, the recursive step can likely be made rigorous.","supporting_citations":[{"cited_title":"Intégration géométrique sur l’espace de Wiener","cited_arxiv_id":null,"evidence_quote":"constructs surface measures on level sets of non-degenerate functionals on Wiener spaces and proves the coarea formula used to define sigma_r."},{"cited_title":"On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions.J","cited_arxiv_id":null,"evidence_quote":"supplies the continuity-equation framework for Sobolev vector fields on Wiener spaces used to prove existence and uniqueness of the flow u_t."},{"cited_title":"Invariant measures for the2d-defocusing nonlinear Schrödinger equa- tion","cited_arxiv_id":null,"evidence_quote":"renormalizes the 2D defocusing cubic NLS and builds the weighted Gibbs measure mu, providing the starting measure and the renormalization pattern for :B:."},{"cited_title":"The two-dimensional Euler equation: a statistical study.Comm","cited_arxiv_id":null,"evidence_quote":"introduces the surface-measure construction on level sets of the enstrophy for the Euler equation, the template the paper adapts to mass level sets."},{"cited_title":"Springer-Verlag, Berlin, 1997","cited_arxiv_id":null,"evidence_quote":"provides the quasi-sure analysis and redefinition technology that allows E, grad E, and :B: to be evaluated on level sets."},{"cited_title":"On invariant Gibbs measures conditioned on mass and momentum.J","cited_arxiv_id":null,"evidence_quote":"establishes earlier invariant Gibbs measures conditioned on mass and momentum in one dimension, the comparison point that this two-dimensional result extends."}],"review_version":1}