{"id":"192eb3ec-053f-4523-9238-30a1309ad2cf","arxiv_id":"2505.12180","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Existence of martingale solutions is established for stochastic evolution equations with pseudo-monotone polynomial drift of arbitrary order and continuous superlinear diffusion, including fractional reaction-diffusion equations.","lead":"This paper proves the existence of martingale solutions for a broad class of stochastic partial differential equations with polynomial-growth drift and superlinear, merely continuous noise, then applies the result to fractional stochastic reaction-diffusion equations. It extends recent existence theory by dropping Lipschitz assumptions on the nonlinear terms and by using a topological representation theorem that works over weak topologies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step (iv) of Theorem 2.3's proof omits the adaptation of [29, Lemma 2.16] to the multi-Banach-space weak/weak-* setting; without it the drift of the limit equation is unidentified and existence of the martingale solution is not established.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the final identification of the drift in Step (iv) of Theorem 2.3 is deferred to Lemma 2.16 of [29] without a proof of adaptation. My reading of the manuscript confirms that this is the single point on which the central claim depends. The paper develops substantial machinery—uniform estimates, tightness in the mixed weak/weak-* topology, the Skorokhod-Jakubowski representation, and the uniform integrability of ψ_n—that plausibly makes the cited argument applicable, but the multi-space extension is not automatic. In particular, the pseudo-monotonicity definition in Lemma 2.1 is pointwise in V, while the proof has only weak Bochner-space convergences; the needed Minty-type argument in the product of L^{q_j}(0,T;V_j) spaces must be supplied. I found no internal contradiction in the rest of the argument, and the application to the fractional stochastic reaction-diffusion system appears to satisfy the abstract hypotheses if Theorem 2.3 is valid. The conditional verdict is therefore appropriate: acceptance should require the missing Step (iv) proof or a precise citation establishing that [29, Lemma 2.16] covers the multi-space setting with weak/weak-* topology. I do not recommend changing the reader's verdict, only emphasizing that the omitted detail is genuinely load-bearing.","tokens_in":31428,"tokens_out":23897,"duration_ms":243249,"concrete_test":"Write out the proof of Step (iv) from (2.51)-(2.54), (2.64), (2.71), and (2.81) without citing [29]; check in particular whether the proof requires ~Z_n → ~Z strongly in L^{q_j}(0,T;V_j) for some or all j, or pointwise weak convergence of ~Z_n(t,ω) in V_j for a.a. (t,ω). If such a condition is indispensable and cannot be obtained from the established tightness and convergence, then the conclusion of Theorem 2.3 is unsupported. If the Bochner Minty argument goes through with only the stated weak and weak-* convergences, the gap is merely expository and conditional acceptance with a supplied proof is appropriate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.4, Step (iv), is the decisive step: after obtaining weak limits ~A_j of A_j(·,Z~_n) in (2.51)-(2.52), the paper must show ∑_j ~A_j = ∑_j A_j(·,Z~) a.e. so that (2.63) becomes the original equation. Instead of a proof, the paper states that 'by the pseudo-monotone argument of Lemma 2.16 in [29], one can prove' this, and that 'the details are omitted here.' The cited lemma was proved in [29] for a single Banach space V with a strong Skorokhod representation in a metric space. The present theorem requires the same conclusion in the product setting V = ∩_{j=1}^J V_j, using only convergence of Z~_n to Z~ in L^1(0,T;H) (via (2.54)), weak convergence in L^{q_j}(0,T;V_j), weak-* convergence in L∞(0,T;H), the liminf inequality (2.64), and the pointwise upper bound (2.71) with ψ_n converging in L^1. None of these by themselves obviously imply the pointwise-in-(t,ω) weak convergence in V_j that the scalar pseudo-monotonicity definition in Lemma 2.1 postulates, nor the strong L^q(0,T;V) convergence used in [29]. If the Bochner-space version of [29, Lemma 2.16] does not transfer, the drift ∑~A_j in (2.63) is unidentified and Theorem 2.3 (hence the application in Theorem 3.2) does not follow.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves an abstract existence result, Theorem 2.3, for martingale solutions of stochastic evolution equations dX = A(t,X)dt + B(t,X)dW with pseudo-monotone drift of polynomial growth and continuous, possibly superlinear, diffusion coefficients that are not locally Lipschitz. The proof uses Galerkin approximations, a priori estimates (Lemmas 2.4 and 2.5), tightness in weak and weak-* topologies (Lemmas 2.6 and 2.7), and the Skorokhod-Jakubowski representation theorem to pass to the limit. The limiting drift is identified in Step (iv) of Section 2.4 by appealing to Lemma 2.16 of [29] without proof. The abstract result is then applied in Theorem 3.2 to the fractional stochastic reaction-diffusion system (1.1)-(1.3) with decreasing polynomial drift and superlinear noise, and a pathwise uniqueness result is given in Theorem 3.3 under additional assumptions.","tokens_in":31588,"tokens_out":13807,"duration_ms":143900,"significance":"If the proof is completed, the paper would make a genuinely useful extension of existing well-posedness results: it removes local Lipschitz continuity from both drift and diffusion and allows superlinear noise with growth order below the drift's polynomial order. The a priori estimates, the tightness argument, and the separation between existence and pathwise uniqueness are valuable and are written out in considerable detail. The main obstacle is that the decisive identification of the limiting drift is deferred to an external lemma in a different setting; this point must be fully resolved before the central claim can be regarded as established. The paper also has the merit of avoiding the disputed strong Skorokhod representation by using the Jakubowski theorem in non-metric topologies.","major_comments":[{"comment":"","section":"Section 2.4, Step (iv)"},{"comment":"","section":"Section 3, verification of (H4) for A_3"}],"minor_comments":[{"comment":"","section":"Section 2.3, Lemma 2.7"},{"comment":"","section":"Section 2.3, Lemma 2.8"},{"comment":"","section":"Section 2.4, Step (iii)"},{"comment":"","section":"Section 3, proof of Theorem 3.2"}],"recommendation":"major_revision","confidential_remarks":"The omission in Step (iv) is explicitly acknowledged by the author, so this is not a hidden gap. The editor should require a full proof of the adapted pseudo-monotone limit argument, not just a citation, before publication. The Section 3 verification of (H4) for the h-term also needs correction or a strengthened assumption; both issues are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper extends Rockner–Shang–Zhang’s martingale solution theory to a finite family of Banach spaces, so the drift can have polynomial growth of arbitrary order and the diffusion can be continuous with subcritical superlinear growth. That is a genuinely useful extension, and the application to fractional reaction-diffusion is worked out carefully. But the proof of the abstract theorem contains a serious omission at the very point where the drift limit is identified, and the paper’s own text concedes the details are not there. That has to be fixed before I would trust the main theorem.\n\nWhat is good: the two-space (V1,V2) structure is the right way to decouple the fractional Laplacian from the polynomial nonlinearity. The a priori estimates in Lemma 2.4 and the growth bounds in Lemma 2.5 are standard but written out. The tightness argument in weak and weak-* topologies (Lemma 2.7) is careful, and using Skorokhod–Jakubowski for the nonmetric path space is appropriate, especially given the authors’ awareness of the issues with strong Skorokhod representation. Section 3, where the abstract conditions are verified for the fractional reaction-diffusion equation, is the strongest part: Lemma 3.1 on continuity of the noise is detailed and convincing. So the machinery is there and most of the proof is solid.\n\nWhere it is soft: Step (iv) of Theorem 2.3. The drift convergence sum_j A_j(·,Z~_n) -> sum_j A_j(·,Z~) is load-bearing. Without it, equation (2.63) is not the original SPDE and existence is not established. The paper supplies the uniform integrability of ψ_n, which is a necessary ingredient, but it does not supply the actual pseudo-monotone transfer, and the cited Lemma 2.16 of [29] was stated for a single Banach space with strong convergence. The adaptation to the product of spaces with weak/weak-* topologies is not automatic. This is not a cosmetic gap. Second, the novelty claim is aimed only at [29]; the relation to [32] and [37], which already treat superlinear noise in different settings, is not discussed, so a referee cannot fully judge the marginal contribution of the application. Minor: the remark that the strong Skorokhod representation theorem is incorrect even in a Polish space is surprising, since the classical theorem is true there; presumably [26] concerns a different strong version, but the paper should say what it means.\n\nBottom line: the paper deserves a serious referee, but the referee should be told to focus on Step (iv). If the gap can be filled, this is a solid, citable extension. As is, it is a conditional accept at best.","headline":"A useful extension of the Rockner–Shang–Zhang existence result to arbitrary polynomial drift and continuous superlinear noise, but the proof of Theorem 2.3 skips the one step that actually identifies the drift limit.","tokens_in":32322,"tokens_out":5164,"would_cite":true,"duration_ms":50727,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60H15","37L55","35R60"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that martingale solutions exist for a broad class of stochastic PDEs with pseudo-monotone drift and superlinear, merely continuous noise, without any local Lipschitz condition.","keywords":["martingale solution","pseudo-monotone operator","superlinear noise","fractional Laplacian","reaction-diffusion equation","Skorokhod-Jakubowski theorem","Galerkin method","pathwise uniqueness"],"falsifier":"One concrete test is to take the scalar equation with $f(u)=-|u|^{p-2}u$ and a continuous superlinear $\\sigma$, and check directly whether the pseudo-monotone step (2.64)-(2.63) identifies the limit drift when the Galerkin sequence converges only weakly in $L^p$; a sequence satisfying the bound (2.71) with two distinct possible limit drifts would falsify the argument.","tokens_in":31002,"feed_emoji":"🧮","tokens_out":9634,"duration_ms":85584,"temperature":0.7,"pith_summary":"This paper proves that a broad class of stochastic partial differential equations has martingale solutions even when the drift is only pseudo-monotone with polynomial growth of arbitrary order and the noise is merely continuous with superlinear growth; neither term has to be locally Lipschitz. The abstract result, Theorem 2.3, covers the evolution equation $dX = A(t,X)\\,dt + B(t,X)\\,dW$ in a framework with several reflexive Banach spaces, and Theorem 3.2 applies it to the fractional stochastic reaction-diffusion equation with decreasing continuous drift and superlinear noise. If correct, this extends earlier existence results that required linear noise growth and differentiable or locally Lipschitz nonlinearities, and it makes the same machinery available for fractional $p$-Laplace and tamed Navier-Stokes equations.","feed_headline":"Superlinear noise still yields martingale solutions","feed_subtitle":"New proof: fractional reaction-diffusion equations admit martingale solutions with merely continuous drift and noise.","key_machinery":"The load-bearing object is the pseudo-monotone drift operator $A(t,\\cdot)=\\sum_{j=1}^J A_j(t,\\cdot)$ acting on the intersection $V=\\bigcap_{j=1}^J V_j$ of several reflexive Banach spaces, together with the compact embedding $V\\subset H$. Pseudo-monotonicity here means that weak convergence of $v_n$ to $v$ plus $\\liminf_n (A(t,v_n),v_n-v)\\ge 0$ lets one pass to the limit in the duality pairing. The proof machinery consists of Galerkin approximations, uniform moment estimates from Itô's formula, tightness in $L^{q_j}_w(0,T;V_j)\\cap L^\\infty_{w*}(0,T;H)\\cap C([0,T],V^*)$, and the Skorokhod-Jakubowski representation theorem for topological spaces, which replaces the invalid strong Skorokhod theorem. The final identification $\\sum_{j=1}^J \\tilde A_j = \\sum_{j=1}^J A_j(\\cdot,\\tilde Z)$ is delegated to the pseudo-monotone argument of [29] adapted to the multi-space setting.","core_discovery":"The central discovery is that the existence of martingale solutions survives the combination of three relaxations: the drift may be only continuous and decreasing with polynomial growth of arbitrary order, the noise may grow superlinearly with order $q