{"id":"12108c9c-25cf-49bd-92b1-66409c2045c0","arxiv_id":"2608.09319","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For McKean-Vlasov stochastic reaction-diffusion equations with rapidly oscillating coefficients, the paper proves finite-time, whole-line, and pullback-attractor averaging, but the proof of the main theorem contains a limiting gap.","lead":"This paper proves averaging principles for a class of random equations describing interacting particles: fast oscillations in the coefficients can be averaged out, giving a simpler effective equation. The authors extend the averaging principle to the whole timeline and to the long-time attractors of the probability laws, which is new for distribution-dependent stochastic partial differential equations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The final double limit in the proof of Theorem 2.3 is unjustified: applying Lemma A.6 with the natural constant B=C_T R^2 makes the prefactor grow like exp(C R^4), so C_{T,R} R^{-2} diverges as R→∞.","rationale":"The reader's weakest-assumption analysis is exactly the point where the proof breaks. The proof needs the limit lim_{ε→0} sup_{[0,T]} E∥uε−¯u∥^2=0, but the established bound is E sup ≤ C_{T,R}(ε^{1/8}+ω_f(ε^{-1/2})^2+ω_γ(ε^{-1/2})^2+ω_g(ε^{-1/2})+R^{-2}) with C_{T,R} growing super-exponentially in R because of the singular Gronwall lemma. The paper's own displayed argument 'Letting ε→0 and then R→∞' is therefore not justified, and no alternative R(ε)→∞ scaling is offered. Since Theorem 2.4 uses the finite-time theorem on shifted intervals (3.43) and Theorem 2.5 uses Lemma 4.12 which relies on Theorem 2.3, the gap propagates to all three main results. I do not see an independent check that rescues the estimate as written. This is a proof gap rather than a disagreement with known results, and it is possible the theorem is true with a more careful stopping-time or moment argument; but as written the central claim is unsupported. Hence the reader's REJECT verdict should stand.","tokens_in":37233,"tokens_out":5129,"duration_ms":51018,"concrete_test":"Re-derive the final estimate in Theorem 2.3 with explicit constants. From (3.34), record B=C_T R^2 in Lemma A.6 and compute the resulting prefactor C_{T,R}=E_{1/2}(C_T R^2 Γ(1/2) T^{1/2}). Then evaluate lim_{R→∞} C_{T,R} R^{-2}; if it is +∞, the double-limit passage is invalid. A complementary check is to replace the H^1-sup bound under τ_R by a bound on L^6 moments from Lemma 3.1 and see whether B can be chosen independent of R; if no R-independent B exists, a new estimate is required for the proof to go through.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The gap identified by the reader is real and load-bearing. In the proof of Theorem 2.3, after (3.39) the fractional Gronwall inequality Lemma A.6 is applied with B=C_{R,T}, where C_{R,T} is inherited from the stopping-time estimates. Under τ_R one only has ∥uε(s)∥_{H^1}+∥¯u(s)∥_{H^1}≤2R, and the term B3 in (3.34) contributes a kernel constant proportional to R^2, so B=C_T R^2. Lemma A.6 then yields the final constant C_{T,R}=E_{1/2}(C_T R^2 Γ(1/2) T^{1/2}). The Mittag-Leffler function E_{1/2}(z) grows like exp(C z^2) for large real z, hence C_{T,R} R^{-2}→∞. Thus the displayed 'Letting ε→0 and then R→∞' is not a valid limiting argument: with R fixed the term C_{T,R} R^{-2} does not vanish as ε→0, and with R=R(ε)→∞ the prefactor C_{T,R} grows faster than any polynomial in R, so the R^{-2} term still fails to vanish. Theorems 2.4 and 2.5 both invoke Theorem 2.3 (via (3.43) and Lemma 4.12), so the entire chain of main results currently rests on this unjustified limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims three averaging results for a McKean–Vlasov stochastic reaction--diffusion equation with rapidly oscillating coefficients on the torus, d ≤ 3: (i) finite-time mean-square convergence to the averaged equation (Theorem 2.3), (ii) whole-line mean-square convergence of unique bounded entire solutions under a contraction condition (Theorem 2.4), and (iii) upper-semicontinuous convergence of pullback attractors in the Wasserstein space P2(H) (Theorem 2.5). The proofs are built on a priori estimates in H and V, a time-discretization/stopping-time argument, a fractional Gronwall lemma, and a uniform second-moment tail estimate for compactness in P2(H). An illustrative mean-field feedback model is given in Section 2.3.","tokens_in":37565,"tokens_out":5607,"duration_ms":54854,"significance":"If the results were correct, they would constitute a substantial advance: they appear to be the first finite-time and whole-line averaging principles for distribution-dependent SPDEs with cubic nonlinearity, and the first attractor-convergence result for the law dynamics in Wasserstein space over a noncompact coefficient hull. The paper also contains several genuinely useful technical ingredients, including the density of P4(V) in P2(H) (Lemma 4.11) and the Wasserstein compactness criterion (Lemma 4.6). However, the central finite-time theorem contains a load-bearing gap in its final limiting argument, and since Theorems 2.4 and 2.5 invoke Theorem 2.3, the significance is conditional on a repair that is not a minor edit.","major_comments":[{"comment":"The claimed double limit in the proof of Theorem 2.3 is not justified. After combining (3.39) and (3.40), the proof obtains y(t) ≤ A_{ε,R,T} + C_{R,T}∫_0^t (t-s)^{-1/2} y(s) ds, with A_{ε,R,T} = C_{T,R}[ε^{1/8} + ω_f(ε^{-1/2})^2 + ω_γ(ε^{-1/2})^2 + ω_g(ε^{-1/2}) + R^{-2}]. Lemma A.6 then yields a bound of the form C_{T,R} times the same bracket. The constant C_{R,T} is inherited from the stopping-time estimates: on {t ≤ τ_R} one only controls ∥uε(s)∥_{H^1} + ∥ū(s)∥_{H^1} ≤ 2R, so the kernel constant in (3.34) is proportional to R^2. Lemma A.6 therefore produces a prefactor E_{1/2}(C R^2 Γ(1/2) T^{1/2}), and since E_{1/2}(z) grows like exp(C z^2) for large real z, the term C_{T,R} R^{-2} diverges as R→∞. Thus the displayed 'Letting ε→0 and then R→∞' is invalid: with R fixed the R^{-2} term does not vanish as ε→0, and with R depending on ε the prefactor grows faster than any polynomial, so the product still fails to vanish. Theorem 2.3 is therefore not proved as written.","section":"Section 3.2, Eqs. (3.39)–(3.40) and Lemma A.6"},{"comment":"The subsequent main results inherit the gap. The proof of Theorem 2.4 uses (3.43), which is exactly the finite-time estimate of Theorem 2.3 applied on the shifted interval, and the proof of Theorem 2.5 uses Lemma 4.12, which invokes Theorem 2.3. Since the finite-time convergence is not established, the whole-line convergence (2.8) and the attractor convergence in Theorem 2.5 are unsupported.","section":"Theorems 2.4 and 2.5"},{"comment":"Remark 3.6(i) claims that the finite-time convergence is uniform over initial distributions with a common V-fourth moment bound. This uniformity is obtained by the same R→∞ argument that is invalid in Theorem 2.3. As written, the assertion does not follow from the preceding estimates, because the uncontrolled dependence of C_{T,R} on R prevents the required uniform limit.","section":"Remark 3.6(i)"}],"minor_comments":[{"comment":"The notation I_{12}^2 after (3.22) is ambiguous: it denotes the third summand in the preceding display, not the square of I_{12}(t). Please rename this term.","section":"Equation (3.22)"},{"comment":"The constant C_{T,R} in Lemma 3.5 is not quantified. Since the main proof depends on the growth of this constant in R, the lemma should either state the dependence explicitly or the proof should avoid relying on it in the final double limit.","section":"Lemma 3.5"},{"comment":"In the passage from (3.39) to the following line, the additive +1 inside the integral is dropped without comment. This is harmless for t ≤ T but should be stated for clarity.","section":"Equation (3.39)"},{"comment":"Lemma A.6 does not record the dependence of C_{T,β,B} on B. Given that this dependence is crucial in the proof of Theorem 2.3, it would be helpful to state the Mittag–Leffler bound explicitly, including the exponential growth of E_{1/2} for large arguments.","section":"Lemma A.6"},{"comment":"There are several typographical and formatting issues, including the title 'A VERAGING PRINCIPLE' and some garbled equation displays in Section 2.1; these should be corrected in a revision.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The central gap is load-bearing and propagates to all main theorems. The proof strategy as presented cannot be repaired by a local change, because the Gronwall constant grows faster than the R^{-2} term; a substantial reworking of Section 3.2 or a different method would be needed. The statements may be true, but the current manuscript does not establish them."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Start with the punchline: the paper's main results are the first whole-line and attractor-level averaging principles for McKean–Vlasov SPDEs, and the architecture is genuinely interesting. But the proof of the finite-time theorem (Theorem 2.3) has a load-bearing gap at the final double limit, and since Theorems 2.4 and 2.5 call on Theorem 2.3, the chain currently doesn't close.\n\nWhat's new: the paper correctly positions itself against the existing literature: prior McKean–Vlasov averaging results stop at the first Bogolyubov theorem, and the authors' earlier SPDE results [15,17] are not law-dependent and give only distributional convergence on bounded intervals. Here they get mean-square convergence on the whole line under a contraction condition, and uniform upper-semicontinuous convergence of pullback attractors in the Wasserstein space. The density argument (P4(V) dense in P2(H)) and the uniform second-moment tail estimate in Lemma 4.7 are solid pieces of work.\n\nThe soft spot is specific and serious. After (3.39), the singular Gronwall lemma A.6 is applied with a constant B that inherits a factor R^2 from the stopping-time H^1 bound. The Mittag-Leffler factor E_{1/2}(C R^2 T^{1/2}) behaves like exp(C R^4), so the term C_{T,R} R^{-2} in the final bound diverges as R→∞. The displayed 'letting ε→0 and then R→∞' is therefore not justified. For fixed R the bound does not vanish as ε→0, and with R=R(ε)→∞ the prefactor grows faster than any polynomial. This isn't a stylistic quibble; the same limit is used in the proofs of Theorems 2.4 and 2.5 (via (3.43) and Lemma 4.12). The gap may be repairable—one could try a different decomposition or a more careful dependence on the cut-off—but as written the main results are unproven.\n\nWho this is for: stochastic analysts working on averaging and mean-field SPDEs. It deserves a serious referee; the question is whether the estimate can be fixed. I don't think a desk rejection is right, but I'd want the authors to address the R-dependence before publication.","headline":"A serious extension of averaging principles to McKean–Vlasov SPDEs, but the finite-time proof has an unjustified double limit that takes down the main theorems.","tokens_in":38076,"tokens_out":2874,"would_cite":false,"duration_ms":24977,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","70K65","37L30","37L55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that rapidly oscillating, distribution-dependent stochastic reaction–diffusion equations on the torus are governed, in the small-oscillation limit, by a single averaged equation; the approximation holds on finite…","keywords":["McKean–Vlasov SPDEs","stochastic reaction–diffusion equations","averaging principle","Wasserstein distance","pullback attractor","attractor convergence","Bogolyubov theorem","mean-field interaction"],"falsifier":"Compute the size of the constant $C_{T,R}$ produced by the singular Gronwall estimate (Lemma A.6) when applied to the kernel $(t-s)^{-1/2}$ in the proof of Theorem 2.3; if $C_{T,R}$ grows faster than quadratically in $R$, the term $C_{T,R}R^{-2}$ diverges as $R\\to\\infty$ and the double limit $\\varepsilon\\to 0$, $R\\to\\infty$ is unjustified, which would invalidate Theorems 2.4 and 2.5 as proved.","tokens_in":37038,"feed_emoji":"🎲","tokens_out":8841,"duration_ms":82082,"temperature":0.7,"pith_summary":"The paper establishes three averaging principles for McKean–Vlasov stochastic reaction–diffusion equations on the torus with rapidly oscillating coefficients, namely equations whose drift and noise depend on the solution's own law. Theorems 2.3–2.5 show that, as the oscillation period tends to zero, solutions converge in mean square to those of an averaged equation, uniformly on finite time intervals and, under a contraction condition, uniformly over the whole real line. At the level of probability laws, the pullback attractors of the nonautonomous original system converge upper-semicontinuously to the global attractor of the averaged system, uniformly over all time translates of the coefficients. A sympathetic reader should care because this moves averaging theory from finite-time statements for distribution-dependent SPDEs to global, attractor-level statements, and provides a class of mean-field feedback models, including a neural-field-like example, where the results apply.","feed_headline":"Three averaging laws proven for mean-field reaction-diffusion SPDEs","feed_subtitle":"Fast oscillations in distribution-dependent stochastic PDEs reduce to one effective equation, on finite time, all real time, and in law.","key_machinery":"The argument is carried by a block-averaging, or freezing, scheme for the mild formulation of (1.1). The rapidly oscillating functions are compared against step processes on small time blocks, with errors controlled by the uniform ergodicity rates $\\omega_f$, $\\omega_g$, $\\omega_\\gamma$ of (H3), by the analytic semigroup estimates of Lemma 3.4, and by the discretization estimate of Lemma 3.5 for the cubic term. The whole-line result uses the exponential contraction $\\rho = 2(\\lambda_1+\\lambda) - 4L_f - 4L_g^2 > 0$ to compare both equations on backward intervals $[t-N,t]$ and then let $N\\to\\infty$. The attractor result uses, in addition, the cocycle structure on the coefficient hull, a uniform second-moment tail estimate yielding relative compactness in $(\\mathcal P_2(H), W_2)$, and the density of $\\mathcal P_4(V)$ in $\\mathcal P_2(H)$, which converts finite-time averaging for smooth initial laws into uniform averaging on compact sets of initial laws.","core_discovery":"The central claim is that the fast time oscillations in the coefficients of equation (1.1), including the coefficient $\\gamma(t/\\varepsilon)$ of the non-Lipschitz cubic term, can be averaged out: the solution $u^\\varepsilon(t)$ is close in mean square to the solution $\\bar u(t)$ of (1.2) with averaged coefficients $\\bar\\gamma$, $\\bar f$, $\\bar g$. Theorem 2.3 proves this uniformly on $[0,T]$ for initial data with four finite moments in the space $V$. Theorem 2.4 adds the contraction condition $\\lambda_1+\\lambda > 2L_f + 2L_g^2$, under which both equations possess unique bounded entire solutions and $\\sup_{t\\in\\mathbb R}\\mathbb E\\|u^\\varepsilon(t)-\\bar u(t)\\|^2\\to 0$. Theorem 2.5 lifts the result to the space $(\\mathcal P_2(H), W_2)$ of probability laws: the pullback attractors $A^\\varepsilon(F)$ of the original cocycle, one for each coefficient symbol $F$ in the hull (the closure of the time translates of the coefficients), converge upper-semicontinuously to the global attractor $\\bar A$ of the averaged semigroup, uniformly in the symbol. The work thereby extends the Bogolyubov averaging principle from finite-dimensional and finite-time settings to global, law-valued asymptotics for distribution-dependent SPDEs.","pith_inferences":["Extending beyond the paper, the block-averaging rates visible in the estimates (an $\\varepsilon^{1/8}$ term from the cutoff and block size, together with $\\omega(\\varepsilon^{-1/2})$ terms) suggest that quantitative, explicit rates could be derived for finite-time and whole-line convergence in this class, though the paper itself does not isolate such rates.","Extending beyond the paper, near the threshold $\\lambda_1+\\lambda = 2L_f + 2L_g^2$ the exponential contraction degenerates; the whole-line and attractor arguments would likely need a polynomial-in-time stability estimate, and the same averaging conclusion may or may not survive.","Extending beyond the paper, the same combination of a sign-definite cubic term, Lipschitz-in-law coefficients, and uniform ergodicity of the fast coefficients could be applied to other locally monotone distribution-dependent SPDEs, such as stochastic Allen–Cahn equations with mean-field coupling, but this is not established here."],"forward_implications":["For any fixed time horizon $T$, the mean-square difference between the original and averaged solutions with the same initial condition tends to zero uniformly in $t\\in[0,T]$.","Under the contraction condition, the unique bounded entire solutions of the two equations are indistinguishable in mean square uniformly over all $t\\in\\mathbb R$, so the averaged equation captures the whole-line statistical dynamics.","The long-time law dynamics of the original nonautonomous system, encoded by its pullback attractors, collapse onto the global attractor of the averaged system as $\\varepsilon\\to 0$, uniformly over the hull of coefficient translations.","For the explicit almost-periodic mean-field feedback model of Example 2.6, all three conclusions hold, so there is a concrete family of neural-field-like interacting systems to which the theory applies."],"supporting_citations":[{"why":"supplies the prior finite-time strong-averaging framework for slow–fast McKean–Vlasov SPDEs that the present work extends to the cubic and law-dependent setting.","marker":"[30]"},{"why":"the closest prior whole-line averaging result for monotone SPDEs, which this paper generalizes to distribution-dependent coefficients and mean-square convergence.","marker":"[15]"},{"why":"provides the bounded-entire-solution argument for complex Ginzburg–Landau equations that is adapted to the law-dependent setting here.","marker":"[17]"},{"why":"gives strong and weak convergence machinery for distribution-dependent SDEs that frames the finite-time result.","marker":"[14]"},{"why":"established whole-line averaging for distribution-independent stochastic ODEs, the baseline the second Bogolyubov theorem improves to mean-square SPDE convergence.","marker":"[13]"},{"why":"recent uniform-in-time quantitative averaging for finite-dimensional multiscale SDEs, contrasted with the paper's whole-real-line uniformity.","marker":"[19]"},{"why":"provides the pullback-attractor construction for asymptotically compact nonautonomous systems used for the law dynamics.","marker":"[7]"},{"why":"supplies the Wasserstein-space compactness criterion (tightness plus uniform second-moment tails) used in Lemma 4.7.","marker":"[56]"}],"fun_headline_variants":["Fast oscillations average out in mean-field reaction-diffusion SPDEs","Three averaging laws for distribution-dependent reaction-diffusion SPDEs","Pullback attractors converge under averaging for mean-field SPDEs","Mean-field SPDEs: oscillatory coefficients reduce to one averaged law","Averaging principle for McKean-Vlasov stochastic reaction-diffusion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole chain of results rests on the limiting step after equation (3.40), where the proof sends the oscillation scale to zero and then the cut-off level to infinity, while the error constant grows with the cut-off level.","fun_headline_variants_meta":{"raw":{"variants":["Fast oscillations average out in mean-field reaction-diffusion SPDEs","Three averaging laws for distribution-dependent reaction-diffusion SPDEs","Pullback attractors converge under averaging for mean-field SPDEs","Mean-field SPDEs: oscillatory coefficients reduce to one averaged law","Averaging principle for McKean-Vlasov stochastic reaction-diffusion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000477,"raw_usage":{"total_tokens":2387,"prompt_tokens":990,"completion_tokens":1397,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":606,"completion_tokens_details":{"reasoning_tokens":1301}},"tokens_in":606,"tokens_out":1397,"duration_ms":10620,"temperature":1.0,"reasoning_tokens":1301,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:24:46.352373+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the size of the constant $C_{T,R}$ produced by the singular Gronwall estimate (Lemma A.6) when applied to the kernel $(t-s)^{-1/2}$ in the proof of Theorem 2.3; if $C_{T,R}$ grows faster than quadratically in $R$, the term $C_{T,R}R^{-2}$ diverges as $R\\to\\infty$ and the double limit $\\varepsilon\\to 0$, $R\\to\\infty$ is unjustified, which would invalidate Theorems 2.4 and 2.5 as proved.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the prior finite-time strong-averaging framework for slow–fast McKean–Vlasov SPDEs that the present work extends to the cubic and law-dependent setting."},{"cited_title":"Cheng and Z","cited_arxiv_id":null,"evidence_quote":"the closest prior whole-line averaging result for monotone SPDEs, which this paper generalizes to distribution-dependent coefficients and mean-square convergence."},{"cited_title":"Cheng, Z","cited_arxiv_id":null,"evidence_quote":"provides the bounded-entire-solution argument for complex Ginzburg–Landau equations that is adapted to the law-dependent setting here."},{"cited_title":"Cheng, Z","cited_arxiv_id":null,"evidence_quote":"gives strong and weak convergence machinery for distribution-dependent SDEs that frames the finite-time result."},{"cited_title":"Cheban and Z","cited_arxiv_id":null,"evidence_quote":"established whole-line averaging for distribution-independent stochastic ODEs, the baseline the second Bogolyubov theorem improves to mean-square SPDE convergence."},{"cited_title":"Crisan, P","cited_arxiv_id":null,"evidence_quote":"recent uniform-in-time quantitative averaging for finite-dimensional multiscale SDEs, contrasted with the paper's whole-real-line uniformity."},{"cited_title":"Caraballo, G","cited_arxiv_id":null,"evidence_quote":"provides the pullback-attractor construction for asymptotically compact nonautonomous systems used for the law dynamics."},{"cited_title":"Villani.Optimal Transport: Old and New, volume 338 ofGrundlehren Math","cited_arxiv_id":null,"evidence_quote":"supplies the Wasserstein-space compactness criterion (tightness plus uniform second-moment tails) used in Lemma 4.7."}],"review_version":1}