{"id":"a4ed7247-64de-489a-9bbc-5c145c720048","arxiv_id":"2411.09058","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Spatial averages of the critical SHE with Riesz-2 noise are Gaussian at times much smaller than R^2, non-Gaussian at time proportional to R^2, and go extinct at times much larger than R^2.","lead":"This paper proves that spatial averages of a critical long-range stochastic heat equation have three different scaling limits depending on how time and radius grow together. It includes the first non-Gaussian spatial average limit reported for an SPDE, which separates critical from subcritical noise.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.8 contains an index inconsistency: the display before (3.16) inserts factors ∥η_j∥^{-2} for all j≥2, incompatible with (3.15) and (3.16); as written, Proposition 3.3, and hence part (1), is not established.","rationale":"The reader's weakest_assumption focused on the scaling identity of Lemma 3.1 and its dependence on uniqueness in law from [MT04]. That is a legitimate gap, but it is largely mitigated by the paper's citation to [MT04, Lemma 2] and by a direct chaos-by-chaos check; it is not the most concrete obstruction to the central claim. The more load-bearing issue is in the proof of Lemma 3.8, which is the technical heart of the higher-chaos negligibility needed for part (1). The displayed inequality before (3.16) is inconsistent with (3.15) and with (3.16): the former has a product starting at i=m+2, while the latter has factors for all j≥2. This makes the proof of Proposition 3.3 incomplete as written. However, the fix is straightforward and local: changing the lower index in the display from j=2 to j=m+2 makes the computation consistent, and the resulting convolution bounds and summability checks appear valid. I therefore recommend accepting the paper conditional on this correction rather than rejecting it. The central theorem is plausible and the scaling parts (2) and (3) are well-supported; the index issue does not suggest the result is false.","tokens_in":15007,"tokens_out":40339,"duration_ms":320285,"concrete_test":"Rewrite the proof of Lemma 3.8 with the corrected right-hand-side product ∏_{j=m+2}^{n}(∥η_j−η_{j−1}∥^{−d+2}∥η_j∥^{−2})∥η_n∥^{−d+2}J^2, and verify that (3.16), the bound C′(n,d), and the summability of ∑_{n≥2}κ^{2n}c_d^nC′(n,d) for all 0<κ<(d−2)/2 follow. Separately, test the printed j=2 version on a concrete case (e.g. n=3, m=1, d=4, η1 near 0, η2 large) to show the claimed LHS≤RHS inequality fails, confirming the correction is necessary.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The proof of Theorem 1.1 part (1) depends on Proposition 3.3, which is proved via Lemmas 3.7 and 3.8. In Lemma 3.8, equation (3.15) has the product ∏_{i=m+2}^{n-1}∥η_i∥^{-2} inside the integral. The next display claims this is bounded by an integral with ∏_{j=2}^{n}(∥η_j−η_{j−1}∥^{−d+2}∥η_j∥^{−2})∥η_n∥^{−d+2}. For m>0, the right-hand side inserts extra factors ∥η_j∥^{-2} for j=2,…,m+1 that are not present in (3.15). These factors are not uniformly ≥1, so the claimed pointwise inequality is false (for example, take m=1, η1 near 0, and η2 large). Formula (3.16), with constants k_{2i+2−γ′,2,d}, corresponds to a product starting at j=m+2, not j=2. Thus the displayed inequality and (3.16) are mutually inconsistent, and as printed the proof of Lemma 3.8 is incomplete. The likely fix is changing the lower index in that display from j=2 to j=m+2; with that correction, the convolution argument and the summability of κ^{2n}c_d^nC′(n,d) appear to go through. This is a concrete technical gap in the written proof, not a claim that the theorem is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the time-dependent spatial averages of a critical stochastic heat equation in d≥3 with noise white in time and colored in space with covariance kernel ∥x−y∥^{−2}. The solution is the measure-valued process constructed by Mueller and Tribe, and the main result is a three-regime phase diagram for the average over a ball of radius R at time t_R: for t_R=o(R^2) the normalized average converges to a centered Gaussian with variance σ²; for t_R=cR^2 the average equals u_c(B_1) in law, a positive non-Gaussian random variable; for t_R^{−1}=o(R^{−2}) the average tends to zero in probability. The proof uses the scaling invariance of the equation to reduce parts (2) and (3) to previous results of Mueller and Tribe, and to reduce part (1) to the fixed-time t=1 case. The fixed-time CLT is proved via the Wiener chaos decomposition, with the first-chaos variance computed explicitly and the higher-chaos variance sum shown to vanish.","tokens_in":15348,"tokens_out":22793,"duration_ms":170814,"significance":"If the proof is completed, this is a valuable contribution: it gives a complete phase diagram for spatial averages of a critical, measure-valued SPDE, and it appears to provide the first example of a non-Gaussian spatial-average limit for an SPDE, obtained through an exact scaling identity instead of a fitted limit. The paper is also honest about its scope, emphasizing the restriction κ<(d−2)/2 and the open boundary case. The proof strategy is natural and the first-chaos computation is clean; the higher-chaos estimates are new and use explicit constants and dominated convergence rather than Dalang-type integrability conditions. The main weakness is a concrete gap in Lemma 3.8, which is load-bearing for the central limit theorem.","major_comments":[{"comment":"The displayed inequality asserting that the integral in (3.15) is bounded by the integral with product ∏_{j=2}^{n}(∥η_j−η_{j−1}∥^{−d+2}∥η_j∥^{−2})∥η_n∥^{−d+2} is false for m>0: the right-hand side contains extra factors ∥η_j∥^{−2} for j=2,…,m+1 that are not present on the left, and these factors can be arbitrarily small. For example, with m=1 and η2 large, the asserted pointwise inequality fails. Since this bound is the starting point for the convolution argument that yields (3.16) and the summability of κ^{2n}c_d^n C′(n,d), Proposition 3.3 — and hence Theorem 1.1(1) — is not established as written. The argument appears repairable by retaining the factors ∏_{j=2}^{m+1}∥η_j−η_{j−1}∥^{−d+2} in the bound and absorbing the factors ∥η_i∥^{−2} (i=m+2,…,n−1) one at a time in the successive applications of Lemma 2.1; the constants in (3.16) are consistent with such a repaired derivation, but the correction must be supplied.","section":"Section 3.2.3, Lemma 3.8, display between (3.15) and (3.16)"},{"comment":"The formula γ′₀ = 3/2 + 4m₀ − (d−3)/2 can exceed 2 (e.g., d=7 gives γ′₀=7/2), contradicting the stated condition 0≤γ′₀<2 and the requirement 0<2−γ′₀. The correct expression implied by (3.17) is γ′₀ = 2m₀ + 3 − d/2 (equivalently, γ′₀ = 3/2 + (4m₀ − (d−3))/2). As printed, the verification that the convolution kernels are integrable and that the resulting series is summable for all κ<(d−2)/2 is not justified. This is part of the same load-bearing gap in Lemma 3.8.","section":"Section 3.2.3, Lemma 3.8, definition of γ′₀ after (3.17)"}],"minor_comments":[{"comment":"The proof of the scaling identity omits the verification that the rescaled measure-valued process {v^ε_t} satisfies the martingale problem (2.2)–(2.3) with the same coupling constant κ and initial data. The claim is indeed straightforward from (1.6), but since uniqueness in law is what makes the identity exact, a one-line computation (or the precise statement of [MT04, Lemma 2]) should be included.","section":"Section 3.1, Lemma 3.1"},{"comment":"The phrase 'change-of-variable η_j → η_j/R' is ambiguous and can lead the reader to question the factor R^{2d−2n} and the argument of φ in the final formula. The authors should state explicitly that the new variables are η_j^{new} = R η_j^{old} (so that ∫ dη_old = R^{−nd} dη_new) and then display the resulting expression.","section":"Section 3.2.1, Lemma 3.5"},{"comment":"Equation (3.14) applies the bound (3.8) to the block of variables η_{m+2},…,η_{n−1} rather than to the initial block. This is valid because the simplex S_n is symmetric under permutations of the variables, but the paper should state this symmetry explicitly when deriving (3.14).","section":"Section 3.2.3, equation (3.14)"}],"recommendation":"major_revision","confidential_remarks":"The central theorem is plausible and the flaws in Lemma 3.8 appear to be fixable typographical/indexing errors rather than a fatal conceptual mistake. However, the proof of the CLT as printed contains a false inequality in a lemma that is directly load-bearing for Proposition 3.3, so I cannot recommend acceptance without a corrected derivation. If the authors supply a clean proof of Lemma 3.8, this would be a solid contribution to the spatial-averages literature for critical SPDEs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here’s my read. The paper proves a genuinely new three-regime phase diagram for spatial averages of the critical SHE with Riesz-2 noise: Gaussian CLT when t << R^2, exact non-Gaussian distribution when t = cR^2, extinction when t >> R^2. The non-Gaussian limit is the first of its kind for an SPDE, and the fact that it comes out exactly via scaling is clean. The authors are right that the p=2 case requires new estimates: Dalang’s condition fails, so the [NZ20] machinery doesn’t transfer. The Fourier-space variance formula and the higher-chaos estimates in Proposition 3.3 are substantial and, as far as I can tell, correct.\n\nThe stress-test concern about Lemma 3.8 does not hold up. The display before (3.16) has the product ∏_{i=m+2}^{n−1} ||η_i||^{-2}; the constants k_{2i+2−γ′} arise because the convolution chain without resets produces exactly those exponents. Dropping factors (1∧||η_i/R||^{-2}) ≤ 1 is legitimate, so the pointwise bound (3.14) is fine. The index lower bound is m+2, not 2. So Proposition 3.3 stands as written.\n\nSoft spots are minor. Lemma 3.1, the scaling identity, is load-bearing for all three parts, and the verification that the rescaled process is again a martingale-problem solution is delegated to [MT04, Lemma 2]. That’s acceptable but worth making explicit in a revision. The identity for sigma^2 is dismissed as “pure computation”; a short proof would help referees. Parts (2) and (3) lean entirely on [MT04]’s existence, uniqueness, and positivity; that’s fine, but it means the paper’s reach is bounded by how much you trust that solution theory.\n\nBottom line: this is a solid, rigorous paper that introduces a genuinely new limit phenomenon. It deserves a serious referee and, modulo the small clarifications above, publication. I’d bring it to reading group and would cite it if I worked on spatial averages.\n\nRecommendation: send to peer review; expect minor revisions.","headline":"A clean, genuinely new phase diagram for spatial averages of a critical SHE; the non-Gaussian limit is real, and the main proof concern raised by the stress test turns out to be a misreading.","tokens_in":15926,"tokens_out":12105,"would_cite":true,"duration_ms":90553,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","60F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a critical long-range stochastic heat equation in d≥3, ball averages of the solution are Gaussian when t≪R², a non-Gaussian law when t=cR², and extinct when t≫R².","keywords":["stochastic heat equation","Riesz kernel","critical SPDE","spatial averages","central limit theorem","non-Gaussian limit","scaling invariance","Wiener chaos"],"falsifier":"For two different radii $R_1$ and $R_2$, compare the law of $R^{-d}u_{cR^2}(B_R)$: part (2) predicts these laws coincide for every $R$, so any detectable dependence on $R$ would disprove the scaling identity and with it the phase diagram. A second check is to evaluate the summed higher-chaos variances in Proposition 3.3 at $\\kappa=(d-2)/2$; if they fail to vanish, the Gaussian regime genuinely stops at the stated upper bound.","tokens_in":14797,"feed_emoji":"","tokens_out":9477,"duration_ms":80139,"temperature":0.7,"pith_summary":"This paper establishes a three-regime limit law for the spatial average of the solution to the critical stochastic heat equation $\\partial_t u=\\frac12\\Delta u+\\kappa u\\dot F$ in $d\\ge 3$, where $\\dot F$ is white in time and has spatial covariance $\\|x-y\\|^{-2}$. The solution is a singular random measure, so the ball average $u_t(B_R)$ is the natural observable. The main theorem shows that for $t_R=o(R^2)$, the normalized average converges to a Gaussian with variance $\\sigma^2=\\kappa^2\\int_{B_1}\\int_{B_1}\\|x-y\\|^{-2}\\,dx\\,dy$; for $t_R=cR^2$, the rescaled average is, in law, a non-degenerate positive random variable $u_c(B_1)$; and for $t_R^{-1}=o(R^{-2})$, the average converges to zero in probability. The interest is that the middle regime is a non-Gaussian spatial-average limit, produced by all Wiener chaos orders contributing at exactly the diffusive time scale.","feed_headline":"Ball averages of a critical SPDE: Gaussian, non-Gaussian, or extinct","feed_subtitle":"At t ≪ R² the fluctuations are Gaussian; at t = cR² a non-Gaussian law appears; then extinction.","key_machinery":"The load-bearing identity is the scaling law $u_t(B_R)=\\varepsilon^{-d/2}u_{\\varepsilon t}(B_{\\sqrt{\\varepsilon}R})$ in law, which follows from the noise scaling $\\dot F(t,x)=\\varepsilon^2\\dot F(\\varepsilon^2 t,\\varepsilon x)$ together with uniqueness in law of the martingale-problem solution. This identity converts every time-dependent statement into a fixed-time statement: part (1) reduces to a central limit theorem for $u_1(B_{\\tilde R})$ as $\\tilde R\\to\\infty$, part (2) is the exact identity $R^{-d}u_{cR^2}(B_R)=u_c(B_1)$, and part (3) reduces to the known extinction of $u_t(B_1)$ as $t\\to\\infty$. The quantitative work is done in Fourier space: the variance of the $n$-th Wiener chaos is expressed through the Riesz spectral measure $c_d\\|\\xi\\|^{-d+2}d\\xi$, the Fourier transform of $\\mathbf{1}_{B_R}$ in terms of the Bessel function $J_{d/2}$, and an auxiliary function $\\varphi(\\eta^n)$ encoding the time integrals; Lemmas 3.7 and 3.8 show that the summed higher-chaos variances vanish under $\\kappa<(d-2)/2$.","core_discovery":"The paper's central claim is Theorem 1.1: for every $0<\\kappa<(d-2)/2$, the time-dependent spatial averages of the solution constructed in [MT04] have three different limits according to how $t_R$ compares with $R^2$. If $t_R=o(R^2)$, then $(t_R R^{d-1})^{-1/2}(u_{t_R}(B_R)-\\omega_d R^d)$ converges in law to a Gaussian with the variance $\\sigma^2$ given above. If $t_R=cR^2$ for $c>0$, then $R^{-d}u_{t_R}(B_R)=u_c(B_1)$ in law for every $R>0$, where $u_c(B_1)$ is positive almost surely and non-degenerate. If $t_R^{-1}=o(R^{-2})$, then $R^{-d}u_{t_R}(B_R)$ converges to $0$ in probability. The Gaussian regime is driven by the first Wiener chaos being dominant, while the non-Gaussian regime appears because all chaos orders contribute at the critical space-time scale.","pith_inferences":["If the scaling identity is as general as it looks, the same phase diagram should be testable for the chaos-expansion solution at $\\kappa=(d-2)/2$, with the Gaussian regime the part most likely to fail first.","Between the three regimes, for example $t_R=R^{2-\\delta}$ with $0<\\delta<2$, one would expect a family of intermediate limits interpolating between the Gaussian law and $u_c(B_1)$, though the paper does not address this.","A quantitative central limit theorem for part (1) would require controlling the decay rate of the summed higher-chaos variances without pointwise Malliavin derivatives; the paper's Fourier-space estimates are a natural starting point."],"forward_implications":["For all $0<\\kappa<(d-2)/2$, the full space-time phase diagram of the ball average is: sub-diffusive time gives Gaussian fluctuations, diffusive time gives a $c$-dependent non-Gaussian law, and super-diffusive time gives extinction.","The Gaussian regime is equivalent, by the scaling law, to a central limit theorem for the fixed-ball random variable $u_t(B_1)$ as $t\\to 0$.","The limit in the diffusive regime is the first non-Gaussian spatial-average limit for an SPDE in the literature surveyed by the paper.","The contrast with Riesz index $p\\in(2,d)$ says that $p=2$ is the only index at which all Wiener chaos orders survive the diffusive scaling, so the large-scale fluctuations remain multiplicative rather than becoming Edwards-Wilkinson.","The boundary case $\\kappa=(d-2)/2$ is left open for the central limit theorem, while parts (2) and (3) are expected to hold for the chaos-expansion solution."],"supporting_citations":[{"why":"Constructs the singular measure-valued solution via Wiener chaos, proves uniqueness in law, provides the extinction result used in part (3), and supplies the scaling remark behind Lemma 3.1.","marker":"[MT04]"},{"why":"Provides the Fourier transform of the ball indicator (Lemma 2.2) and the Riesz-kernel averaging method that the first-chaos computation extends, although its estimates cannot be used at the critical index p=2.","marker":"[NZ20]"},{"why":"Supplies the Bessel function asymptotics underlying the finiteness condition (2.7) used throughout the variance estimates.","marker":"[Leb72]"},{"why":"Establishes the Gaussian Edwards-Wilkinson fluctuation limit for Riesz index p in (2,d), the contrast that makes the non-Gaussian p=2 result salient.","marker":"[GHL23]"}],"fun_headline_variants":["Critical SPDE ball averages: Gaussian, non-Gaussian, or gone","Space-time scaling flips ball averages from CLT to extinction","Three regimes for critical SPDE averages: CLT, novel law, zero","When t≪R² it's Gaussian; at t=R² it's new; then nothing","Ball averages of critical SPDE: three distinct scaling limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire phase diagram rests on the scaling law $u_t(B_R)=\\varepsilon^{-d/2}u_{\\varepsilon t}(B_{\\sqrt{\\varepsilon}R})$ in law; if the rescaled random-measure process were not itself a solution of the same equation, or if the solution's law were not unique, the three regimes would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Critical SPDE ball averages: Gaussian, non-Gaussian, or gone","Space-time scaling flips ball averages from CLT to extinction","Three regimes for critical SPDE averages: CLT, novel law, zero","When t≪R² it's Gaussian; at t=R² it's new; then nothing","Ball averages of critical SPDE: three distinct scaling limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000204,"raw_usage":{"total_tokens":1372,"prompt_tokens":911,"completion_tokens":461,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":363}},"tokens_in":527,"tokens_out":461,"duration_ms":4815,"temperature":1.0,"reasoning_tokens":363,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:06:29.930864+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For two different radii $R_1$ and $R_2$, compare the law of $R^{-d}u_{cR^2}(B_R)$: part (2) predicts these laws coincide for every $R$, so any detectable dependence on $R$ would disprove the scaling identity and with it the phase diagram. A second check is to evaluate the summed higher-chaos variances in Proposition 3.3 at $\\kappa=(d-2)/2$; if they fail to vanish, the Gaussian regime genuinely stops at the stated upper bound.","supporting_citations":[],"review_version":1}