{"id":"97129834-f139-4b53-bdcc-eda76579a809","arxiv_id":"1908.09331","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the 2D stochastic Allen-Cahn equation, the Galerkin approximation converges in a negative Besov space with rate N^{-(α-δ)} for any α in (0,2/9) and δ>0.","lead":"This paper proves a convergence rate for Galerkin approximations of the stochastic Allen-Cahn equation on the 2D torus driven by space-time white noise. It is the first quantitative convergence rate for a singular SPDE with superlinear nonlinearity in two dimensions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Equation (3.11) is not merely borrowed; as printed its frequency bookkeeping is wrong (n=2 gives K(m1)∑K(m2−m1) rather than K⋆K(m1)), so the N-uniform bound and the N^{−α} rate do not follow as written.","rationale":"Good-faith reading: the paper tests a plausible rate and has a coherent plan (heat-equation rate, then pathwise error via stopping times). The import of [38] is legitimate, and the rest of the argument after a correct covariance bound is mostly standard. However, the one equation that turns the spectral decay into an N-rate is (3.11). The Reader called it the weakest external input; checking it literally reveals an index error: for n=2 it is not the isometry expression for the covariance. This is an internal inconsistency, not merely a missing proof: if (3.11) were true, the N-uniform bounds (3.10) would fail. Since the whole linear-rate step depends on this, the central claim lacks a written proof. The fix is likely to correct the phase/index convention and to include a self-contained derivation of the truncated convolution estimate; until then the verdict should stay conditional. Agreement with the reader is partial: we flag the same equation but a different defect.","tokens_in":20688,"tokens_out":24357,"duration_ms":249016,"concrete_test":"Recompute the n=2 case of (3.11) from (3.1)-(3.5) using the isometry (A.1): for s=t, the true covariance coefficient at output frequency q is ∑_{|k|,|q−k|≤N} K(k)K(q−k) with K(m)=1/(2I_m), while the printed RHS is K(q)∑_{|l−q|≤N}K(l−q). Evaluate at q=0 (block j=−1): the true coefficient is ∑_{|k|≤N}K(k)^2=O(1), the printed one is K(0)∑_{|l|≤N}K(l)=O(log N). If the two disagree, (3.11) must be corrected and Lemma 3.4 re-derived from the corrected identity; if they agree under a different index convention, that convention should be stated and the cancellation of the log N verified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.3's rate hinges on (3.11). As printed, (3.11) is not the covariance of the Wick powers, so the chain (3.11) → Lemma A.3 → (3.10) → Lemma 3.4 → Theorem 4.4 breaks. For n=2, s=t, the Wiener isometry gives Cov(Δ_j:(Z^N_{-∞,t})^{:2:}(x), Δ_j:(Z^N_{-∞,t})^{:2:}(y)) = 2∑_{q∈A_{2^j}} (∑_{|k|,|q−k|≤N} K(k)K(q−k)) e_q(x−y), with K(m)=e^{-I_m|s−t|}/(2I_m). The right side displayed in (3.11) is instead ∑_{m1∈A_{2^j}} K(m1)(∑_{|m2|≤N} K(m2−m1)) e_{m1}(x−y). These differ already at q=0: the correct coefficient is ∑_{|k|≤N}K(k)^2=O(1), whereas the printed coefficient is K(0)∑_{|l|≤N}K(l)=O(log N). Hence (3.11) cannot be the identity quoted from [38], and the N-uniform bound (3.10) would fail if (3.11) were used literally. The rate N^{-α_-} in Lemmas 3.4/3.5 and Theorem 4.4 is therefore not established as written; a corrected and proved version of the covariance identity is needed.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a convergence rate for the Galerkin approximation of the two-dimensional stochastic Allen-Cahn equation driven by space-time white noise. The solution is understood in the Da Prato-Debussche renormalized sense, with the stochastic heat equation part \\(\\bar Z\\) and a more regular remainder \\(Y\\). The authors first establish a rate \\(N^{-(\\alpha-\\delta)}\\) for the Galerkin projection of the stochastic heat equation in the Besov space \\(C^{-\\alpha}\\) (Section 3, Lemmas 3.4 and Theorem 3.5). They then combine this with Besov and Schauder estimates, stopping times, and Gronwall arguments to prove Theorem 4.4: for \\(\\alpha\\in(0,2/9)\\), \\(\\gamma'>3\\alpha/2\\), and any \\(\\delta>0\\), the probability that \\(\\sup_{t\\le T} t^{\\gamma'}\\|X_t-X^N_t\\|_{-\\alpha}\\) is of order \\(N^{\\delta-\\alpha}\\) tends to zero as \\(N\\to\\infty\\).","tokens_in":21050,"tokens_out":26978,"duration_ms":262700,"significance":"If the proof is correct, this is the first convergence-rate result for a spatial approximation of a singular two-dimensional SPDE with superlinearly growing nonlinearity; the Da Prato-Debussche strategy and the use of Wick powers in Besov spaces are natural and the overall architecture is coherent. The paper is also honest in relying on a limited number of prior regularity and well-posedness inputs rather than assuming the convergence rate. However, the central covariance identity on which the linear rate rests is not proved and, as displayed, is false; moreover, the parameter bookkeeping in the final Gronwall step does not support the stated range \\(\\alpha<2/9\\). These issues are load-bearing, so the current manuscript does not establish the advertised theorem.","major_comments":[{"comment":"Section 3.3, Eqs. (3.11), (3.18), (3.19): the covariance identities are false as printed. For n=2 and s=t, the Fourier coefficient of Cov(Delta_j :(Z^N_{-infinity,t})^{:2:}(x), Delta_j :(Z^N_{-infinity,t})^{:2:}(y)) at mode m1 is proportional to the truncated convolution sum_k K(k)K(m1-k) with |k| and |m1-k| bounded by N, while the displayed formula instead has K(m1) sum_{m2} K(m2-m1). For n=2 this gives O(log N) at small m1 instead of O(1); for n=1 the Galerkin covariance should have Fourier support |m1|<=N, which is absent from the display. In the untruncated identity (3.18) the displayed inner sum over m2 diverges logarithmically. Since Lemma 3.4 derives (3.16) from these displays, and Theorem 4.4 depends on (3.16) through Theorem 3.5 and Theorem 4.3, the rate N^{-(alpha-delta)} is not established as written. The correct spectral covariance of the Wick powers must be stated and proved, including the N-dependence of the truncated convolutions.","section":"Section 3.3"},{"comment":"Section 3.3, Eq. (3.10): the uniform-in-N bound for the truncated Wick powers is not proved. The text says it follows from the proofs of [38, Theorem 2.1, Proposition 2.3] using (A.1) and the semigroup property, but those cited results concern the untruncated stationary process; the truncated process and the uniformity in N are precisely the new content needed here. Since Eq. (3.11) is the only N-dependent covariance estimate supplied, the authors should prove (3.10) directly from a correct covariance identity and Lemma A.3 rather than citing the untruncated case.","section":"Section 3.3"},{"comment":"Section 4.1, proof of Theorem 4.3: the rate attribution is inconsistent. The stopping time and the norm ||.||_barL are defined with a parameter kappa>0; the final use of (3.24) in the display before (4.15) has exponent ((alpha+beta)(n-1)+n alpha)/2 and error N^{-alpha p}, which corresponds to taking kappa=alpha in (3.24). But the preceding Gronwall estimate requires (alpha+beta)/2 + kappa + 2 gamma' < 1. With kappa=alpha, beta close to alpha, and gamma' close to 3 alpha/2, this forces 5 alpha < 1, i.e. alpha < 1/5; hence the stated range alpha<2/9 is not covered by the argument as written. If instead kappa<alpha, the \\bar Z-difference term decays only as N^{-kappa}, which is slower than the claimed N^{-alpha}. The authors need to state a consistent choice of kappa and verify the integrability conditions for that choice, or adjust the range and rate.","section":"Section 4.1, proof of Theorem 4.3"}],"minor_comments":[{"comment":"The stopping time nu^{M,epsilon}_N is defined with the norm ||.||_alpha, but the processes \\bar Z^{:n:} are only controlled in C^{-alpha}; the subsequent displays in the same section use ||.||_{-alpha}. This appears to be a typo and should be corrected to -alpha.","section":"Section 4.1, Eq. (4.5)"},{"comment":"In the proof of (4.18), the projection estimate (2.11) is used with lambda<alpha; the text should state explicitly that lambda>0 and that the optimal choice is lambda close to alpha, since otherwise the rate attribution is not transparent.","section":"Section 4.2, proof of Theorem 4.4"},{"comment":"The notation 'm1 in A_{2^j}, m1 in Z^2' is redundant, and the convention m0=0 should be stated before the formula rather than after; this would help a reader verify the intended frequency bookkeeping.","section":"Section 3.3, Eq. (3.11)"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is plausible and the proof strategy is recoverable, but the covariance identity in Section 3.3 must be corrected and reproved, and the kappa-parameter issue in the Gronwall argument must be resolved. I would ask for a full re-derivation of Section 3.3 and a careful re-verification of the final rate before considering the paper further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give it to you straight. This paper proves the first convergence rate for a Galerkin approximation of the 2D renormalized stochastic Allen-Cahn equation (dynamical Φ^4_2) driven by space-time white noise. That's a genuine gap in the numerical analysis literature. The rate is weak (α<2/9 in a negative Besov norm, in probability), but it's the first one in a singular setting with superlinear nonlinearity. The heat-equation rate (α<1 in C^{-α}) is a building block, and the stopping-time argument in Section 4 is coherent. The exponent bookkeeping checks out. The self-citations to [27,38] are for existence and regularity, not the rate, so there's no circularity.\n\nThe soft spot is serious: equation (3.11) is the foundation of the whole linear-rate argument, and as printed it is not the covariance of the truncated Wick powers. For n=2, the identity should read 2∑_{q∈A_{2^j}} (∑_{|k|,|q-k|≤N} K(k)K(q-k)) e_q(x-y), but (3.11) gives ∑_{m1∈A_{2^j}} K(m1)(∑_{|m2|≤N} K(m2-m1)) e_{m1}(x-y). At q=0 the two differ: the correct coefficient is O(1), the printed one is O(log N). That is not a cosmetic typo. The chain (3.11) → Lemma 3.4 → Lemma 3.5 → Theorem 4.4 depends on this identity, and the N-uniform bound (3.10) would not follow if (3.11) were used literally. The authors quote it as 'using the results in the proofs of [38]', but as written it cannot be what [38] proves. So the rate N^{-α_-} is not established by the current text.\n\nThat said, I think the result is probably true. The correct covariance would still yield the same kind of bounds, perhaps with extra logs absorbed into ε. The fix is to rewrite (3.11) with the correct convolution constraint and prove it (or give a precise statement from [38] and verify it). The final theorem should also state explicitly that the rate is in probability, not strong.\n\nWho is this for: people working on numerical approximation of singular SPDEs, particularly Φ^4_2. It deserves a serious referee, but the referee will need to dig into Section 3.3. I'd conditionally accept: require the correction of (3.11) and a self-contained derivation or exact reference.","headline":"First convergence rate for a 2D renormalized stochastic Allen-Cahn equation, but the load-bearing covariance identity (3.11) is wrong as printed; the result is probably true and worth refereeing, but not as is.","tokens_in":21576,"tokens_out":16337,"would_cite":false,"duration_ms":144252,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","82C28"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that Galerkin approximations of the two-dimensional stochastic Allen-Cahn equation converge to the renormalized solution in the Besov space $C^{-\\alpha}$ with rate $N^{-(\\alpha-\\delta)}$ for every $\\alpha<2/9$.","keywords":["stochastic Allen-Cahn equations","Galerkin approximation","convergence rate","Besov space","space-time white noise","Wick renormalization","2D torus","dynamic Phi^4_2 model"],"falsifier":"Compute the exact second moment of the dyadic-block error for the Wick square: using (3.18)--(3.19), $$E\\big|\\Delta_j\\big((Z^N_{-\\infty,t})^{:2:} - $Z^{{:2:}}$_{-\\infty,t}\\big)\\big|^2$$ reduces to a fixed constant times $\\sum_{m\\in A_{2^j}} K_0^{\\star 2}_{>N}K_0(m)$. Lemma A.3 bounds this tail by $(1+N^2)^{-1+\\varepsilon}2^{2j(\\varepsilon+\\lambda)}$, and the theorem requires this bound to hold uniformly in $N$. A direct summation or numerical check for $N=10,100,1000$ that finds a slower decay, or an $N$-dependent constant that spoils the dyadic sum, would falsify the rate $N^{-(\\alpha-\\delta)}$ in Theorems 3.5 and 4.4.","tokens_in":20496,"feed_emoji":"📐","tokens_out":13498,"duration_ms":115064,"temperature":0.7,"pith_summary":"The paper proves a quantitative convergence rate for Galerkin (Fourier-mode) approximation of the two-dimensional stochastic Allen-Cahn equation driven by space-time white noise, a singular SPDE whose cubic nonlinearity must be renormalized. For initial data in the Besov space $C^{-\\alpha}$ with $\\alpha \\in (0,2/9)$, the error between the $N$-mode Galerkin solution $X^N$ and the renormalized solution $X$, measured in $C^{-\\alpha}$ with a small time weight, is of order $N^{-(\\alpha-\\delta)}$ for any $\\delta>0$: with probability tending to one, $\\sup_{t\\in[0,T]} t^{\\gamma'} \\|X_t - X^N_t\\|_{-\\alpha} \\lesssim N^{\\delta-\\alpha}$. This is the first convergence rate reported for a spatial approximation of a singular SPDE with superlinearly growing nonlinearity in dimension two, upgrading earlier qualitative convergence statements. The proof obtains the rate first for the linear stochastic heat equation and its Wick powers, then transfers it to the nonlinear equation through a stopping-time and Gronwall argument.","feed_headline":"2-D Allen-Cahn Galerkin error decays as N^{-α}","feed_subtitle":"A renormalized Wick-power estimate turns qualitative convergence into a polynomial rate.","key_machinery":"The engine is the Wick renormalization of the truncated heat process. For the projected heat solution $Z^N$, the renormalized powers $(Z^N)^{:2:}=(Z^N)^2-R_N$ and $(Z^N)^{:3:}=(Z^N)^3-3R_N Z^N$, with $R_N=\\|\\mathbf{1}_{[0,\\infty)}H_N\\|^2_{L^2(\\mathbb{R}\\times \\mathbb{T}^2)}$ diverging logarithmically in $N$, are the objects that converge. The quantitative control comes from the covariance identity (3.11) for the Littlewood--Paley blocks $\\Delta_j (Z^N_{-\\infty,\\cdot})^{:n:}$, which expresses the second moment as a truncated convolution $\\sum_{m\\in A_{2^j}} K_0^{\\star n}_{>N}K_0(m)$ of the kernel $K_0(m)=(1+|m|^2)^{-1}$. Lemma A.3 bounds these convolution tails by $(1+N^2)^{-1+\\varepsilon}$, and summing dyadic blocks with a Besov embedding yields the $N^{-\\alpha}$ rate; the nonlinear equation is reached by splitting $X=Y+\\bar{Z}$, controlling $Y$ on stopping times, and applying Gronwall's inequality.","core_discovery":"The central result, Theorem 4.4, states that for $\\alpha\\in(0,2/9)$, $X_0\\in C^{-\\alpha}$, any $\\delta>0$, and any $\\gamma'>3\\alpha/2$, the Galerkin solutions of the renormalized equation satisfy $$\\lim_{N\\to\\infty} P\\Big( \\sup_{t\\in[0,T]} $t^{{\\gamma'}}$ \\|P_N X_t - X^N_t\\|_{-\\$\\alpha$} \\gtrsim $N^{{\\delta-\\alpha}}$\\Big) = 0$$ and the same with $X_t$ in place of $P_N X_t$. In other words, the finite-dimensional projections converge to the renormalized solution in $C^{-\\alpha}$ with rate $N^{-(\\alpha-\\delta)}$, after a time weight $t^{\\gamma'}$. The paper proves this by establishing in Theorem 3.5 that the renormalized Wick powers $(\\bar{Z}^N_t)^{:n:}$ of the Galerkin heat process converge to $\\bar{Z}^{:n:}_t$ in $L^p(C([0,T];C^{-\\alpha}))$ at the same polynomial rate for $n=1,2,3$, and then showing that the interaction term $Y^N$ inherits this rate on a set of high probability bounded by stopping times. The same rate for $P_N X_t - X^N_t$ shows the dominant error is the truncated nonlinear interaction rather than the projection of the initial data.","pith_inferences":["Inference: the range $\\alpha<2/9$ is likely an artifact of the condition $\\gamma'>3\\alpha/2$ imposed by the $Y$-component; a sharper estimate of the interaction term would plausibly extend the same mechanism to all $\\alpha<1/3$, the range where the renormalized Wick powers are defined.","Inference: the same covariance-tail machinery should transfer to other singular SPDEs on the 2D torus whose renormalization constants diverge logarithmically, giving a rate determined by the singularity exponent of the noise.","Inference: a testable extension is to replace the $C^{-\\alpha}$ norm with the stronger $B^{-\\alpha}_{p,q}$ norms; the proof's use of Besov embedding suggests the rate $N^{-(\\alpha-\\delta)}$ may persist for $p$ large enough, with an explicit $p$-dependent loss."],"forward_implications":["For every $\\alpha<2/9$, the Galerkin solution with $N$ Fourier modes is, after a time weight $t^{\\gamma'}$, within $N^{-(\\alpha-\\delta)}$ of the renormalized solution in $C^{-\\alpha}$ with probability tending to one.","The same rate holds for the projected trajectory: $P_N X_t - X^N_t$ converges at $N^{-(\\alpha-\\delta)}$, so the approximation error is governed by the truncated nonlinear interaction rather than by the projection of the initial data.","The Wick powers of the Galerkin heat process converge to the renormalized Wick powers at rate $N^{-(\\alpha-\\delta)}$ in $L^p(C([0,T];C^{-\\alpha}))$ for every $\\alpha\\in(0,1)$ and $n=1,2,3$; the nonlinear theorem inherits this linear rate.","The result provides the first quantitative convergence rate for a spatial discretization of a singular SPDE with superlinear nonlinearity driven by space-time white noise in dimension two."],"supporting_citations":[{"why":"supplies the covariance and hypercontractivity estimates for the Wick powers of the truncated heat process that Lemmas 3.2 and 3.4 rely on.","marker":"[38, Theorem 2.1, Proposition 2.3]"},{"why":"gives the convolution-tail bounds for the kernels $K_\\gamma$ that Lemma A.3 adapts to produce the $N^{-\\alpha}$ decay.","marker":"[38, Corollary C.3]"},{"why":"is the Kolmogorov-type criterion used to show the Wick powers are continuous in $C([0,T]; \\mathcal{C}^{-\\alpha})$ with uniform moment bounds.","marker":"[27, Lemma 5.2]"},{"why":"provides existence and uniqueness for the shifted equation (1.4) defining the interaction term $Y$.","marker":"[27, Theorem 6.2]"},{"why":"introduces the decomposition $X=\\bar Z+Y$ and the renormalization constants $R_N$ used in the Galerkin Wick powers.","marker":"[29, p.4]"},{"why":"gives well-posedness of the equation for $Y$ on the torus, needed for the nonlinear convergence argument.","marker":"[31, Theorem 3.10]"},{"why":"gives existence and uniqueness for the finite-dimensional Galerkin equation (1.11).","marker":"[26, Theorem 5.1]"},{"why":"bounds the projection error $P_N f-f$ in Besov spaces, used to control the initial-value contribution in Theorem 3.5.","marker":"[37, Proposition A.11]"},{"why":"is the technique for estimating convolution tails on $\\mathbb{Z}^2$ adapted in Lemma A.3.","marker":"[18, Lemma 10.14]"}],"fun_headline_variants":["2D stochastic Allen-Cahn Galerkin rate N^{−α+δ}","Galerkin Allen-Cahn converges as N^{−α+δ} on 2D torus","Polynomial rate for Galerkin Allen-Cahn in 2D white noise","Allen-Cahn Galerkin error: N^{−α+δ} in Besov norm","Convergence rate N^{−α+δ} for 2D Allen-Cahn Galerkin"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof inherits, without reproving, a uniform-in-$N$ covariance estimate for the renormalized Wick powers of the truncated heat process; if that estimate is not uniform in $N$ or decays differently, the claimed rate $N^{-(\\alpha-\\delta)}$ does not follow.","fun_headline_variants_meta":{"raw":{"variants":["2D stochastic Allen-Cahn Galerkin rate N^{−α+δ}","Galerkin Allen-Cahn converges as N^{−α+δ} on 2D torus","Polynomial rate for Galerkin Allen-Cahn in 2D white noise","Allen-Cahn Galerkin error: N^{−α+δ} in Besov norm","Convergence rate N^{−α+δ} for 2D Allen-Cahn Galerkin"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001121,"raw_usage":{"total_tokens":4671,"prompt_tokens":958,"completion_tokens":3713,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":574,"completion_tokens_details":{"reasoning_tokens":3593}},"tokens_in":574,"tokens_out":3713,"duration_ms":24927,"temperature":1.0,"reasoning_tokens":3593,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:14:49.301205+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the exact second moment of the dyadic-block error for the Wick square: using (3.18)--(3.19), $$E\\big|\\Delta_j\\big((Z^N_{-\\infty,t})^{:2:} - $Z^{{:2:}}$_{-\\infty,t}\\big)\\big|^2$$ reduces to a fixed constant times $\\sum_{m\\in A_{2^j}} K_0^{\\star 2}_{>N}K_0(m)$. Lemma A.3 bounds this tail by $(1+N^2)^{-1+\\varepsilon}2^{2j(\\varepsilon+\\lambda)}$, and the theorem requires this bound to hold uniformly in $N$. A direct summation or numerical check for $N=10,100,1000$ that finds a slower decay, or an $N$-dependent constant that spoils the dyadic sum, would falsify the rate $N^{-(\\alpha-\\delta)}$ in Theorems 3.5 and 4.4.","supporting_citations":[],"review_version":1}