Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus

Rongchan Zhu; Ting Ma

REVIEW 3 major objections 3 minor 41 references

Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read 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 $\alpha<2/9$.

desk verdict 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. read the letter →

arxiv 1908.09331 v1 pith:NGZ544RK submitted 2019-08-25 math.PR cs.NAmath.APmath.NA

Ting Ma , Rongchan Zhu This is my paper · ORCID

classification math.PRcs.NAmath.APmath.NA MSC 60H1582C28

keywords stochasticAllen-CahnequationsGalerkinapproximationconvergencerateBesovspacespace-timewhitenoiseWickrenormalization2DtorusdynamicPhi^4_2model

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

That 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.

Who 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.

Referee Report

3 major / 3 minor

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 $N\to\infty$.

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 (3)

[Section 3.3] 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 3.3] 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 4.1, proof of Theorem 4.3] 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.

minor comments (3)

[Section 4.1, Eq. (4.5)] 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 4.2, proof of Theorem 4.4] 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 3.3, Eq. (3.11)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence-rate theorems are derived via covariance estimates and Besov-space bounds, not assumed or fitted; self-citations are not load-bearing.

full rationale

The paper's derivation chain is self-contained in the relevant sense: it defines renormalized Wick powers, imports a covariance estimate (3.11) from [38], proves uniform Besov moment bounds, derives the linear heat-equation rate by comparing the covariances of Z and Z^N and applying the convolution estimates of Lemma A.3, and then extends the rate to the Allen-Cahn equation via pathwise estimates and Gronwall's inequality. No step fits a parameter to the quantity being predicted, and no result is renamed as a new prediction: the rate N^{-(alpha-delta)} is obtained by explicit estimates, not assumed as an input. The well-posedness of Y and Y^N is cited from [31, Theorem 3.10], [27, Theorem 6.2], and [26, Theorem 5.1]; although [31] is co-authored by R. Zhu, that citation is paired with the independent result [27] and supplies existence and uniqueness, not the convergence rate. The self-citation [40] appears only as background in the introduction. Thus the central claim does not reduce to a self-citation chain or to a fitted input. The most load-bearing external input is the covariance identity (3.11), quoted from [38, Theorem 2.1, Proposition 2.3] rather than proved; a referee should verify its frequency bookkeeping, since as printed the n=2 case appears to place the convolution indices differently from the Wiener-isometry expression. That is a correctness or verification risk, not a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard Besov-space inequalities and on published well-posedness and covariance results. No free parameters are fitted to data, and no new entities are introduced; the Wick renormalization constants are standard and not invented.

assumptions (3)

standard math Besov embedding, Schauder estimates, and multiplicative inequalities on the 2D torus (Lemmas 2.1-2.3)
Imported from [4,27,16,34]; these provide the analytic backbone for all norm estimates in the paper.
domain assumption Existence and uniqueness of mild solutions Y, Y^N to (1.4) and (1.11) from [31, Theorem 3.10] and [27, Theorem 6.2]
The paper does not prove well-posedness but relies on prior theorems; this is the foundation for the solution decomposition X=Y+Z and for the uniform moment bounds used in the stopping time argument.
domain assumption The covariance identity (3.11) for the Wick powers of the truncated heat process, quoted from [38, Theorem 2.1 and Proposition 2.3]
This identity is the key estimate that yields the linear convergence rate in Lemma 3.4 and Theorem 3.5; it is not proved in this paper.

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Pith. "Pith review of Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus." pith.science (2026). https://pith.science/paper/NGZ544RK

@misc{pith2026190809331,
  author       = {Pith},
  title        = {Pith review of: Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NGZ544RK}},
  note         = {Machine review of arXiv:1908.09331}
}

abstract

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on $\T$. First we prove that the convergence rate for stochastic 2D heat equation is of order $\alpha-\delta$ in Besov space $\C^{-\alpha}$ for $\alpha\in(0,1)$ and $\delta>0$ arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order $\alpha-\delta$ in $\C^{-\alpha}$ for $\alpha\in(0,2/9)$ and $\delta>0$ arbitrarily small.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [38]

Tsatsoulis, H

P. Tsatsoulis, H. Weber. Spectral gap for the stochastic qua ntization equation on the 2-dimensional torus. Ann. Inst. Henri Poincare Probab. Stat ., 54(3): 1204-1249, 2018

work page 2018

[2] [1]

Albeverio, M

S. Albeverio, M. R¨ ockner. Stochastic diﬀerential equations in in ﬁnite dimen- sions: Solutions via Dirichlet forms. Probab. Theory Related Field . 89 (1991) 347-386

work page 1991

[3] [2]

Space-time approximation of stochastic $p$-Laplace systems

D. Breit, M. Hofmanova. Space-time approximation of stochast ic p-Laplace type systems. arXiv:1904.03134, 2019

work page Pith review arXiv 1904

[4] [3]

Becker, A

S. Becker, A. Jentzen. Strong convergence rates for nonline arity- truncated Euler-type approximations of stochastic Ginzburg-La ndau equa- tions. Stochastic Processes and their Applications . Volume 129, Issue 1, January 2019, Pages 28-69

work page 2019

[5] [4]

Bahouri, J

H. Bahouri, J. Y. Chemin, and R. Danchin. Fourier analysis and non lin- ear partial diﬀerential equations. Grundlehren der Mathematischen Wis- senschaften 343, Springer , 2011

work page 2011

[6] [5]

Becker, B

S. Becker, B. Gess, A. Jentzen, and P. E. Kloeden. Strong con vergence rates for explicit space-time discrete numerical approximations of stoch astic Allen- Cahn equations. arXiv:1711. 02423, 2017. 22

work page 2017

[7] [6]

Beccari, M

M. Beccari, M. Hutzenthaler, A. Jentzen, R. Kurniawan, F. Lind ner, and D. Salimova. Strong and weak divergence of exponential and linear-imp licit Euler approximations for stochastic partial diﬀerential equations with s uperlinearly growing nonlinearities. arXiv:1903.06066

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Strong and weak convergence rates of ﬁn ite element method for stochastic partial diﬀerential equation with non-globa lly Lipschitz coeﬃcients

Jianbo Cui, Jialin Hong. Strong and weak convergence rates of ﬁn ite element method for stochastic partial diﬀerential equation with non-globa lly Lipschitz coeﬃcients. arXiv:1806.0156, 2018

work page arXiv 2018