Pith. sign in

REVIEW 2 major objections 4 minor 34 references

McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper establishes that, in two dimensions, a broad class of scaling-critical reaction-diffusion equations with white-noise initial data and logarithmically attenuated reactions converge, as the mollification scale vanishes, to a Gaussi

desk verdict A real generalization of GRZ with a genuinely different proof method, but the cutoff function in Section 6 violates its own condition (6.2), so the main theorem's approximation step needs a fix. read the letter →

arxiv 2509.06260 v1 pith:5B4O7V5T submitted 2025-09-08 math.PR math.AP

Bryan Castillo , Alexander Dunlap This is my paper · ORCID
classification math.PRmath.AP MSC 60H1560H0735K57
keywords scaling-criticalreaction-diffusionAllen-CahnequationMcKean-VlasovwhitenoiseinitialdataMalliavinderivativelogarithmicattenuationGaussianfluctuationstwospatialdimensions
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 that the leading-order statistics of a large class of two-dimensional reaction-diffusion equations with random initial data are Gaussian and are governed by a McKean–Vlasov equation, not by the original nonlinearity directly. This gives a general mechanism for the limit, and it recovers the known Allen–Cahn result as a special case. The proof uses Malliavin calculus and comparison principles instead of the Wild-expansion method of earlier work. A curious reader should care because the result reduces a complex stochastic PDE limit to an explicit one-dimensional ODE, and it extends the known behavior to non-polynomial reaction terms.

What carries the argument

The central object is the McKean–Vlasov equation (1.15), obtained from the original reaction term by replacing f(t+ε²,v) with its projection onto the first Wiener chaos, E[f'(t+ε²,v)]v. The proof's workhorse is a pointwise Malliavin-derivative bound, 0 ≤ D_z u^ε ≤ e^{3L1+mt} G_{t+ε²}(x-z), which provides decorrelation and Gaussian concentration. Around this, the authors build iterative error estimates over exponentially spaced time intervals t_m = ε^{2-mδ} - ε², using Grönwall-type lemmas and approximating the monotone part of the nonlinearity by a cutoff that diverges very slowly as ε→0.

What would settle it

Compute, at a fixed small ε, the normalized mean-square error (T+ε²)^{1/2} |u^ε(T,X) - (1 + 3(2 - log_ε T)λ²/(2π))^{-1/2} G^m_T*η(X)| for the cubic Allen–Cahn equation; if this does not tend to zero as ε→0 for some T>0, the theorem is false. A sharper test: choose a reaction whose derivative grows exactly like u² (the excluded threshold γ₂=2) and check whether the normalized error to the McKean–Vlasov solution still vanishes; failure there would confirm the threshold.

Watch

Extended reading notes

Core claim

For any reaction term f in the class S' (an odd, differentiable function that splits into a bounded smooth part and an increasing part whose derivative grows slower than u^2), the solution u^ε of the mollified, logarithmically attenuated equation converges in normalized mean-square error to the solution v^ε of the McKean–Vlasov equation (1.15). The McKean–Vlasov solution is exactly Gaussian, of the form σ_ε(t) G^m_t * η_ε, and σ_ε solves the ODE (1.17). Under the self-similar scaling f(t,u)=t^{-3/2}F(t^{1/2}u), the rescaled amplitude converges to the solution of σ'(q) = -E[F'(σ(q) G_1 * η)] σ(q). For the cubic Allen–Cahn case F(u)=λ²u³, this gives σ(q)=(1+3qλ²/(2π))^{-1/2}, reproducing the l

Load-bearing premise

The reaction term must split into a bounded smooth piece and an odd increasing piece whose derivative grows more slowly than u²; the monotonicity of the second piece is what lets the comparison principle control the error, and the growth bound is what lets the cutoff approximation close.

Editorial extensions

If this is right

  • For the logarithmically attenuated Allen–Cahn equation with white-noise initial data, the limiting law is the Gaussian heat flow e^{mT}G_T*η multiplied by the constant (1+3λ²/π)^{-1/2}, uniformly on compact time intervals.
  • The same Gaussian description holds for any self-similar reaction term in the class S', with the amplitude governed by the explicit ODE σ'(q) = -E[F'(σ(q)G_1*η)]σ(q).
  • The result removes the restriction on mT that appeared in earlier work, allowing any real mass parameter m.
  • Because the convergence is uniform in space and time, the Gaussian description applies to spatial pointwise statistics and two-point correlations of the solution.
  • The McKean–Vlasov equation is shown to be the actual effective equation for the leading-order behavior, not merely a formal reformulation.

Reading between the lines

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

  • The proof's mechanism suggests that Gaussian first-chaos projection is a universal attractor for scaling-critical reaction terms with subquadratic derivative growth; testing a reaction whose derivative grows exactly like u² log u would likely locate the threshold where this description breaks down.
  • The exponential timescale t = ε^{2-q} - ε² indicates that all nonlinear effects are generated in an o(1) layer near t=0; in other spatial dimensions, analogous logarithmic factors and ODEs would presumably take different explicit forms.
  • The iterated-logarithm cutoff g(ε) yields a convergence rate no faster than (log log 1/ε)/(log 1/ε)^{1/4}; a sharper analysis might remove the iterated logarithm and identify the true rate.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies a class of scaling-critical reaction-diffusion equations in two dimensions with white-noise initial data mollified at scale ε² and reaction term attenuated by (log ε^{-1})^{-1}. The main result, Theorem 1.3, asserts that for f in a broad class \bar S′, the normalized L² error between the solution u^ε and the corresponding McKean–Vlasov solution v^ε vanishes uniformly on compact time intervals. Theorem 1.4 identifies the limiting amplitude σ̄(q) via an ODE, and for f(u)=λ²u³ the authors recover the Allen–Cahn limit of Gabriel–Rosati–Zygouras. The proof strategy combines Malliavin derivative bounds, Gaussian Poincaré inequalities, short-time iterative estimates, and an approximation of the non-Lipschitz reaction term by cutoffs belonging to the smoother class \bar S^1.

Significance. If the main theorem were established, the paper would provide a genuinely new proof of the GRZ Allen–Cahn limit and a substantial generalization to non-polynomial reaction terms, while also clarifying the role of the McKean–Vlasov equation. The paper's conceptual framework — Malliavin calculus plus iterative decorrelation estimates — is attractive and contains several correct and useful ingredients, including the well-posedness of the McKean–Vlasov equation and the explicit ODE for σ̄. However, the proof of Theorem 1.3 contains a load-bearing error in the cutoff approximation of Section 6: the stated asymptotic condition on the cutoff is unsatisfiable, and the final bounds do not tend to zero. This is not a local technical gap but an obstruction in the central argument as written.

major comments (2)
  1. [Section 6.1, Eq. (6.2)] The asymptotic condition (6.2) is unsatisfiable. Since δ_ε=(log ε^{-1})^{-1/2}, the ratio e^{3g(ε)^{γ1}}/(g(ε)^{γ2}δ_ε) equals e^{3g(ε)^{γ1}}(log ε^{-1})^{1/2}/g(ε)^{γ2}. For any g(ε)→∞, the exponential in g^{γ1} dominates the polynomial g^{γ2} and the logarithmic factor only grows, so the ratio tends to infinity. In particular, the choice (6.57) gives a ratio of order (log log log log ε^{-1})^{3/(14ℓ1)} (log ε^{-1})^{1/2} / (log log log log ε^{-1})^{γ2/γ1} → ∞, contradicting the text's claim that (6.2) holds. This condition is used to justify the boundedness of denominators in Corollaries 6.5–6.6 and the application of Proposition 1.6 to the cutoff nonlinearity, so the approximation step is invalid as written.
  2. [Section 6.4, Eqs. (6.55)–(6.58)] Even if one replaced (6.2) by a feasible condition such as g^{γ1} δ_ε → 0, the final step still fails. The bound (6.55) contains the factor exp{36ℓ1 g^{γ1} + 6 e^{12ℓ1 g^{γ1}+|m|T0} ℓ2 g^{γ2}}/(log ε^{-1})^{1/4}; for any divergent g, this diverges or at best tends to a nonzero constant. With the chosen g in (6.57), inequality (6.56) is plainly false: the right-hand side is log log log log ε^{-1}, while the left-hand side is approximately exp{6 (log log log log ε^{-1})^{6/7}} for ℓ1=ℓ2=1. Consequently (6.58) does not follow. The fundamental obstruction is that the cutoff nonlinearity has stated C^{1,1} constants \tilde L1∼g^{γ1}, \tilde L2∼g^{γ2}, and Proposition 1.6's bound has constants of order e^{O(\tilde L1)}; no choice of divergent g can make the final estimate tend to zero. Thus Theorem 1.3 is not proved by the given argument.
minor comments (4)
  1. [Section 3, Lemma 3.1] The extension from globally Lipschitz f to general f ∈ \bar S is only sketched. Since Lemma 3.1 underpins the concentration estimates used throughout, the approximation argument for non-Lipschitz f should be written out in detail or supplied as an appendix.
  2. [Section 6.3, Lemma 6.7] The proof of the final interval in Lemma 6.7 is omitted with 'the argument is essentially the same ... so we omit the details.' Because the final interval uses δ_f^ε instead of δ_ε and requires a separate check of the conditions (1.28), this step should be shown explicitly.
  3. [Section 6.1] The notation \bar S′ is used interchangeably with \bar S' in a few places; please standardize in a revision. Also, in Lemma 6.4 the reference to 'Proposition 3.2' should be to Lemma 3.2.
  4. [Throughout] Several equations contain minor typographical issues: for example, the interval notation in (5.33) is malformed, and in (6.38) the factor e^{3L1(δ_ε+1)+|m|T0} should be checked for consistency with the preceding definition of β. These are presentation issues and do not affect the main concerns above.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the McKean–Vlasov limit is derived, not assumed; the possible gap in condition (6.2) is a correctness concern, not circularity.

full rationale

The paper's central claim, Theorem 1.3, is not obtained by fitting a parameter or by assuming the limiting McKean–Vlasov equation. The McKean–Vlasov solution v^epsilon is defined independently by (1.15), with sigma_epsilon solving the ODE (1.17). The proof of convergence u^epsilon ≈ v^epsilon is carried out through Malliavin derivative bounds (Lemma 3.1), Gaussian Poincaré inequalities, and Gronwall-type estimates that are stated and proved in Section 4. The extension from S1 nonlinearities to general S' nonlinearities in Section 6 uses explicit cutoff functions, and the approximation errors are bounded using the concentration estimate (Lemma 3.2); no term in the final error bound is set equal to zero by definition. Theorem 1.4 derives the limiting ODE (1.22) from the already-established representation (1.16)–(1.17) via the change of variables (1.19); the cubic solution (1.23) is obtained by ordinary integration, and the comparison to the GRZ result is an independent consistency check, not an input. Citations to prior work, including [21] and [19], are used for well-posedness conventions and technical estimates; these are not load-bearing in the sense of substituting for the target derivation. The skeptical observation about condition (6.2) — that the stated limit e^{3g^{γ1}}/(g^{γ2}δ_ε)→0 appears unsatisfiable for any g→∞ — is a potential mathematical gap in the proof, but it is not a circularity: it concerns whether the cutoff argument closes, not whether the conclusion is assumed. Accordingly, the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the existence and Malliavin regularity of solutions to the mollified PDE (assumed, standard for this class), the comparison/maximum principle, Gaussian Poincare inequalities, and the imported well-posedness framework for the McKean-Vlasov equation from GRZ. The cutoff function g(epsilon) and the time-step delta_epsilon are chosen by hand to make the approximation and iteration arguments work.

free parameters (2)
  • time-step parameter delta_epsilon = (log epsilon^{-1})^{-1/2}
    Defined in (1.32) to define the exponential mesh t_m; chosen small enough to make the Gronwall iteration converge (conditions (5.15), (5.19)). Not fitted to data.
  • cutoff scale g(epsilon) = ( log log log log(epsilon^{-1}) / (14 ell_1) )^{1/gamma_1}
    Chosen in Section 6.4 to truncate f' in the approximation; growth is engineered so that exp(3g^{gamma1}) g^{gamma2} delta_epsilon tends to 0 while the C^{1,1} constants remain under control. This is an ad hoc choice for the proof.
assumptions (5)
  • domain assumption Existence and Malliavin regularity of solutions to (1.5) for f in S-bar (or S-bar-prime) with mollified white noise initial data.
    The proof of Lemma 3.1 shows the Malliavin derivative bound conditional on existence; the paper does not give a self-contained well-posedness proof for (1.5) for the full class. Standard semilinear heat equation theory is assumed.
  • standard math Comparison principle and maximum principle for parabolic operators, including with expectation terms.
    Used in Lemma 3.1, Lemma 2.2, and Lemma 6.3 to get signs and bounds on Malliavin derivatives and sub-/super-solutions.
  • standard math Gaussian Poincare inequality and Malliavin calculus covariance bounds hold for functionals of Gaussian white noise.
    Used to convert L^2(P) bounds on Malliavin derivatives into bounds on the solution error (Corollaries 5.3 and 6.5) and to bound the covariance in Proposition 5.1.
  • domain assumption The McKean-Vlasov equation (1.15) has a unique solution in the sense of Definition 2.1, with solution space and epsilon-dependent L^p bounds imported from GRZ [21, Proposition 6.2] and regularity from [21, Proposition 1.2].
    Lemma 2.2 proves uniqueness and Lemma 2.3 gives the explicit form, but the solution class and bound (2.3) are cited from [21]. The paper states that the full solution property follows from the proof of Proposition 1.2 in [21].
  • standard math Standard white noise covariance and heat kernel mollification produce smooth stationary Gaussian initial data.
    Foundation of the model in (1.3)-(1.5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data." pith.science (2026). https://pith.science/paper/5B4O7V5T

@misc{pith2026250906260,
  author       = {Pith},
  title        = {Pith review of: McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5B4O7V5T}},
  note         = {Machine review of arXiv:2509.06260}
}
abstract

We study a large class of scaling-critical reaction-diffusion equations in two spatial dimensions, where the initial data is white noise mollified at scale $\varepsilon^2$ and the reaction term is attenuated by a factor of $(\log\varepsilon^{-1})^{-1}$. We show that as $\varepsilon\to 0$, the solution converges to the solution of a McKean-Vlasov equation, which is Gaussian with standard deviation given by the solution to an ODE. Our result covers the case of the reaction term $f(u)=u^3$, and thus gives a new proof of the limiting behavior for the Allen-Cahn equation discovered in the recent work of Gabriel, Rosati, and Zygouras (Probab. Theory Related Fields 192: 1373-1446, 2025).

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 21 canonical work pages

  1. [1]

    Marginal triviality of the scaling limits of critical 4D Ising andϕ 4 4 models.Ann

    Michael Aizenman and Hugo Duminil-Copin. Marginal triviality of the scaling limits of critical 4D Ising andϕ 4 4 models.Ann. of Math. (2), 194(1):163–235, 2021.doi: 10.4007/annals.2021. 194.1.3

    work page doi:10.4007/annals.2021 2021

  2. [2]

    The two-dimensional stochastic heat equation: renor- malizing a multiplicative noise.J

    Lorenzo Bertini and Nicoletta Cancrini. The two-dimensional stochastic heat equation: renor- malizing a multiplicative noise.J. Phys. A, 31(2):615–622, 1998.doi: 10.1088/0305- 4470/ 31/2/019

    doi:10.1088/0305- 1998

  3. [3]

    A.J. Bray. Theory of phase ordering kinetics.Physica A: Statistical Mechanics and its Appli- cations, 194(1):41–52, 1993.doi: 10.1016/0378-4371(93)90338-5

    work page doi:10.1016/0378-4371(93)90338-5 1993

  4. [4]

    2D anisotropic KPZ at station- arity: scaling, tightness and nontriviality.Ann

    Giuseppe Cannizzaro, Dirk Erhard, and Philipp Sch¨ onbauer. 2D anisotropic KPZ at station- arity: scaling, tightness and nontriviality.Ann. Probab., 49(1):122–156, 2021.doi: 10.1214/20- AOP1446

    doi:10.1214/20- 2021

  5. [5]

    Weak coupling limit of the anisotropic KPZ equation.Duke Math

    Giuseppe Cannizzaro, Dirk Erhard, and Fabio Toninelli. Weak coupling limit of the anisotropic KPZ equation.Duke Math. J., 172(16):3013–3104, 2023.doi: 10.1215/00127094-2022-0094

    work page doi:10.1215/00127094-2022-0094 2023

  6. [6]

    Gaussian fluctuations for the stochastic Burgers equation in dimensiond≥2.Comm

    Giuseppe Cannizzaro, Massimiliano Gubinelli, and Fabio Toninelli. Gaussian fluctuations for the stochastic Burgers equation in dimensiond≥2.Comm. Math. Phys., 405(4):Paper No. 89, 60, 2024.doi: 10.1007/s00220-024-04966-z. 34

    work page doi:10.1007/s00220-024-04966-z 2024

  7. [7]

    The critical 2dstochastic heat flow is not a Gaussian multiplicative chaos.Ann

    Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras. The critical 2dstochastic heat flow is not a Gaussian multiplicative chaos.Ann. Probab., 51(6):2265–2300, 2023.doi: 10.1214/23- aop1648

    doi:10.1214/23- 2023

  8. [8]

    The critical 2d stochastic heat flow.Invent

    Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras. The critical 2d stochastic heat flow.Invent. Math., 233(1):325–460, 2023.doi: 10.1007/s00222-023-01184-7

    doi:10.1007/s00222-023-01184-7 2023

Show all 34 references
  1. [9]

    The two-dimensional KPZ equation in the entire subcritical regime.Ann

    Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras. The two-dimensional KPZ equation in the entire subcritical regime.Ann. Probab., 48(3):1086–1127, 2020.doi: 10 . 1214 / 19 - AOP1383

    2020

  2. [10]

    Universality in marginally relevant disordered systems.Ann

    Francesco Caravenna, Rongfeng Sun, and Nikos Zygouras. Universality in marginally relevant disordered systems.Ann. Appl. Probab., 27(5):3050–3112, 2017.doi: 10.1214/17-AAP1276

    2017 doi

  3. [11]

    Constructing a solution of the (2 + 1)-dimensional KPZ equation.Ann

    Sourav Chatterjee and Alexander Dunlap. Constructing a solution of the (2 + 1)-dimensional KPZ equation.Ann. Probab., 48(2):1014–1055, 2020.doi: 10.1214/19-AOP1382

    2020 doi

  4. [12]

    Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method.Stoch

    Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu. Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein’s method.Stoch. Partial Differ. Equ. Anal. Comput., 11(1):122–176, 2023.doi: 10.1007/s40072-021-00224-8

    2023 doi

  5. [13]

    Norm inflation for the cubic nonlinear heat equation above the scaling critical regularity, 2024

    Ilya Chevyrev, Tadahiro Oh, and Yuzhao Wang. Norm inflation for the cubic nonlinear heat equation above the scaling critical regularity, 2024. arXiv: 2205.14488[math.AP]

    2024 arXiv

  6. [14]

    Conditional GMC within the stochastic heat flow, 2025

    Jeremy Clark and Li-Cheng Tsai. Conditional GMC within the stochastic heat flow, 2025. arXiv: 2507.16056[math.PR]

    2025 arXiv

  7. [15]

    Weak-disorder limit at criticality for directed polymers on hierarchical graphs.Comm

    Jeremy Thane Clark. Weak-disorder limit at criticality for directed polymers on hierarchical graphs.Comm. Math. Phys., 386(2):651–710, 2021.doi: 10.1007/s00220-021-04149-0

    2021 doi

  8. [16]

    Ergodicity of infinite volume Φ 4 3 at high temperature, 2025

    Pawe l Duch, Martin Hairer, Jaeyun Yi, and Wenhao Zhao. Ergodicity of infinite volume Φ 4 3 at high temperature, 2025. arXiv: 2508.07776[math.PR]

    2025 arXiv

  9. [17]

    Non-perturbative approach to the Bourgain-Spencer conjecture in stochastic homogenization.J

    Mitia Duerinckx. Non-perturbative approach to the Bourgain-Spencer conjecture in stochastic homogenization.J. Math. Pures Appl. (9), 176:183–225, 2023.doi: 10.1016/j.matpur.2023. 06.005

    2023 doi

  10. [18]

    The 2D nonlinear stochastic heat equation: pointwise statistics and the decoupling function, 2023

    Alexander Dunlap and Cole Graham. The 2D nonlinear stochastic heat equation: pointwise statistics and the decoupling function, 2023. arXiv: 2308.11850[math.PR]

    2023

  11. [19]

    A forward-backward SDE from the 2D nonlinear stochastic heat equation.Ann

    Alexander Dunlap and Yu Gu. A forward-backward SDE from the 2D nonlinear stochastic heat equation.Ann. Probab., 50(3):1204–1253, 2022.doi: 10.1214/21-aop1563

    2022 doi

  12. [20]

    Prentice-Hall, Inc., Engle- wood Cliffs, NJ, 1964, pages xiv+347

    Avner Friedman.Partial differential equations of parabolic type. Prentice-Hall, Inc., Engle- wood Cliffs, NJ, 1964, pages xiv+347

    1964

  13. [21]

    The Allen–Cahn equation with weakly critical random initial datum.Probability Theory and Related Fields, 192(3-4):1373–1446, 2025.doi: 10.1007/s00440-024-01312-1

    Simon Gabriel, Tommaso Rosati, and Nikos Zygouras. The Allen–Cahn equation with weakly critical random initial datum.Probability Theory and Related Fields, 192(3-4):1373–1446, 2025.doi: 10.1007/s00440-024-01312-1

    2025 doi

  14. [22]

    Gaussian fluctuations from the 2D KPZ equation.Stoch

    Yu Gu. Gaussian fluctuations from the 2D KPZ equation.Stoch. Partial Differ. Equ. Anal. Comput., 8(1):150–185, 2020.doi: 10.1007/s40072-019-00144-8

    2020 doi

  15. [23]

    Stochastic heat flow is a black noise, 2025

    Yu Gu and Li-Cheng Tsai. Stochastic heat flow is a black noise, 2025. arXiv: 2506 . 16484 [math.PR]

    2025

  16. [24]

    The Allen-Cahn equation with generic initial datum.Probab

    Martin Hairer, Khoa Lˆ e, and Tommaso Rosati. The Allen-Cahn equation with generic initial datum.Probab. Theory Related Fields, 186(3-4):957–998, 2023.doi: 10 . 1007 / s00440 - 023 - 01198-5. 35

    2023

  17. [25]

    Ryser, and Hendrik Weber

    Martin Hairer, Marc D. Ryser, and Hendrik Weber. Triviality of the 2D stochastic Allen-Cahn equation.Electron. J. Probab., 17:no. 39, 14, 2012.doi: 10.1214/EJP.v17-1731

    2012 doi

  18. [26]

    Kohn and Felix Otto

    Robert V. Kohn and Felix Otto. Upper bounds on coarsening rates.Communications in Mathematical Physics, 229(3):375–395, 2002.doi: 10.1007/s00220-002-0693-4

    2002 doi

  19. [27]

    Global well-posedness of the dynamic Φ 4 model in the plane.Ann

    Jean-Christophe Mourrat and Hendrik Weber. Global well-posedness of the dynamic Φ 4 model in the plane.Ann. Probab., 45(4):2398–2476, 2017.doi: 10.1214/16-AOP1116

    2017 doi

  20. [28]

    Second order Poincar´ e inequalities and CLTs on Wiener space.J

    Ivan Nourdin, Giovanni Peccati, and Gesine Reinert. Second order Poincar´ e inequalities and CLTs on Wiener space.J. Funct. Anal., 257(2):593–609, 2009.doi: 10.1016/j.jfa.2008.12.017

    2009 doi

  21. [29]

    Probability and its Applications (New York)

    David Nualart.The Malliavin calculus and related topics. Probability and its Applications (New York). Springer-Verlag, Berlin, second edition, 2006, pages xiv+382

    2006

  22. [30]

    Universal scaling in the motion of random interfaces.Phys

    Takao Ohta, David Jasnow, and Kyozi Kawasaki. Universal scaling in the motion of random interfaces.Phys. Rev. Lett., 49:1223–1226, 17, October 1982.doi: 10.1103/PhysRevLett.49. 1223

    1982 doi

  23. [31]

    Parisi and Yong Shi Wu

    G. Parisi and Yong Shi Wu. Perturbation theory without gauge fixing.Sci. Sinica, 24(4):483– 496, 1981

    1981

  24. [32]

    Stochastic heat flow by moments, 2025

    Li-Cheng Tsai. Stochastic heat flow by moments, 2025. arXiv: 2410.14657[math.PR]

    2025

  25. [33]

    Frederi G. Viens. Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent.Stochastic Process. Appl., 119(10):3671–3698, 2009.doi: 10. 1016/j.spa.2009.07.002

    2009

  26. [34]

    Viens and Andrew B

    Frederi G. Viens and Andrew B. Vizcarra. Supremum concentration inequality and modulus of continuity for sub-nth chaos processes.J. Funct. Anal., 248(1):1–26, 2007.doi: 10.1016/j. jfa.2007.03.019. 36

    2007 doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.