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REVIEW 2 major objections 3 minor 72 references

Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The 2D stochastic Burgers equation diffuses as (log t)^{2/3}, with prefactor (3/2π)^{2/3}λ^{4/3}.

desk verdict A serious proof paper that plausibly settles the 1986 vBKS conjecture with the sharp constant, but the whole edifice rests on one uniform integral estimate in Appendix C that a specialist must check line by line. read the letter →

arxiv 2501.00344 v2 pith:VLTEY2XL submitted 2024-12-31 math.PR math-phmath.APmath.MP

Giuseppe Cannizzaro , Quentin Moulard , Fabio Toninelli This is my paper · ORCID
classification math.PRmath-phmath.APmath.MP MSC 60H1582C2260F05
keywords stochasticBurgersequationsuperdiffusioncriticaldimensioncentrallimittheoremrenormalizationgroupdiffusionmatrixasymmetricsimpleexclusionprocess
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

This paper proves a long-standing conjecture for the stochastic Burgers equation in two space dimensions: at large times the effective diffusion in the direction of the nonlinearity grows like $(\log t)^{2/3}$, not like a constant, and the prefactor is exactly $(3/2\pi)^{2/3}\lambda^{4/3}$. It further shows that under the logarithmically adjusted rescaling the solution converges to the solution of a linear stochastic heat equation with an explicit anisotropic diffusion matrix, a Gaussian fixed point in the renormalization-group sense. If correct, this is the first scaling-limit theorem for a critical singular SPDE beyond the weak-coupling regime, and it identifies the precise large-scale behavior expected for other driven diffusive systems at their critical dimension.

What carries the argument

The argument is carried by the resolvent of the generator $\mathcal{L}^\tau$ acting on the Fock space built from Gaussian white noise. The key object is a scale-dependent diagonal operator $G_\tau^M=(1-S_\tau+\mathfrak{g}_\tau^M)^{-1}$, where $S_\tau$ is the linear diffusion part and $\mathfrak{g}_\tau^M$ has Fourier multiplier $\frac12|e_1\cdot p_{1:n}|^2\, g_\tau^M(L_\tau(\frac12|\sqrt{R_\tau}p_{1:n}|^2))$, built from $g_\tau(x)=((3/2)x+\nu_\tau^{3/2})^{2/3}-\nu_\tau$ with $\nu_\tau=(\log\tau)^{-2/3}$. The function $g_\tau$ solves the ODE $\dot y=1/\sqrt{\nu_\tau+y}$, and integrating that square-root law produces both the $2/3$ exponent and the constant $(3/2\pi)^{2/3}\lambda^{4/3}$. The proof splits the antisymmetric generator into $A^{\sharp,\tau}$ (singular in Fourier, with no growth in the chaos index) and $A^{\flat,\tau}$ (regular in Fourier, growing in chaos), then proves Replacement and Recursive Replacement Lemmas that control the error of the diagonal ansatz. The resolvent is represented as $s_\tau=\sum_{k\ge0}(G_\tau^M A^{\sharp,\tau}_+)^k G_\tau^M\phi$, and the recursive lemma shows that its anisotropic $\Gamma H^1_\tau$-norm, a Sobolev norm controlling derivatives in the $e_1$ direction, vanishes like $\nu_\tau^{1/4}\sqrt{|\log\nu_\tau|}$, exactly small enough to close the argument.

What would settle it

If the central claim is right, $D_{1,1}(t)/(\log t)^{2/3}$ converges to $(3/2\pi)^{2/3}\lambda^{4/3}$ as $t\to\infty$ for the regularized 2D SBE with fixed mollifier and coupling, while $D_{2,2}(t)\to1$. Measuring this ratio in a direct simulation and seeing a different limit, or a transverse diffusivity that does not approach 1, would falsify the claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is the sharp, complete large-time asymptotics of the regularized 2D SBE. Theorem 1.1 states that $D(t) = \operatorname{diag}(C_{\mathrm{eff}}(\lambda)(\log t)^{2/3}(1+o(1)), 1)$ as $t\to\infty$, with $C_{\mathrm{eff}}(\lambda)=(3/2\pi)^{2/3}\lambda^{4/3}$. Theorem 1.3 states that, after the superdiffusive rescaling $u_\tau(t,x)=\tau^{1/2}(\log\tau)^{1/6}u(\tau t, \tau^{1/2}R_\tau^{-1/2}x)$ with $R_\tau=\operatorname{diag}((\log\tau)^{-2/3},1)$, the finite-dimensional distributions converge to those of the anisotropic linear SHE $\partial_t u_{\mathrm{eff}} = \frac12\nabla\cdot D_{\mathrm{eff}}\nabla u_{\mathrm{eff}} + \nabla\cdot\sqrt{D_{\mathrm{eff}}}\,\xi$ with $D_{\mathrm{eff}}=\operatorname{diag}(C_{\mathrm{eff}}(\lambda),1)$. Theorem 1.4 upgrades this to convergence of the full semigroup in $L^\theta(P)$, uniformly in time, for initial laws absolutely continuous with respect to the stationary white-noise law. Thus the nonlinearity does more than renormalize diffusion: in the active direction it creates the effective diffusion coefficient and the effective noise, while the transverse direction remains diffusive with coefficient 1.

Load-bearing premise

The load-bearing premise is the Replacement Lemma's uniform integral approximation: the exact diagonal kernel of $-A^{\sharp,\tau}_-G_\tau^M A^{\sharp,\tau}_+$ is replaced by $\int_0^{L_\tau} d\ell/\sqrt{1+g_\tau^M(\ell)}$ up to an error of order $\nu_\tau$ in the $\Gamma H_\tau^{-1}$ norm; if this approximation fails at the stated rate, the ODE, the $2/3$ exponent, and the constant $(3/2\pi)^{2/3}\lambda^{4/3}$ all collapse.

Editorial extensions

If this is right

  • In the active direction the effective diffusivity diverges as $C_{\mathrm{eff}}(\lambda)(\log t)^{2/3}$, while the transverse entry stays at 1; the system is superdiffusive only along the direction forced by the nonlinearity.
  • The prefactor is independent of the microscopic diffusion coefficient in the active direction and scales as $\lambda^{4/3}$, the same power as in the one-dimensional stochastic Burgers case; the paper derives this from scaling plus the vanishing microscopic diffusivity in that direction.
  • Under the superdiffusive rescaling $R_\tau=\operatorname{diag}((\log\tau)^{-2/3},1)$, the limiting object is an anisotropic linear stochastic heat equation, so all nonlinear effects are absorbed into the renormalized diffusion matrix and noise.
  • The earlier Tauberian bounds for the 2D SBE and 2D ASEP, which left diverging subleading corrections, are replaced by an exact asymptotic with explicit error; for the active entry the paper obtains $D_{1,1}(t)=C_{\mathrm{eff}}(\lambda)(\log t)^{2/3}+O((\log t)^{1/2+o(1)})$.
  • The authors conjecture that 2D ASEP and similar driven diffusive systems at the critical dimension share the same Gaussian fixed point as the SBE.

Reading between the lines

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

  • Beyond the paper: the same diagonal-ansatz method should transfer to 2D ASEP, the natural lattice counterpart, and would predict the same $(\log t)^{2/3}$ correlation-length growth with the same $\lambda^{4/3}$ coupling dependence; this is a testable simulation prediction.
  • Beyond the paper: the square root in the flow equation $\dot y=1/\sqrt{\nu_\tau+y}$ appears to be the mechanism that fixes the logarithmic exponent $2/3$, so other anisotropic critical models with the same flow should share the exponent even if their fixed points are not Gaussian.
  • Beyond the paper: the $L^\theta(P)$ uniform-in-time semigroup convergence is stronger than annealed convergence, and under additional ergodicity assumptions it may yield a quenched central limit theorem, although almost-sure convergence is not claimed in the paper.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper studies the regularized two-dimensional Stochastic Burgers Equation at the critical dimension d=2. Its first main result, Theorem 1.1, identifies the sharp large-time asymptotics of the diffusion matrix, D(t) = diag(C_eff(lambda) (log t)^{2/3}(1+o(1)), 1), with explicit constant C_eff(lambda) = (3/2pi)^{2/3} lambda^{4/3}. Its second main result, Theorems 1.3 and 1.4, proves that under the logarithmically superdiffusive rescaling (1.8) the finite-dimensional distributions and, in L^theta(P), the full semigroup converge to those of the anisotropic linear Stochastic Heat Equation with effective diffusion matrix D_eff = diag(C_eff(lambda), 1). The proof uses the Fock-space representation of the generator, a sharp/flat splitting of the antisymmetric part, an approximate diagonal fixed point g_tau^M governed by the ODE y' = 1/sqrt(nu_tau + y), Replacement Lemmas, a recursive replacement lemma, a resolvent ansatz, and a Trotter-Kato argument.

Significance. If correct, this is a major advance: it appears to be the first rigorous scaling-limit result for a critical singular SPDE beyond the weak-coupling regime, and it gives an explicit universal constant rather than only Tauberian bounds with subleading corrections. The paper also contains several valuable ingredients of independent interest: a rigorous derivation of the Green-Kubo formula (Proposition 2.5), stationarity and skew-reversibility of the mollified dynamics (Proposition 2.3), and a systematic renormalization-group-style control of all length scales. The proof is elaborate and largely self-contained, and the sharp constant emerges from the fixed-point ODE rather than from a fit. However, the central quantitative estimate, Lemma C.1 in Appendix C, is not present in the review copy, and the replacement lemmas and the sharp constant depend on its claimed uniform O(nu_tau) error. The overall assessment is therefore conditional on that missing proof.

major comments (2)
  1. [Appendix C / Lemma C.1] Both Replacement Lemmas 3.6 and 3.9 are proved by invoking Lemma C.1, which replaces the exact diagonal kernel of -A^{sharp,-}_tau G^M_tau A^{sharp,+}_tau by the double integral displayed near (3.24) and (3.32), with a claimed uniform O(nu_tau) error. This uniformity in the external momenta p_{1:n} and in the chaos index n is load-bearing: it is what removes the N-growth from the replacement error, stabilizes the Picard iteration in Lemma 4.12, and ultimately controls the rate in Theorem 4.1 and the sharp constant C_eff in (1.6). The review copy I was asked to assess does not contain the proof of Lemma C.1; the text stops at the heading of Appendix C. I therefore cannot verify this step from the displayed material. Please supply the complete statement and proof, with explicit dependence on the function Phi, on M, and on all arguments, and with confirmation that the error is uniform in n and p_{1:n}.
  2. [Section 3.3, Eqs. (3.24)-(3.26), (3.32)-(3.33)] Assuming Lemma C.1, the algebra after (3.24) is transparent: the integral in the variable 1/ζ evaluates to 1/sqrt(1+g^M_tau(ell)), and the lower bound (3.26) using the cutoff at M nu_tau yields the O(M nu_tau) comparison with g^M_tau(ell_tau). The same ODE mechanism is used in Lemma 3.9. However, the current presentation leaves the entire burden of the uniform rate on the unstated Lemma C.1. Since a polynomial factor in n or a dependence on p_{1:n} through L_tau(1/2|sqrt(R_tau)p_{1:n}|^2) would invalidate the estimate (3.21) and the convergence ||r_tau - G^M_tau phi|| -> 0 in Theorem 5.1, I ask that the statement of Lemma C.1 be given in the main text with its error constants made fully explicit.
minor comments (3)
  1. [Section 4] There are typographical slips: Proposition 4.3 begins with 'Assumte', and in (4.9) and the following line 's_epsilon' appears where 's_tau' is clearly intended; Equations (4.11) and (4.23) contain similar 'G_epsilon'/'G^M_tau' slips.
  2. [Section 4.2] The bound on f^tau_w(L_tau(x)) in the proof of Theorem 4.1 is compressed; the intermediate inequality f^tau_w(L_tau(x)) <~ nu_tau^{1/2} sqrt(nu_tau + g_tau(L_tau(x))) max{n_0, log(sqrt(nu_tau + g_tau(L_tau(x)))/nu_tau)} should be displayed before the final nu_tau^{1/2}|log nu_tau| bound, since this is where the sqrt(|log nu_tau|) factor originates.
  3. [Notation] The review copy contains many OCR artifacts in equation labels (e.g., '/one.taboldstyle./two.taboldstyle/zero.taboldstyle'); if this reflects the submitted file, a cleanly typeset version should be provided to allow verification of all cross-references.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the sharp constant and the 2/3 exponent are derived from the fixed-point ansatz and Replacement Lemmas, not imported from the conjecture or from a fit.

full rationale

The derivation chain is self-contained. The approximate fixed point g_tau^M is fixed by the exact integral identity behind the Replacement Lemma: the ODE dot(y) = 1/sqrt(nu_tau + y) is chosen so that ∫_0^ell dℓ/sqrt(nu_tau + g_tau(ℓ)) = g_tau(ℓ), and this identity is then used to show that -A^sharp,- G_tau^M A^sharp,+ is close to g_tau^M. The constant C_eff = (3/(2π))^{2/3} λ^{4/3} is obtained by evaluating the same function at L_tau(0) = λ²/π + o(1), not by matching a target value. The choice nu_tau = (log τ)^{-2/3} is part of the theorem's scaling statement; the proof does not use the conjectured (log t)^{2/3} asymptotic as an input. The Replacement Lemma and Recursive Replacement Lemma are proved in the paper via Appendix C rather than imported from the vBKS conjecture. Self-citations such as [CT24] and [CGT24] are heuristic or methodological and are not the justification of the main estimates. The only step that cannot be checked from the text displayed here is the uniform O(nu_tau) estimate in Appendix C's Lemma C.1, which is a verification/correctness concern rather than circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new physical entities are postulated: no new particles, fields, forces, dimensions, or conserved quantities. The effective diffusion operator D_eff and the approximate fixed point g_τ^M are mathematical limit objects constructed from the generator kernels, with explicit Fourier multipliers; the Gaussian fixed point is a theorem about the limiting semigroup, not an imported mechanism. The two free parameters listed (M and ν_τ) are technical choices whose influence is shown to vanish or be self-consistent in the final statements.

free parameters (2)
  • M (artificial cutoff in the approximate fixed point g_τ^M) = any M ≥ M_0(λ) such that C_M^{(1)} ∨ C_M^{(2)} ≤ 1/2; final results independent of M
    Introduced in (3.16)-(3.18) to force the approximate fixed point into a perturbative regime. The constants track explicitly like λ^2 (1∨log M)/(1∨M^{1/2}) and vanish in the limit (Remarks 3.7, 3.10); Theorem 4.1 and the final theorems do not depend on M.
  • ν_τ = (1∨log τ)^{-2/3}, the superdiffusive rescaling rate = (log τ)^{-2/3}
    Chosen in (1.16) so that the diffusion matrix entries are O(1) under rescaling (6.1). The 2/3 exponent is then derived from the fixed-point ODE ẏ = 1/√(ν_τ+y) whose solution is g_τ (3.19), so it is confirmed rather than assumed; but the ansatz shape is informed by the vBKS conjecture being proved.
assumptions (6)
  • standard math Fock-space isometry between L_2(P_τ) and ΓL_2_τ, including surjectivity of ι_τ via analyticity of Θ_τ with empty zero set
    Invoked in Section 1.3 and Lemma 1.7; requires the mollifier's Fourier transform to vanish only on a Lebesgue-null set, stated in the proof of Lemma 1.7.
  • domain assumption Stationarity and skew-reversibility of the mollified dynamics (Prop. 2.3)
    Proved in Appendix A.3 by periodic approximation; used to derive the Green-Kubo formula (2.18), the correlation decay (2.5), and the Markov semigroup properties (Lemma 2.6).
  • domain assumption Polynomial spatial decay of correlations (Prop. 2.4)
    Justifies integration by parts in the Green-Kubo derivation and the δ→0 regularization in Section 6.1 via the comparison with ψ_δ (6.2).
  • ad hoc to paper Sharp/flat splitting A_τ = A_τ^{♯} + A_τ^{♭} with χ^♯ in (3.9)
    A technical decomposition of the generator: A^♯ produces no off-diagonal chaos terms (Lemma 3.3) while A^♭ is uniformly bounded in τ with an extra N factor (Prop. 3.4). It is constructed to make the estimates work.
  • ad hoc to paper Artificial cutoff M with condition (4.2): C_M^{(1)} ∨ C_M^{(2)} ≤ 1/2
    Stabilizes the recursive iteration I_τ(f) + ν_τ C_f C_M^{(1)} in Lemma 3.9; the final results are stated to be independent of M.
  • domain assumption Initial law Q absolutely continuous with respect to the white noise law P in Theorems 1.3 and 1.4
    Used in Theorem 1.3 for the finite-dimensional convergence; Theorem 1.4 is stated for F ∈ L^θ(P). The body discloses this restriction.

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Pith. "Pith review of Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension." pith.science (2026). https://pith.science/paper/VLTEY2XL

@misc{pith2026250100344,
  author       = {Pith},
  title        = {Pith review of: Superdiffusive Central Limit Theorem for the Stochastic Burgers Equation at the critical dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VLTEY2XL}},
  note         = {Machine review of arXiv:2501.00344}
}
abstract

The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity. In space dimension d = 2, the SBE is critical, being formally scale invariant under diffusive scaling. As such, it falls outside of the domain of applicability of the theories of Regularity Structures and paracontrolled calculus. In apparent contrast with the formal scale invariance, we fully prove the conjecture first appeared in [H. van Beijeren, R. Kutner, & H. Spohn, Phys. Rev. Lett., 1986] according to which the 2d-SBE is logarithmically superdiffusive, i.e. its diffusion coefficient diverges like $(\log t)^{2/3}$ as $t\to\infty$, thus removing subleading diverging multiplicative corrections in [D. De Gaspari & L. Haunschmid-Sibitz, Electron. J. Probab., 2024] and in [H.-T. Yau, Ann. of Math., 2004] for 2d-ASEP. We precisely identify the constant prefactor of the logarithm and show it is proportional to $\lambda^{4/3}$, for $\lambda>0$ the coupling constant, which, intriguingly, turns out to be exactly the same as for the one-dimensional Stochastic Burgers/KPZ equation. More importantly, we prove that, under super-diffusive space-time rescaling, the SBE has an explicit Gaussian fixed point in the Renormalization Group sense, by deriving a superdiffusive central limit-type theorem for its solution. This is the first scaling limit result for a critical singular SPDE, beyond the weak coupling regime, and is obtained via a refined control, on all length-scales, of the resolvent of the generator of the SBE. We believe our methods are well-suited to study other out-of-equilibrium driven diffusive systems at the critical dimension, such as 2d-ASEP, which, we conjecture, have the same large-scale Fixed Point as SBE.

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Works this paper leans on

72 extracted references · 37 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type url doi eprint volume year label extra.label sort.label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := ...

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

    work page

  3. [3]

    Armstrong , A

    S. Armstrong , A. Bou-Rabee , and T. Kuusi . Superdiffusive central limit theorem for a B rownian particle in a critically-correlated incompressible random drift. arXiv preprint arXiv:2404.01115 (2024)

    arXiv 2024

  4. [4]

    Aizenman and H

    M. Aizenman and H. Duminil-Copin . Marginal triviality of the scaling limits of critical 4d Ising and _4^4 models. Ann. Math. (2) 194, no. 1, (2021), 163--235. doi:10.4007/annals.2021.194.1.3 http://dx.doi.org/10.4007/annals.2021.194.1.3

    doi:10.4007/annals.2021.194.1.3 2021

  5. [5]

    Bertini and N

    L. Bertini and N. Cancrini . The two-dimensional stochastic heat equation: renormalizing a multiplicative noise. J. Phys. A, Math. Gen. 31, no. 2, (1998), 615--622. doi:10.1088/0305-4470/31/2/019 http://dx.doi.org/10.1088/0305-4470/31/2/019

    work page doi:10.1088/0305-4470/31/2/019 1998

  6. [6]

    , Comtet, A

    Bouchaud, J.P. , Comtet, A. , Georges, A. , and Le Doussal, P. Anomalous diffusion in random media of any dimensionality. J. Phys. France 48, no. 9, (1987), 1445--1450. doi:10.1051/jphys:019870048090144500 http://dx.doi.org/10.1051/jphys:019870048090144500

    work page doi:10.1051/jphys:019870048090144500 1987

  7. [7]

    Bal \'a zs , J

    M. Bal \'a zs , J. Quastel , and T. Sepp \"a l \"a inen . Fluctuation exponent of the KPZ /stochastic Burgers equation. J. Am. Math. Soc. 24, no. 3, (2011), 683--708. doi:10.1090/S0894-0347-2011-00692-9 http://dx.doi.org/10.1090/S0894-0347-2011-00692-9

    work page doi:10.1090/s0894-0347-2011-00692-9 2011

  8. [8]

    Barab\' a si and H

    A.-L. Barab\' a si and H. E. Stanley . Fractal concepts in surface growth. Cambridge University Press, Cambridge, 1995. doi:10.1017/CBO9780511599798 http://dx.doi.org/10.1017/CBO9780511599798

    doi:10.1017/cbo9780511599798 1995

Show all 72 references
  1. [9]

    Caravenna and F

    F. Caravenna and F. Cottini . Gaussian limits for subcritical chaos. Electron. J. Probab. 27, (2022), 35. Id/No 81. doi:10.1214/22-EJP798 http://dx.doi.org/10.1214/22-EJP798

    2022 doi

  2. [10]

    Chandra , I

    A. Chandra , I. Chevyrev , M. Hairer , and H. Shen . Langevin dynamic for the 2d Yang - Mills measure. Publ. Math., Inst. Hautes \'E tud. Sci. 136, (2022), 1--147. doi:10.1007/s10240-022-00132-0 http://dx.doi.org/10.1007/s10240-022-00132-0

    2022 doi

  3. [11]

    Chandra , I

    A. Chandra , I. Chevyrev , M. Hairer , and H. Shen . Stochastic quantisation of Yang - Mills - Higgs in 3d. Invent. Math. 237, no. 2, (2024), 541--696. doi:10.1007/s00222-024-01264-2 http://dx.doi.org/10.1007/s00222-024-01264-2

    2024 doi

  4. [12]

    Comets , C

    F. Comets , C. Cosco , and C. Mukherjee . Space-time fluctuation of the Kardar - Parisi - Zhang equation in d 3 and the Gaussian free field. Ann. Inst. Henri Poincar \'e , Probab. Stat. 60, no. 1, (2024), 82--112. doi:10.1214/22-AIHP1272 http://dx.doi.org/10.1214/22-AIHP1272

    2024 doi

  5. [13]

    Chatterjee and A

    S. Chatterjee and A. Dunlap . Constructing a solution of the (2+1) -dimensional KPZ equation. Ann. Probab. 48, no. 2, (2020), 1014--1055. doi:10.1214/19-AOP1382 http://dx.doi.org/10.1214/19-AOP1382

    2020 doi

  6. [14]

    Cannizzaro , D

    G. Cannizzaro , D. Erhard , and P. Sch \"o nbauer . 2d anisotropic KPZ at stationarity: scaling, tightness and nontriviality. Ann. Probab. 49, no. 1, (2021), 122--156. doi:10.1214/20-AOP1446 http://dx.doi.org/10.1214/20-AOP1446

    2021 doi

  7. [15]

    Cannizzaro , D

    G. Cannizzaro , D. Erhard , and F. Toninelli . The Stationary AKPZ equation: logarithmic superdiffusivity. Commun. Pure Appl. Math. 76, no. 11, (2023), 3044--3103. doi:10.1002/cpa.22108 http://dx.doi.org/10.1002/cpa.22108

    2023 doi

  8. [16]

    Cannizzaro , D

    G. Cannizzaro , D. Erhard , and F. Toninelli . Weak coupling limit of the anisotropic KPZ equation. Duke Math. J. 172, no. 16, (2023), 3013--3104. doi:10.1215/00127094-2022-0094 http://dx.doi.org/10.1215/00127094-2022-0094

    2023 doi

  9. [17]

    Cannizzaro and H

    G. Cannizzaro and H. Giles . An invariance principle for the 2d weakly self-repelling B rownian polymer, 2024. ://arxiv.org/abs/2403.06730

    2024 arXiv

  10. [18]

    Cannizzaro , M

    G. Cannizzaro , M. Gubinelli , and F. Toninelli . Gaussian fluctuations for the stochastic Burgers equation in dimension d 2 . Commun. Math. Phys. 405, no. 4, (2024), 60. Id/No 89. doi:10.1007/s00220-024-04966-z http://dx.doi.org/10.1007/s00220-024-04966-z

    2024 doi

  11. [19]

    Cannizzaro , L

    G. Cannizzaro , L. Haunschmid-Sibitz , and F. Toninelli . t -superdiffusivity for a Brownian particle in the curl of the 2d GFF . Ann. Probab. 50, no. 6, (2022), 2475--2498. doi:10.1214/22-AOP1589 http://dx.doi.org/10.1214/22-AOP1589

    2022 doi

  12. [20]

    Chang , C

    C.-C. Chang , C. Landim , and S. Olla . Equilibrium fluctuations of asymmetric simple exclusion processes in dimension d 3 . Probab. Theory Related Fields 119, no. 3, (2001), 381--409. doi:10.1007/PL00008764 http://dx.doi.org/10.1007/PL00008764

    2001 doi

  13. [21]

    Chatzigeorgiou , P

    G. Chatzigeorgiou , P. Morfe , F. Otto , and L. Wang . The gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off. arXiv preprint arXiv:2212.14244 (2022)

    2022

  14. [22]

    Cosco , S

    C. Cosco , S. Nakajima , and M. Nakashima . Law of large numbers and fluctuations in the sub-critical and L^2 regions for she and kpz equation in dimension d 3 . Stochastic Processes Appl. 151, (2022), 127--173. doi:10.1016/j.spa.2022.05.010 http://dx.doi.org/10.1016/j.spa.2022.05.010

    2022 doi

  15. [23]

    Caravenna , R

    F. Caravenna , R. Sun , and N. Zygouras . Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27, no. 5, (2017), 3050--3112. doi:10.1214/17-AAP1276 http://dx.doi.org/10.1214/17-AAP1276

    2017 doi

  16. [24]

    Caravenna , R

    F. Caravenna , R. Sun , and N. Zygouras . The two-dimensional KPZ equation in the entire subcritical regime. Ann. Probab. 48, no. 3, (2020), 1086--1127. doi:10.1214/19-AOP1383 http://dx.doi.org/10.1214/19-AOP1383

    2020 doi

  17. [25]

    Caravenna , R

    F. Caravenna , R. Sun , and N. Zygouras . The critical 2d stochastic heat flow. Invent. Math. 233, no. 1, (2023), 325--460. doi:10.1007/s00222-023-01184-7 http://dx.doi.org/10.1007/s00222-023-01184-7

    2023 doi

  18. [26]

    Cannizzaro and F

    G. Cannizzaro and F. Toninelli . Lecture notes on stationary critical and super-critical SPDE s, 2024. ://arxiv.org/abs/2403.15006

    2024 arXiv

  19. [27]

    Dunlap and Y

    A. Dunlap and Y. Gu . A forward-backward SDE from the 2d nonlinear stochastic heat equation. Ann. Probab. 50, no. 3, (2022), 1204--1253. doi:10.1214/21-AOP1563 http://dx.doi.org/10.1214/21-AOP1563

    2022 doi

  20. [28]

    Dunlap , Y

    A. Dunlap , Y. Gu , L. Ryzhik , and O. Zeitouni . Fluctuations of the solutions to the KPZ equation in dimensions three and higher. Probab. Theory Relat. Fields 176, no. 3-4, (2020), 1217--1258. doi:10.1007/s00440-019-00938-w http://dx.doi.org/10.1007/s00440-019-00938-w

    2020 doi

  21. [29]

    Da Prato and J

    G. Da Prato and J. Zabczyk . Stochastic equations in infinite dimensions., vol. 44 of Encycl. Math. Appl. Cambridge: Cambridge University Press, reprint of the 1992 hardback ed. ed., 2008. doi:10.1017/CBO9780511666223 http://dx.doi.org/10.1017/CBO9780511666223

    1992 doi

  22. [30]

    P. Duch . Renormalization of singular elliptic stochastic PDEs using flow equation (2022)

    2022

  23. [31]

    S. N. Ethier and T. G. Kurtz . Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. Characterization and convergence

    1986

  24. [32]

    D. E. Edmunds and H. Triebel . Function spaces, entropy numbers, differential operators, vol. 120 of Camb. Tracts Math. Cambridge: Cambridge Univ. Press, 1996

    1996

  25. [33]

    Funaki and J

    T. Funaki and J. Quastel . KPZ equation, its renormalization and invariant measures. Stoch. Partial Differ. Equ., Anal. Comput. 3, no. 2, (2015), 159--220. doi:10.1007/s40072-015-0046-x http://dx.doi.org/10.1007/s40072-015-0046-x

    2015 doi

  26. [34]

    Gerolla , M

    L. Gerolla , M. Hairer , and X.-M. Li . Fluctuations of stochastic PDEs with long-range correlations. Preprint, arXiv :2303.09811 [math. PR ] (2023), 2023. ://arxiv.org/abs/2303.09811

    2023 arXiv

  27. [35]

    D. D. Gaspari and L. Haunschmid-Sibitz . ( t)^ 2/3 -superdiffusivity for the 2d stochastic Burgers equation . Electronic Journal of Probability 29, no. none, (2024), 1 -- 34. doi:10.1214/24-EJP1249 http://dx.doi.org/10.1214/24-EJP1249

    2024 doi

  28. [36]

    Gubinelli , P

    M. Gubinelli , P. Imkeller , and N. Perkowski . Paracontrolled distributions and singular PDE s. Forum Math. Pi 3, (2015), e6, 75. doi:10.1017/fmp.2015.2 http://dx.doi.org/10.1017/fmp.2015.2

    2015 doi

  29. [37]

    Goncalves and M

    P. Goncalves and M. Jara . Nonlinear fluctuations of weakly asymmetric interacting particle systems. Arch. Ration. Mech. Anal. 212, no. 2, (2014), 597--644. doi:10.1007/s00205-013-0693-x http://dx.doi.org/10.1007/s00205-013-0693-x

    2014 doi

  30. [38]

    Gu and J

    Y. Gu and J. Li . Fluctuations of a nonlinear stochastic heat equation in dimensions three and higher. SIAM J. Math. Anal. 52, no. 6, (2020), 5422--5440. doi:10.1137/19M1296380 http://dx.doi.org/10.1137/19M1296380

    2020 doi

  31. [39]

    Gubinelli and N

    M. Gubinelli and N. Perkowski . KPZ reloaded. Commun. Math. Phys. 349, no. 1, (2017), 165--269. doi:10.1007/s00220-016-2788-3 http://dx.doi.org/10.1007/s00220-016-2788-3

    2017 doi

  32. [40]

    Gubinelli and N

    M. Gubinelli and N. Perkowski . Energy solutions of KPZ are unique. J. Am. Math. Soc. 31, no. 2, (2018), 427--471. doi:10.1090/jams/889 http://dx.doi.org/10.1090/jams/889

    2018 doi

  33. [41]

    Gubinelli and N

    M. Gubinelli and N. Perkowski . The infinitesimal generator of the stochastic Burgers equation. Probab. Theory Relat. Fields 178, no. 3-4, (2020), 1067--1124. doi:10.1007/s00440-020-00996-5 http://dx.doi.org/10.1007/s00440-020-00996-5

    2020 doi

  34. [42]

    Gr \"a fner , N

    L. Gr \"a fner , N. Perkowski , and S. Popat . Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions. Preprint, arXiv :2411.07680 [math. PR ] (2024), 2024. ://arxiv.org/abs/2411.07680

    2024 arXiv

  35. [43]

    Y. Gu , L. Ryzhik , and O. Zeitouni . The Edwards - Wilkinson limit of the random heat equation in dimensions three and higher. Commun. Math. Phys. 363, no. 2, (2018), 351--388. doi:10.1007/s00220-018-3202-0 http://dx.doi.org/10.1007/s00220-018-3202-0

    2018 doi

  36. [44]

    Gabriel , T

    S. Gabriel , T. Rosati , and N. Zygouras . The Allen - Cahn equation with weakly critical random initial datum. Probab. Theory Relat. Fields (2024). doi:10.1007/s00440-024-01312-1 http://dx.doi.org/10.1007/s00440-024-01312-1

    2024 doi

  37. [45]

    Y. Gu . Gaussian fluctuations from the 2 D KPZ equation. Stoch. Partial Differ. Equ. Anal. Comput. 8, no. 1, (2020), 150--185. doi:10.1007/s40072-019-00144-8 http://dx.doi.org/10.1007/s40072-019-00144-8

    2020 doi

  38. [46]

    M. Hairer . Solving the KPZ equation. Ann. Math. (2) 178, no. 2, (2013), 559--664. doi:10.4007/annals.2013.178.2.4 http://dx.doi.org/10.4007/annals.2013.178.2.4

    2013 doi

  39. [47]

    M. Hairer . A theory of regularity structures. Invent. Math. 198, no. 2, (2014), 269--504. doi:10.1007/s00222-014-0505-4 http://dx.doi.org/10.1007/s00222-014-0505-4

    2014 doi

  40. [48]

    D. Henry . Geometric theory of semilinear parabolic equations, vol. 840 of Lect. Notes Math. Springer, Cham, 1981

    1981

  41. [49]

    Hairer and C

    M. Hairer and C. Labb \'e . A simple construction of the continuum parabolic Anderson model on R^2 . Electron. Commun. Probab. 20, (2015), 11. Id/No 43. doi:10.1214/ECP.v20-4038 http://dx.doi.org/10.1214/ECP.v20-4038

    2015 doi

  42. [50]

    Horv\'ath , B

    I. Horv\'ath , B. T\'oth , and B. Vet o . Relaxed sector condition. Bull. Inst. Math. Acad. Sin. (N.S.) 7, no. 4, (2012), 463--476

    2012

  43. [51]

    Ito and F

    K. Ito and F. Kappel . Evolution equations and approximations, vol. 61 of Ser. Adv. Math. Appl. Sci. Singapore: World Scientific, 2002

    2002

  44. [52]

    Komorowski , C

    T. Komorowski , C. Landim , and S. Olla . Fluctuations in Markov processes. Time symmetry and martingale approximation. , vol. 345 of Grundlehren Math. Wiss. Berlin: Springer, 2012. doi:10.1007/978-3-642-29880-6 http://dx.doi.org/10.1007/978-3-642-29880-6

    2012 doi

  45. [53]

    Kupiainen

    A. Kupiainen . Renormalization group and stochastic PDEs . Ann. Henri Poincar \'e 17, no. 3, (2016), 497--535. doi:10.1007/s00023-015-0408-y http://dx.doi.org/10.1007/s00023-015-0408-y

    2016 doi

  46. [54]

    Landim , S

    C. Landim , S. Olla , and H. Yau . Diffusive behaviour of the equilibrium fluctuations in the asymmetric exclusion processes. Adv. Stud. Pure Math. 39, (2004), 307--324. doi:10.2969/aspm/03910307 http://dx.doi.org/10.2969/aspm/03910307

    2004

  47. [55]

    Landim and H

    C. Landim and H. T. Yau . Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab. Theory Relat. Fields 108, no. 3, (1997), 321--356. doi:10.1007/s004400050112 http://dx.doi.org/10.1007/s004400050112

    1997 doi

  48. [56]

    Lygkonis and N

    D. Lygkonis and N. Zygouras . E dwards-- W ilkinson fluctuations for the directed polymer in the full L ^2 -regime for dimensions d 3 . Ann. Inst. H. Poincar\'e (B): Prob. Stat. 58(1), (2022), 65--104

    2022

  49. [57]

    Morfe , F

    P. Morfe , F. Otto , and C. Wagner . A Critical Drift - Diffusion Equation : Connections to the Diffusion on SL(2) . Preprint, arXiv :2410.15983 [math. PR ] (2024), 2024. ://arxiv.org/abs/2410.15983

    2024

  50. [58]

    Martin and N

    J. Martin and N. Perkowski . Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model. Ann. Inst. Henri Poincar \'e , Probab. Stat. 55, no. 4, (2019), 2058--2110. doi:10.1214/18-AIHP942 http://dx.doi.org/10.1214/18-AIHP942

    2019 doi

  51. [59]

    Matetski , J

    K. Matetski , J. Quastel , and D. Remenik . The KPZ fixed point. Acta Math. 227, no. 1, (2021), 115--203. doi:10.4310/ACTA.2021.v227.n1.a3 http://dx.doi.org/10.4310/ACTA.2021.v227.n1.a3

    2021 doi

  52. [60]

    D. Nualart . The M alliavin calculus and related topics . Probability and its Applications (New York). Springer-Verlag, Berlin, second ed., 2006

    2006

  53. [61]

    Otto and C

    F. Otto and C. Wagner . A critical drift-diffusion equation: intermittent behavior. Preprint, arXiv :2404.13641 [math. PR ] (2024), 2024. ://arxiv.org/abs/2404.13641

    2024

  54. [62]

    Quastel and H

    J. Quastel and H. Spohn . The one-dimensional KPZ equation and its universality class. J. Stat. Phys. 160, no. 4, (2015), 965--984. doi:10.1007/s10955-015-1250-9 http://dx.doi.org/10.1007/s10955-015-1250-9

    2015 doi

  55. [63]

    Quastel and S

    J. Quastel and S. Sarkar . Convergence of exclusion processes and the KPZ equation to the KPZ fixed point. J. Am. Math. Soc. 36, no. 1, (2023), 251--289. doi:10.1090/jams/999 http://dx.doi.org/10.1090/jams/999

    2023 doi

  56. [64]

    R. Tao . Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two. Stoch. Partial Differ. Equ., Anal. Comput. 12, no. 1, (2024), 220--246. doi:10.1007/s40072-022-00282-6 http://dx.doi.org/10.1007/s40072-022-00282-6

    2024 doi

  57. [65]

    L.-C. Tsai . Stochastic heat flow by moments. Preprint, arXiv :2410.14657 [math. PR ] (2024), 2024. ://arxiv.org/abs/2410.14657

    2024

  58. [66]

    T\' o th and B

    B. T\' o th and B. Valk\' o . Superdiffusive bounds on self-repellent B rownian polymers and diffusion in the curl of the G aussian free field in d=2 . J. Stat. Phys. 147, no. 1, (2012), 113--131. doi:10.1007/s10955-012-0462-5 http://dx.doi.org/10.1007/s10955-012-0462-5

    2012 doi

  59. [67]

    van Beijeren , R

    H. van Beijeren , R. Kutner , and H. Spohn . Excess noise for driven diffusive systems. Phys. Rev. Lett. 54, (1985), 2026--2029. doi:10.1103/PhysRevLett.54.2026 http://dx.doi.org/10.1103/PhysRevLett.54.2026

    1985 doi

  60. [68]

    B. Virag . The heat and the landscape I . arXiv preprint (2020). arXiv:2008.07241 http://arxiv.org/abs/2008.07241

    2020 arXiv

  61. [69]

    J. B. Walsh . An introduction to stochastic partial differential equations. \'E cole d' \'e t \'e de probabilit \'e s de Saint - Flour XIV - 1984, Lect . Notes Math . 1180, 265-437 (1986)., 1986

    1986

  62. [70]

    Wolf, Dietrich

    E. Wolf, Dietrich . Kinetic roughening of vicinal surfaces. Phys. Rev. Lett. 67, (1991), 1783--1786

    1991

  63. [71]

    H.-T. Yau . ( t)^ 2/3 law of the two dimensional asymmetric simple exclusion process. Ann. of Math. (2) 159, no. 1, (2004), 377--405. doi:10.4007/annals.2004.159.377 http://dx.doi.org/10.4007/annals.2004.159.377

    2004 doi

  64. [72]

    Yang and Z

    H. Yang and Z. Yang . Weak coupling limit of a B rownian particle in the curl of the 2D GFF . arXiv preprint arXiv:2405.05778 (2024)

    2024 arXiv

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