pith. sign in

arxiv: 2606.07353 · v1 · pith:5VRLMIUEnew · submitted 2026-06-05 · 🧮 math.PR

The Gaussian structure of a perturbed KPZ

Yu Gu , Tomasz Komorowski This is my paper

Pith reviewed 2026-06-27 21:07 UTC · model grok-4.3

classification 🧮 math.PR
keywords KPZ equationinvariant measureBrownian bridgerelative entropyRadon-Nikodym derivativestochastic PDEwhite noise
0
0 comments X

The pith

The KPZ equation with small spatial perturbation V has a unique invariant measure absolutely continuous to the Brownian bridge.

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

The paper examines the KPZ equation on a circle with an added smooth spatial function V. It shows that when the integral of V squared is small enough, there exists a unique invariant measure absolutely continuous with respect to the Brownian bridge. This measure has finite relative entropy to the law of the bridge, and the Radon-Nikodym derivative belongs to every L^p space for p greater than 1. The argument extends a known discretization and mollification scheme and applies log-Sobolev and spectral gap inequalities that hold for the underlying Gaussian measure.

Core claim

The KPZ equation on the circle with additive perturbation V admits a unique invariant measure that is absolutely continuous with respect to the Brownian bridge; the measure has finite relative entropy with respect to the bridge law, and its Radon-Nikodym derivative lies in L^p for every p in (1, infinity), provided the integral of V squared is sufficiently small.

What carries the argument

Discretization and mollification scheme of the perturbed equation, combined with log-Sobolev and spectral gap inequalities for the underlying Gaussian measure.

If this is right

  • The long-time distribution of solutions converges to this unique perturbed invariant measure.
  • Observables under the invariant measure admit moment bounds from the L^p membership of the density.
  • The relative entropy finiteness quantifies how close the perturbed stationary law remains to the unperturbed Brownian bridge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scheme might apply to other additive perturbations if the functional inequalities continue to hold after mollification.
  • For large perturbations the absolute continuity could break, suggesting a phase transition in the structure of the invariant measure.
  • The result supplies a concrete reference measure that could be used to study fluctuations or large deviations in the perturbed growth model.

Load-bearing premise

The discretization and mollification scheme extends directly to the perturbed equation while preserving the applicability of log-Sobolev and spectral gap inequalities for the Gaussian measure.

What would settle it

An explicit V with small integral of V squared for which either no unique invariant measure exists that is absolutely continuous to the Brownian bridge, or the Radon-Nikodym derivative fails to belong to some L^p.

read the original abstract

We study the KPZ equation on a circle with an additive spatial perturbation $\partial_t h=\tfrac12\Delta h+\tfrac12|\nabla h|^2+\xi+ V$, where $\xi$ is a spacetime white noise and $V$ is a smooth spatial function. When $V=0$, it is well-known that the unique invariant measure is the Brownian bridge. In the presence of the perturbation, we show that the equation admits a unique invariant measure that is absolutely continuous with respect to the Brownian bridge. We further prove the measure has a finite relative entropy with respect to the law of the bridge and that, for any $p\in(1,\infty)$, the corresponding Radon-Nikodym derivative belongs to $L^p$, provided that $\int V^2$ is sufficiently small. The proof uses the discretization and mollification scheme of \cite{FQ}, together with an application of the log-Sobolev and spectral gap inequalities for the underlying Gaussian measure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the KPZ equation on the circle with additive smooth spatial perturbation V: ∂_t h = (1/2)Δh + (1/2)|∇h|^2 + ξ + V. It claims that when ∫V² is sufficiently small, there exists a unique invariant measure μ absolutely continuous with respect to the Brownian bridge μ₀, with finite relative entropy H(μ|μ₀) and Radon-Nikodym derivative dμ/dμ₀ belonging to L^p(μ₀) for all p ∈ (1,∞). The argument extends the discretization/mollification scheme of [FQ] and invokes log-Sobolev and spectral-gap inequalities on the underlying Gaussian measure μ₀.

Significance. If the central claim holds, the result establishes stability of the Gaussian invariant measure under small deterministic perturbations, extending the well-known V=0 case. The proof architecture reuses the [FQ] scheme together with standard functional inequalities for Gaussians, which supplies quantitative control and avoids new ad-hoc constructions; this is a clear strength when the smallness condition on ∫V² is verified to close the estimates.

major comments (1)
  1. [§4] §4 (extension of the [FQ] scheme): the text must explicitly verify that the additive perturbation V, after mollification, produces error terms whose L² norms remain controlled by the smallness assumption without requiring additional renormalization; otherwise the applicability of the log-Sobolev inequality on the perturbed dynamics is not immediate.
minor comments (2)
  1. The smallness threshold on ∫V² is stated but its explicit dependence on the mollification parameter is not displayed; adding a short remark after the statement of the main theorem would clarify the range of applicability.
  2. [§2] Notation for the mollified noise and the regularized drift should be introduced once in §2 and used consistently thereafter to avoid minor confusion in the estimates of §5.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the recommendation of minor revision. We address the point on the mollification of V below.

read point-by-point responses

  1. Referee: [§4] §4 (extension of the [FQ] scheme): the text must explicitly verify that the additive perturbation V, after mollification, produces error terms whose L² norms remain controlled by the smallness assumption without requiring additional renormalization; otherwise the applicability of the log-Sobolev inequality on the perturbed dynamics is not immediate.

    Authors: We agree that an explicit verification is useful for clarity. In the extension of the [FQ] scheme in §4, the mollification is performed on the space-time white noise ξ exactly as in the unperturbed case, while the fixed smooth function V is replaced by its mollification V_ε. Because V is smooth (hence bounded and continuous), the L² difference ||V - V_ε||_{L²(𝕋)} vanishes as ε → 0, independently of the smallness parameter. The assumption that ∫V² is sufficiently small therefore directly controls ||V_ε||_{L²} by a constant multiple of the original quantity (via the triangle inequality), with no renormalization required: V enters the equation additively and does not generate singular products with the noise that would necessitate counterterms. The resulting perturbation remains small in the appropriate norms, allowing the log-Sobolev and spectral-gap inequalities on the unperturbed Gaussian μ₀ to be applied with constants independent of V. We will insert a short paragraph in §4 that records this L²-control estimate together with its dependence on the mollification scale. revision: yes

Circularity Check

0 steps flagged

No significant circularity; relies on external scheme and standard inequalities

full rationale

The derivation extends the discretization/mollification from the external reference [FQ] to the perturbed KPZ dynamics and invokes standard log-Sobolev and spectral-gap inequalities on the Gaussian Brownian bridge measure. These are independent external inputs with no reduction of the claimed invariant measure properties (absolute continuity, finite relative entropy, L^p Radon-Nikodym) to fitted parameters or self-citations within the paper. The smallness condition on ∫V² is an explicit hypothesis, not a fitted output. The central claims remain non-circular and externally benchmarked.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the proof rests on applicability of an external discretization scheme and standard Gaussian inequalities whose precise hypotheses are not restated here.

axioms (2)
  • domain assumption The discretization and mollification scheme of [FQ] applies to the perturbed KPZ equation.
    Cited as the main proof tool in the abstract.
  • standard math Log-Sobolev and spectral gap inequalities hold for the underlying Gaussian measure in the perturbed setting.
    Invoked to control the Radon-Nikodym derivative.

pith-pipeline@v0.9.1-grok · 5689 in / 1298 out tokens · 17373 ms · 2026-06-27T21:07:42.589780+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

37 extracted references · 2 linked inside Pith

  1. [1]

    Bakhtin and L

    Y . Bakhtin and L. Li,Thermodynamic limit for directed polymers and stationary solutions of the Burgers equation, Comm. Pure Appl. Math.72(2019), no. 3, 536–619. 1

    2019

  2. [2]

    Bakry, I

    D. Bakry, I. Gentil, and M. Ledoux,Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften, vol.348, Springer, Cham, 2014. 29, 30

    2014

  3. [3]

    R. Basu, S. Sarkar, and A. Sly,Invariant measures for TASEP with a slow bond, arXiv:1704.07799, 2017. 2

    Pith/arXiv arXiv 2017

  4. [4]

    R. Basu, V . Sidoravicius, and A. Sly,Last passage percolation with a defect line and the solution of the slow bond problem, arXiv:1408.3464, 2014. 2

    Pith/arXiv arXiv 2014

  5. [5]

    Bauerschmidt, B

    R. Bauerschmidt, B. Dagallier, and H. Weber,Holley–Stroock uniqueness method for theΦ 4 2 dynamics, arXiv:2504.08606, 2025. 3

    arXiv 2025

  6. [6]

    Bertini and G

    L. Bertini and G. Giacomin,Stochastic Burgers and KPZ equations from particle systems, Comm. Math. Phys.183(1997), no. 3, 571–607. 1

    1997

  7. [7]

    Balázs, J

    M. Balázs, J. Quastel, and T. Seppäläinen,Fluctuation exponent of the KPZ/stochastic Burgers equation, J. Amer. Math. Soc.24(2011), 683–708. 25

    2011

  8. [8]

    V . I. Bogachev,Weak Convergence of Measures, Mathematical Surveys and Monographs, vol.234, American Mathematical Society, Providence, RI, 2018

    2018

  9. [9]

    J. Coe, M. Hairer, and L. Tolomeo,Quasi-Gaussianity of the 2D stochastic Navier–Stokes equations, arXiv:2510.13460, 2025. 2

    arXiv 2025

  10. [10]

    Comets,Directed Polymers in Random Environments, Lecture Notes in Mathematics, vol.2175, Springer, Berlin, 2017

    F. Comets,Directed Polymers in Random Environments, Lecture Notes in Mathematics, vol.2175, Springer, Berlin, 2017. 9

    2017

  11. [11]

    P. Duch, M. Hairer, J. Yi, and W. Zhao,Ergodicity of infinite volumeΦ 4 3 at high temperature, arXiv:2508.07776, 2025. 3

    arXiv 2025

  12. [12]

    R. M. Dudley,Convergence of Baire measures, Studia Math.27(1966), 251–268

    1966

  13. [13]

    Dunlap, C

    A. Dunlap, C. Graham, and L. Ryzhik,Stationary solutions to the stochastic Burgers equation on the line, Comm. Math. Phys.382(2021), no. 2, 875–949. 1

    2021

  14. [14]

    Dunlap and E

    A. Dunlap and E. Sorensen,Viscous shock fluctuations in KPZ, Comm. Math. Phys.407(2026), no. 4, Paper No. 63. 1

    2026

  15. [15]

    Dunlap, Y

    A. Dunlap, Y . Gu, and T. Rosati,Invariant measures for the open KPZ equation: an analytic perspective, arXiv:2512.03328, 2025. 2

    arXiv 2025

  16. [16]

    Dupuis and R

    P. Dupuis and R. S. Ellis,A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997. 27

    1997

  17. [17]

    M. G. R. Flores,On the strict positivity of solutions of the stochastic heat equation, Ann. Probab.42(2014), 1635–1643. 5

    2014

  18. [18]

    S. R. Foguel,The Ergodic Theory of Markov Processes, Van Nostrand Reinhold, New York, 1969

    1969

  19. [19]

    Funaki and J

    T. Funaki and J. Quastel,KPZ equation, its renormalization and invariant measures, Stoch. Partial Differ. Equ. Anal. Comput.3(2015), no. 2, 159–220. 1, 5, 7, 14, 21, 22, 23, 24, 31

    2015

  20. [20]

    Gu and T

    Y . Gu and T. Komorowski,KPZ on torus: Gaussian fluctuation, Ann. Inst. Henri Poincaré Probab. Stat.60 (2024), no. 3, 1570–1618. 4, 7, 8, 9 34 YU GU, TOMASZ KOMOROWSKI

    2024

  21. [21]

    Gubinelli, M

    M. Gubinelli, M. Hofmanová, and N. Rana,Decay of correlations in stochastic quantization: the exponen- tial Euclidean field in two dimensions, Stoch. Partial Differ. Equ. Anal. Comput.13(2025), no. 1, 107–145. 3

    2025

  22. [22]

    Gubinelli and N

    M. Gubinelli and N. Perkowski,KPZ reloaded, Communications in Mathematical Physics 349.1 (2017): 165-269. 12

    2017

  23. [23]

    Gubinelli and N

    M. Gubinelli and N. Perkowski,Energy solutions of KPZ are unique, J. Amer. Math. Soc.31(2018), 427–

    2018

  24. [24]

    Gubinelli and N

    M. Gubinelli and N. Perkowski,The infinitesimal generator of the stochastic Burgers equation, Probab. Theory Related Fields178(2020), no. 3–4, 1067–1124. 3, 4, 12

    2020

  25. [25]

    Hairer,Solving the KPZ equation, Annals of mathematics (2013): 559-664

    M. Hairer,Solving the KPZ equation, Annals of mathematics (2013): 559-664. 12

    2013

  26. [26]

    Hairer and J

    M. Hairer and J. C. Mattingly,The strong Feller property for singular stochastic PDEs, Ann. Inst. Henri Poincaré Probab. Stat.54(2018), no. 3, 1314–1340. 1

    2018

  27. [27]

    Janjigian, F

    C. Janjigian, F. Rassoul-Agha, and T. Seppäläinen,Ergodicity and synchronization of the Kardar–Parisi– Zhang equation, arXiv:2211.06779, 2022. 1

    arXiv 2022

  28. [28]

    S. A. Janowsky and J. L. Lebowitz,Finite-size effects and shock fluctuations in the asymmetric simple- exclusion process, Phys. Rev. A45(1992), no. 2, 618–625. 2

    1992

  29. [29]

    S. A. Janowsky and J. L. Lebowitz,Exact results for the asymmetric simple exclusion process with a block- age, J. Stat. Phys.77(1994), no. 1–2, 35–51. 2

    1994

  30. [30]

    Kipnis and C

    C. Kipnis and C. Landim,Scaling Limits of Interacting Particle Systems, Grundlehren der Mathematischen Wissenschaften, vol.320, Springer, Berlin, 1999. 17

    1999

  31. [31]

    Komornik,Asymptotic periodicity of the iterates of weakly constrictive Markov operators, Tôhoku Math

    J. Komornik,Asymptotic periodicity of the iterates of weakly constrictive Markov operators, Tôhoku Math. J.38(1986), 15–27. 28

    1986

  32. [32]

    J. C. Mattingly, M. Romito, and L. Su,The Gaussian structure of the singular stochastic Burgers equation, Forum Math. Sigma10(2022), Paper No. e46. 2

    2022

  33. [33]

    J. C. Mattingly and T. M. Suidan,The small scales of the stochastic Navier–Stokes equations under rough forcing, J. Stat. Phys.118(2005), no. 1–2, 343–364. 2

    2005

  34. [34]

    J. C. Mattingly and T. M. Suidan,Transition measures for the stochastic Burgers equation, Contemp. Math. 458(2008), 409–418. 2

    2008

  35. [35]

    Mueller,On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep.37 (1991), 225–245

    C. Mueller,On the support of solutions to the heat equation with noise, Stochastics Stochastics Rep.37 (1991), 225–245. 5

    1991

  36. [36]

    Sinai,Two results concerning asymptotic behavior of solutions of the Burgers equation with force, Journal of Statistical Physics 64.1 (1991): 1-12

    Ya G. Sinai,Two results concerning asymptotic behavior of solutions of the Burgers equation with force, Journal of Statistical Physics 64.1 (1991): 1-12. 8

    1991

  37. [37]

    J. B. Walsh,An introduction to stochastic partial differential equations, inÉcole d’Été de Probabilités de Saint-Flour XIV–1984, Lecture Notes in Mathematics, vol.1180, Springer, Berlin, 1986, pp. 265–439. 5, 7 (Yu Gu) DEPARTMENT OFMATHEMATICS, UNIVERSITY OFMARYLAND, COLLEGEPARK, MD, 20742 USA (Tomasz Komorowski) INSTITUTE OFMATHEMATICS, POLISHACADEMY OFS...

    1984