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Dynamical interface above a hard wall and reflected SPDE on the half-line

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Pith's one-line read This paper proves that stationary fluctuations of a hard-wall corner-flip interface, after diffusive rescaling, converge in law to the reflected stochastic heat equation on the half-line, with the 3D Bessel law as an invariant measure.

desk verdict Half-line hard-wall interface convergence to reflected SHE is novel and essentially correct; two proof details need fixing before acceptance. read the letter →

arxiv 2509.03328 v1 pith:72HTHJKP submitted 2025-09-03 math.PR

Pierre Faugère , Cyril Labbé This is my paper · ORCID
classification math.PR MSC 60H1560K3560F17
keywords reflectedstochasticheatequationhardwallcornerflipdynamicsrandominterface3-dimensionalBesselprocessinvariantmeasurescalinglimithalf-line
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

Consider a height function on the nonnegative integers, starting at zero, whose neighboring heights always differ by one, and which flips local corners as long as the flip does not push it below zero, with an extra pinning at the origin. This paper proves that, starting from the stationary law of this process and rescaling space by $\epsilon$, time by $\epsilon^2$, and heights by $\sqrt{\epsilon}$, the interface fluctuations converge in law to the solution of the reflected stochastic heat equation on the half-line. The convergence is joint: the discrete noise and the discrete reflection term produced by the wall also converge, to space-time white noise and to the continuum reflection measure. As a corollary, the law of the 3-dimensional Bessel process starting from zero is invariant for the limiting SPDE. The result matters because it establishes a dynamical invariance principle through the wall constraint: the discrete exclusion effect becomes a reflection measure that pushes the continuum solution upward exactly when it touches zero.

What carries the argument

The load-bearing identity is the rescaled semimartingale decomposition, which rewrites the discrete dynamics as $\langle h^\epsilon_t, \varphi \rangle_\epsilon = \langle h^\epsilon_0, \varphi \rangle_\epsilon + \int_0^t \epsilon^{-2} \langle \Delta^\epsilon h^\epsilon_s, \varphi \rangle_\epsilon\, ds + \sqrt{2}\, W^\epsilon_t(\varphi) + \int \varphi\, d\eta^\epsilon$, with the discrete noise $W^\epsilon$ and discrete reflection measure $\eta^\epsilon$ defined explicitly from the corner-flip rates. This is the exact discrete mirror of the weak form of the reflected SPDE. Around this identity, three tools carry the proof: static moment estimates for the conditioned random walk that gives the invariant measure $\pi$, whose scaling limit is the 3-dimensional Bessel process; the Lyons-Zheng decomposition, which uses reversibility under $\pi$ to reduce time-increment bounds on the Fourier transform of $h^\epsilon$ to martingale moment estimates; and martingale convergence criteria applied to $W^\epsilon$, whose bracket process is shown to converge to $t\,\|\varphi\|^2_{L^2}$.

What would settle it

A decisive check is to apply the generator to that infinite weighted sum explicitly and see whether the result is integrable under the stationary measure; if it diverges for some frequency, the time-increment bound breaks and the tightness proof collapses. A numerical surrogate: simulate the stationary interface at small $\epsilon$ and estimate the worst-case Fourier increment in time; any growth faster than $(t-s)^{1/2}+\epsilon^{3/4}$ contradicts the paper's core estimate.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: for an interface $h$ started from its stationary measure $\pi$ and then rescaled, the triple $(h^\epsilon, W^\epsilon, \eta^\epsilon)$ converges in law as $\epsilon \to 0$ to $(u, W, \eta)$, where $W$ is a cylindrical Wiener process and $(u, \eta)$ is the solution of the reflected stochastic heat equation $\partial_t u = \partial^2_{xx} u + \sqrt{2}\,\dot W + \eta$ on $[0,\infty)$, with Dirichlet condition $u(t,0)=0$, $u \ge 0$, $\eta \ge 0$, and $\int u\, d\eta = 0$, starting from an independent 3-dimensional Bessel-distributed initial condition. The proof shows that any limit point of the rescaled semimartingale equation satisfies the weak formulation of this SPDE, and then invokes strong uniqueness for the continuum equation to identify the limit. Corollary 1.3 states that the law of the 3-dimensional Bessel process starting from zero is invariant for the reflected SPDE.

Load-bearing premise

The proof's most delicate premise is that a standard calculus identity (Dynkin's formula) can be applied to an infinite weighted sum of the interface heights, with no approximation argument showing that the sum lies in the domain of the generator; if that identity fails, the central bound on time increments of the interface, and hence the compactness of the rescaled interfaces, loses its main support.

Editorial extensions

If this is right

  • If the theorem is right, the reflected stochastic heat equation on the half-line is the exact scaling limit of the stationary corner-flip interface with a hard wall and pinning at the origin.
  • The joint convergence gives a continuum description of the wall: the reflection measure $\eta$ is supported on the zero set of $u$, so in the limit the interface touches zero only at exceptional times but is still pushed upward by that measure.
  • Corollary 1.3 provides an explicit invariant measure: the Bessel-3 law is stationary for the reflected SPDE, giving a concrete starting point for studying long-time behavior and correlations.
  • The convergence of the discrete noise identifies the noise strength in the limiting SPDE, fixing the coefficient $\sqrt{2}$ in the equation as the correct fluctuation scale.

Reading between the lines

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

  • A natural extension the authors leave implicit: varying the pinning strength or adding a slope at infinity should produce the same reflected SPDE with different boundary conditions, and the invariant measure should arise from a different conditioning transform of the simple random walk.
  • The explicit joint convergence of noise and reflection measure suggests that occupation-time and current fluctuations in the discrete model could be studied through the continuum pair $(u, \eta)$, with $\eta$ acting as a local time at the wall.
  • One could test universality numerically by running the stationary dynamics at small $\epsilon$ and checking that the one-time spatial law matches the Bessel-3 law while the reflection measure sits on rare zero-set times; if another discrete interface model with the same constraint converges to the same SPDE, that would corroborate a universality conjecture the paper does not state.
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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 / 4 minor

Summary. The paper studies the stationary fluctuations of a corner-flip interface model on the half-line lattice N above a hard wall, with pinning at the origin. Under diffusive scaling, it claims joint convergence in law of the rescaled interface h^ε, the discrete noise W^ε, and the discrete reflection measure η^ε to (u, W, η), where W is a cylindrical Wiener process and (u, η) is the solution of the reflected stochastic heat equation on [0,∞) with Dirichlet condition at 0, starting from a 3-dimensional Bessel initial condition independent of W. Corollary 1.3 concludes that the 3D Bessel law is invariant for this SPDE. The strategy is modular: Section 3 proves tightness of the interface via weighted Sobolev/Hölder estimates and a Lyons–Zheng decomposition; Section 4 proves convergence of the discrete noise to white noise via martingale brackets and Rebolledo’s theorem; Section 5 proves tightness of the reflection measure from the semi-discrete PDE; Section 6 identifies any joint limit point as the unique strong solution of the SPDE. The proof relies on an external invariance principle for the conditioned random walk and on external strong uniqueness for the reflected SPDE.

Significance. The result, if completed, is significant: it provides the first scaling limit of a discrete interface above a hard wall on an infinite spatial domain, connecting it to a Nualart–Pardoux type reflected stochastic heat equation on the half-line, and it gives a dynamical proof of invariance of the 3-dimensional Bessel law. The paper is well structured and contains several reusable quantitative estimates (Lemmas 2.4, 3.3, 3.5–3.8, and the bracket computations in Section 4). The identification of limit points is not circular: it uses external strong uniqueness and an external invariance principle, and the self-reference to prior work is limited to a technical segment result. The main concerns are localized technical gaps in the tightness proof of the interface (Section 3.2), both of which appear repairable; they do not undermine the overall strategy.

major comments (2)
  1. [Section 3.2, Lemma 3.5, Eqs. (3.8)–(3.9)] Dynkin's formula is applied to the non-cylindrical functional f_ζ(h) = ⟨h, e^{-ρx - iζx}⟩ on the infinite lattice, and to its backward-time counterpart, without showing that f_ζ belongs to the domain of the generator of the Feller process or supplying a cylindrical approximation argument. The generator L was defined only on cylindrical functions, and for this f_ζ one must justify passing from finite sums over sites to the infinite sum defining Lf_ζ and the associated martingale. This is not a cosmetic point: the bound (3.7) on Fourier increments, and hence Lemma 3.6, Proposition 3.4, Lemma 3.7, Theorem 3.1, and ultimately the tightness of η^ε via (5.4), all rest on this step. The gap is likely repairable: truncating f_ζ to finitely many sites gives a cylindrical function, and the estimates (3.10)–(3.11) are uniform in the truncation, so a dominated-convergence or uniform-tightness argument should close it; the manuscript should provide that argument explicitly.
  2. [Section 3.2, proof of Proposition 3.4] The displayed moment chain E∏_{j=1}^p |A_j|^2 ≤ ∏_{j=1}^p (E|A_j|^{2j+1})^{1/(2j)} ≤ ∏_{j=1}^p (c_{2j+1}(t-s)^{3/8})^2 is not justified. The first inequality is not a valid Hölder or Cauchy-Schwarz estimate (it already fails in general for p=2), and the second does not follow from Lemma 3.6: Lemma 3.6 controls the L^{2j+1} norm, i.e. (E|A|^{2j+1})^{1/(2j+1)}, not the displayed 1/(2j)-th power. A correct route is to apply Hölder with a common exponent p, obtaining (E|A_j|^{2p})^{1/p}, and then use Lemma 3.6 with m=2p; the claimed (t-s)^{3p/4} bound can be recovered that way. As written, however, the proof of the crucial estimate (3.6) is invalid, and (3.6) is used in Lemma 3.7 and Theorem 3.1.
minor comments (4)
  1. [Equations (1.8) and (4.1)] The displayed normalization of W^ε appears inconsistent: equation (1.9) and the bracket computations in Section 4 correspond to a coefficient ε/√2, whereas the displayed equations suggest ε√2. Please check and correct the displayed coefficient if it is not a typesetting artifact.
  2. [Section 6, proof of item (v), after (6.5)] The statement that h^ε is equal to √ε on the support of η^ε is not correct in general: a site with h^ε=0 and Δ^ε h^ε = -2√ε satisfies the reflection condition, so h^ε can vanish on the support. The inequality h^ε ≤ √ε on the support is sufficient for the argument, since it still gives F(h^ε,η^ε) ≤ √ε ∫ xψ dη^ε.
  3. [Section 4, proof of Theorem 4.1] In the sentence 'for all φ∈S′([0,∞))' the space should be S([0,∞)) (Schwartz functions), not the space of distributions S′; the same correction applies in the surrounding argument where φ is used as a test function.
  4. [Section 3.2, Lemma 3.5] The proof applies Dynkin's formula to a complex-valued function without comment; this is harmless after splitting into real and imaginary parts, but a brief remark would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the limit identification is built on external uniqueness and invariance results, not on the paper's own conclusions.

full rationale

The paper's central derivation is self-contained against external inputs. The stationary law of the discrete interface is shown to converge to the 3D Bessel law using the functional limit theorem of Bryn-Jones and Doney [3], which is an external, parameter-free result. The reflected SPDE limit is identified by checking Definition 1.1 and invoking strong uniqueness from Hambly and Kalsi [8], again external to the authors. The tightness argument uses the Lyons–Zheng decomposition [15] and martingale estimates; the only self-citation, Etheridge and Labbé [6], is used for a double-BDG technique and not for the target convergence. Corollary 1.3 follows from the established convergence together with stationarity of h^epsilon, not by assuming the invariant measure. The concern raised about Lemma 3.5 (application of Dynkin's formula to a non-cylindrical functional) is a possible rigor gap in the proof, but it is not a case of a conclusion being equivalent to its inputs by construction; it therefore does not constitute circularity under the criteria.

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

No constants are fitted and no new entities are introduced. The central claim rests on standard references for Markov generator theory, the conditioned random walk, the invariance principle, and strong uniqueness of the reflected SPDE on the half-line.

assumptions (6)
  • domain assumption Strong existence and uniqueness of the reflected stochastic heat equation (1.4) on the half-line (Hambly-Kalsi [8, Theorem 2.6]).
    Invoked in the proof of Theorem 1.2 to conclude that every subsequential limit (u, eta) equals the unique solution, thereby identifying the law of the limit point (Section 1.2 and Section 6).
  • domain assumption Functional central limit theorem for a random walk conditioned to stay nonnegative, giving convergence of pi^epsilon to the law of the 3-dimensional Bessel process (Bryn-Jones-Doney [3, Theorem 2.1]).
    Used for tightness of the one-time marginals in Section 3.5 and to identify the initial law in Theorem 1.2 and Corollary 1.3.
  • domain assumption Transitions (2.4) and moment bounds for the conditioned random walk (Bertoin-Doney [2], Lamperti [13]).
    Provides the quantitative estimates on the stationary measure pi used throughout Sections 2.3, 3.1, and 4.1.
  • standard math The flip dynamics on the infinite lattice is well defined as a Feller process via the Markov pregenerator L and its closure (Liggett [14, Theorem I, 3.9 and 5.2]).
    Used in Section 2.1 to construct the process and to justify the martingale problem and reversibility.
  • standard math Interpolation between Sobolev spaces and the embedding W^{delta,r} into C^b (Triebel [20, p. 182]).
    Used in Lemma 3.7 to convert W^{s1,r1} and H^{-s0} moment bounds into Holder norm estimates.
  • standard math Tightness criteria for D([0,infinity), C) and Mitoma's criterion for D([0,infinity), S') (Kallenberg [11], Mitoma [16]).
    Used in Sections 3.5 and 4.3 to reduce tightness of the interface and of the noise to moment estimates and real-valued projections.

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Pith. "Pith review of Dynamical interface above a hard wall and reflected SPDE on the half-line." pith.science (2026). https://pith.science/paper/72HTHJKP

@misc{pith2026250903328,
  author       = {Pith},
  title        = {Pith review of: Dynamical interface above a hard wall and reflected SPDE on the half-line},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/72HTHJKP}},
  note         = {Machine review of arXiv:2509.03328}
}
abstract

We consider a dynamical random interface on the infinite lattice $\mathbb{N}$ evolving according to a "corner flip" dynamic above a hard wall, with an additional pinning at the origin. We study the stationary fluctuations under a diffusive scaling and prove convergence in law towards the solution of an SPDE of Nualart-Pardoux's type, namely the Reflected Stochastic Heat Equation on the half-line. We also obtain that the law of the 3-dimensional Bessel process is an invariant measure for this SPDE.

Figures

Figures reproduced from arXiv: 2509.03328 by the authors.

Figure 1
Figure 1. Graphical representation of the jump rates for the discrete dy￾namic. To each is site is associated a random Poisson clock of parameter one, independent from those of the other sites. Every time a clock rings, if the cor￾responding site forms a corner, we flip it, except if the flipped interface takes negative values. Here, non-crossed arrows represent possible transitions with their associated rate, while the cross… view at source ↗

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