Matching Large Deviation Bounds of the Zero-Range Process in the whole space

Benjamin Fehrman; Benjamin Gess; Daniel Heydecker

REVIEW 3 major objections 4 minor 16 references

Matching Large Deviation Bounds of the Zero-Range Process in the whole space

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The zero-range process on Z^d satisfies a full large deviation principle in the whole space for every dimension d≥1, with rate function given by relative entropy plus an optimal L2 control, under only the canonical jump-rate hypotheses…

desk verdict Strong paper with a genuinely new probabilistic technique and a real gap in the infinite-volume uniqueness proof; the main theorem is conditional on a fix. read the letter →

arxiv 2507.23452 v1 pith:CDNUFUU3 submitted 2025-07-31 math.PR math.AP

Benjamin Fehrman , Benjamin Gess , Daniel Heydecker This is my paper · ORCID

classification math.PRmath.AP MSC 60F1060K3582C2235K55

keywords zero-rangeprocesslargedeviationshydrodynamiclimitskeletonequationrenormalisedkineticsolutionssuperexponentialestimateinfinitevolumeentropydissipation

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 aims to close the long-standing gap between the upper and lower large-deviation bounds for the hydrodynamic rescaling of the zero-range process on Z^d, for every d≥1 and in the whole space. The authors claim that, under the canonical jump-rate hypotheses (A1)–(A2), the large deviation rate function can be taken to be the relative entropy of the initial state plus half the squared L2 norm of the optimal control driving the skeleton equation. To get there, they prove that the superexponential estimate holds in arbitrary dimension and infinite volume, and that the skeleton equation has a well-posed theory on the whole space for initial data with finite relative entropy but infinite mass. A sympathetic reader would care because matching bounds are exactly what turns a variational candidate into a genuine large deviation principle, and the proof removes the convexity/concavity restrictions that limited earlier treatments.

What carries the argument

The load-bearing object is the skeleton equation ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g), together with its kinetic formulation, in which the kinetic function χ(ρ,ξ)=1_{0<ξ<ρ} satisfies a transport equation with a nonnegative parabolic defect measure p satisfying δ0(ξ−ρ)φ′(ρ)|∇ρ|² ≤ p. Uniqueness for this equation is proved in the class of renormalised kinetic solutions by a relative-entropy/kinetic comparison that cuts off large velocity and distant space, using interpolation and Sobolev estimates to control the spatial cutoffs at infinity. The probabilistic half of the argument is carried by a new restriction lemma (Lemma 3.5) that limits the superexponential variational problem to densities whose Dirichlet-form contributions are uniformly typical, dx,y(f) ≤ $zN^{{-2}}$, with an error vanishing as z→∞; this substitutes for the translation averaging that previously forced compact geometry. A 'defective concavity' estimate, ($φ^{{1/2}}$(u))^ε ≤ ϑ $φ^{{1/2}}$(u^ε), replaces the global convexity or concavity of $φ^{{1/2}}$ and makes the weak-to-kinetic passage work for zero-range nonlinearities.

What would settle it

Compute, for a jump rate satisfying (A1)–(A2), two renormalised kinetic solutions of ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g) on R^d with the same initial datum ρ0 ∈ Ent_{φ,γ} and the same control g, and show their L1 distance on a bounded ball is positive at some time; Theorem 6.5 asserts this distance is zero. A more targeted check is whether the terms in (6.22) involving Δφ_R and |∇φ_R| fail to vanish as R→∞ for some admissible φ; the equality of rate functions in Theorem 8.3 depends on that vanishing.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: if the local jump rate λ satisfies (A1)–(A2) and the zero-range process starts from the invariant measure ν_γ, then for every open U, liminf $N^{{-d}}$ log P(π^N ∈ U) ≥ -inf_{π∈U} I(π), and for every closed A, limsup $N^{{-d}}$ log P(π^N ∈ A) ≤ -inf_{π∈A} I(π). The rate function I assigns to a trajectory the relative entropy H_Φ(π0|γ) plus 1/2 of the infimum of ||g||²_{L²} over controls for which ρ solves the skeleton equation ∂tρ = Δφ(ρ) − ∇·($φ^{{1/2}}$(ρ)g). This equality of upper and lower rate functions is obtained by proving that the weak and renormalised kinetic solution theories of the skeleton equation coincide on the whole space, so that the lower bound can be extended from smooth fluctuations to all trajectories, and by removing the global convexity/concavity assumptions on $φ^{{1/2}}$ through a defective-concavity estimate.

Load-bearing premise

The load-bearing premise is that the spatial cutoffs at infinity in the uniqueness proof of the skeleton equation vanish using only local entropy dissipation, the L2 control g, and interpolation, even though the initial data have infinite total mass.

Editorial extensions

If this is right

The full large deviation principle for the zero-range process in the whole space holds in every dimension d≥1, with the explicit rate function I of Definition 1.1.
The superexponential estimate is now available in arbitrary dimension and infinite volume, so the two-block estimate no longer requires the one-dimensional Sobolev embedding or a compact torus.
The skeleton equation admits unique renormalised kinetic solutions for initial data with finite relative entropy but infinite total mass, a class forced by the infinite-volume equilibrium.
Weak solutions and renormalised kinetic solutions coincide under hypotheses (A1)–(A2), so the upper-bound rate function, the lower-bound rate function, and the variational rate function are all equal.
No separate global convexity or concavity condition on φ^{1/2} is needed; the mild spectral-gap and Lipschitz conditions on the jump rate suffice.

Reading between the lines

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

Editorial inference: the defective-concavity estimate is not tied to zero-range structure and could plausibly be carried over to other parabolic-hyperbolic equations whose diffusivity is only locally elliptic.
Editorial inference: Lemma 3.5's typicality cutoff suggests a general recipe for infinite-volume interacting particle systems: replace global averaging over a transitive group by local averaging across the scale of the test function; one could test this on symmetric exclusion or Kac models.
Editorial inference: because the rate-function equality is proved through uniqueness of kinetic solutions, any future model where weak and kinetic solutions diverge would automatically exhibit a mismatch between upper and lower large-deviation bounds, along the lines of earlier counterexamples.
Editorial inference: in dimensions d≥3 the oscillating-profile example in Proposition 3.4 indicates that the two-block estimate genuinely fails without extra regularity; a quantitative version of that failure could inform whether higher-order corrections appear at the level of prefactors.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper makes a serious run at a longstanding open problem and introduces tools that are worth keeping. The probabilistic side is the most convincing part: Lemma 3.5 is a clever replacement for the lost torus averaging in the two-block estimate, and Proposition 3.4 nicely explains why the standard path argument must fail in d≥2. The defective concavity lemma (Lemma 7.4) is also new and plausibly removes the convexity/concavity assumptions that blocked earlier work. If the analytic machinery holds up, Theorem 1.2 would be a major result.

The soft spot is Theorem 6.5. The proof tracks a localized difference with cutoffs in space and in the kinetic variable, and it shows, after letting ε,δ→0, that the time derivative of this localized quantity is bounded by terms that vanish as the cutoffs are removed. What is missing is a legitimate passage to the limit R→∞, M→∞. The initial data live in Ent_{Φ,γ}, so they need not be globally L1, and the final line claiming max_t ∥ρ1−ρ2∥_{L1(Rd)}=0 is not even meaningful for that class. More concretely, the derivative bounds do not give a uniform-in-R (or later in-M) control that would justify integrating the differential inequality and then taking the cutoff limits. The phrase "monotone convergence" is doing more work than the estimates support. That is a load-bearing gap because Theorem 8.3 inherits it: without genuine uniqueness of renormalised kinetic solutions, the rate-function equality does not follow.

I do not think this is a cosmetic error. But it may be repairable. A local L1 uniqueness statement, or a weighted-L1 contraction, would likely be enough for the applications in Section 8, and the existing estimates (Proposition 5.5, local entropy bounds) could be reorganized to yield it. There are also small typos—Theorem 3.3 sends ϵ→∞ instead of ϵ→0—that should be swept up.

Bottom line: the paper deserves serious refereeing, not desk rejection. The probabilistic contributions and the PDE framework are substantial enough that even a conditional acceptance with a request for a repaired uniqueness proof is reasonable. I would want to see the gap closed before relying on Theorem 1.2.

Referee Report

3 major / 4 minor

Summary. The paper proves matching large-deviation upper and lower bounds for the hydrodynamic rescaling of the zero-range process on the whole space Z^d, in every dimension d≥1, under the canonical hypotheses (A1)-(A2) on the jump rate. The proof has two main parts. The first is probabilistic: a superexponential estimate (Theorem 3.3) adapted to infinite volume and d≥2, based on a new local-averaging restriction (Lemma 3.5) that replaces the unavailable global averaging used on the torus. The second is analytic: an extension of the skeleton equation theory of [FG23] to initial data with finite relative entropy with respect to a constant density, including well-posedness of renormalised kinetic solutions (Theorem 6.5), equivalence of weak and kinetic solutions under a novel 'defective concavity' estimate (Lemma 7.4), and equality of the variational rate functions I=I^lo=I^up (Theorem 8.3). Theorem 1.2 is then deduced from the partial large deviation principle of Theorem 3.1 and the rate-function identification.

Significance. If the central claims hold, this resolves a long-standing open problem from [KL99] by giving matching bounds in the whole space without global convexity/concavity assumptions on the nonlinearity. The probabilistic contribution is substantial: the two-block estimate in arbitrary dimension is genuinely new in infinite volume, and the local-typicality Lemma 3.5 is an elegant replacement for the torus averaging argument. The analytic contribution is also significant: the defective-concavity inequality (7.7) and the infinite-volume renormalised-solution theory extend [FG23] in a non-obvious way. The paper is detailed, self-contained in its main estimates, and free of fitted parameters. However, the proof of the central uniqueness theorem contains a gap in the passage to the global L^1 limit, and the statement of Theorem 3.3 contains a limit-direction typo. These issues are local and likely repairable, but they currently affect the identification of the rate functions.

major comments (3)

[Section 6.2, proof of Theorem 6.5] The conclusion 'max_{t∈[0,T]} ∥ρ1(·,t)-ρ2(·,t)∥_{L^1(R^d)}=0' is not justified for initial data in Ent_{Φ,γ}(R^d), since such data need not lie in L^1(R^d). The quantity I^{ε,δ,M,R}_t defined in (6.6) is only localised by φ_R and ζ_M; for fixed M the integral ∫ |χ1-χ2|^2 ζ_M can be infinite after R→∞ because the level set where both densities lie in the support of ζ_M may have infinite Lebesgue measure under only a finite relative entropy condition. The argument removes the cutoffs ε,δ, then R, then M using 'monotone convergence', but monotone convergence does not apply to a limit that is infinite and no uniform-in-R bound or integrable majorant is supplied. The derivative estimates (6.19)-(6.23) only show that cutoff errors vanish asymptotically; they do not provide the integrated inequality I^{M,R}(t) ≤ I^{M,R}(0)+∫_0^t error(R,s) ds from which one could let R→∞ and obtain local uniqueness, e.g. ∫_{B_K}|ρ1-ρ2|=0 for each K. As written, uniqueness of renormalised kinetic solutions is therefore not established, and Theorem 8.3, which uses Theorem 6.5 to identify the weak limit in Step 3, inherits this gap. The proof should be repaired by proving a localised uniqueness statement and then using it in Theorem 8.3; for the rate-function equality, local L^1_loc uniqueness is sufficient.
[Theorem 3.3, Eq. (3.10)] The limit in (3.10) is written as lim sup_{ϵ→∞}, but the averaging scale ϵ is macroscopic and should go to 0; this is consistent with (3.13), with the proof's discussion of ℓ,ε,N→∞, and with the use of ε→0 in Lemma 4.2. As printed, the statement is the wrong direction for the cutoff.
[Theorem 8.3, Step 3] The proof asserts that the minimising control g in the definition of I(ρ) is unique. This is not proved and does not obviously follow from convexity of the admissible set without a closedness argument. The proof actually only needs that the weak limit g̃ is a minimizer, not that it equals the original g, so the uniqueness assertion is unnecessary; it should be removed or replaced by a direct argument that ∥g̃∥_{L^2}=∥g∥_{L^2} makes g̃ admissible and hence a minimizer.

minor comments (4)

[Section 8, Step 1] The initial data ρ0,n,R are defined using a spatial cutoff φ_R, but the displayed formula writes 'ρ0,n' without the R dependence; the notation should be consistent throughout Step 1 and Step 2.
[Section 8, Step 3] In the line 'I(ρn,R) = 1/2 ∫ Φ(ρn)|∇Hn,R|^2', the argument of Φ should be ρn,R, not ρn.
[Section 6.2, Eq. (6.22)] The text says 'as in the proof of Theorem 6.5 we first pass to the limit R→∞' inside the proof of Theorem 6.5 itself; this self-reference should be corrected.
[Definition 6.1] The kinetic measure p is defined on R^d × (0,∞) × [0,T], but the equation in item (3) integrates p over ξ∈R; the domain of ξ-integration should be stated consistently, with p extended by zero or restricted to positive ξ.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the LDP is derived from in-paper analytic estimates; reliance on [FG23] is on published independent results, with the infinite-volume and non-convex extensions proven here.

full rationale

The derivation chain of Theorem 1.2 is not circular. The partial large deviation bounds in Theorem 3.1 are proven from the microscopic process via the superexponential estimate, entropy dissipation control, and exponential tightness, with no fitted parameters. The rate-function identification in Theorem 8.3 is an analytic statement: I is defined as an infimum over controls g and is shown to equal the lower semicontinuous envelope of I restricted to smooth fluctuations, using the well-posedness and comparison theory developed in Sections 5-8. That theory is largely proven in this paper: existence via Galerkin approximation, uniqueness of renormalised kinetic solutions via a relative-entropy comparison, and weak-kinetic equivalence via the new 'defective concavity' Lemma 7.4 derived directly from the ellipticity condition (7.1). Where the paper cites the authors' earlier work [FG23], e.g., in Proposition 5.6 and equation (8.4), it cites published, peer-reviewed results whose assumptions do not include the present target theorem, and it explicitly extends them to infinite volume and to general nonlinearities without global convexity. The skeptical concern about Theorem 6.5 is a correctness gap about the legitimacy of removing spatial cutoffs for non-L1 initial data, not a circularity: even if that limit fails, the failure is an unjustified analytic passage, not an assumption of the conclusion. No step could be exhibited in which a prediction reduces by construction to a fitted input, a definition, or a self-citation chain. The paper is accordingly assigned a low score reflecting only minor reliance on the authors' prior work, which is not load-bearing in a circular sense.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no fitted constants and no new physical entities. It relies on the canonical assumptions (A1)-(A2) on the jump rate, the derived uniform ellipticity (2.4), and the defective concavity lemma as a replacement for convexity. The kinetic defect measure p is a standard technical device, not an invented entity.

assumptions (4)

domain assumption Jump rate lambda satisfies (A1) Lipschitz growth with lambda(0)=0 and (A2) spectral gap and nondegeneracy, equations (1.3)-(1.4).
Canonical hypotheses on the microscopic model, stated in Section 1, from which ellipticity (2.4) and other properties of phi are derived.
domain assumption Uniform ellipticity of phi: a <= phi'(xi) <= A, equation (2.4).
Folklore consequence of (A2) cited at (2.4); used throughout Sections 5-8 and to obtain defective concavity.
ad hoc to paper Defective concavity bound (Phi^{1/2}(u))^epsilon(x) <= theta Phi^{1/2}(u^epsilon(x)) in Lemma 7.4, with theta = sqrt(A/a).
This is the paper's replacement for the convexity or concavity of Phi^{1/2} in [FG23]; it is proven from (2.4), but the whole matching result rests on it.
ad hoc to paper Existence of renormalised kinetic solutions with a kinetic defect measure p satisfying vanishing at infinity, Definition 6.1 and Proposition 7.6.
The extension to the Ent space with cutoff errors at infinity is the fragile step; if the defect measure fails to vanish at infinity, uniqueness and rate equivalence break.

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Pith. "Pith review of Matching Large Deviation Bounds of the Zero-Range Process in the whole space." pith.science (2026). https://pith.science/paper/CDNUFUU3

@misc{pith2026250723452,
  author       = {Pith},
  title        = {Pith review of: Matching Large Deviation Bounds of the Zero-Range Process in the whole space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CDNUFUU3}},
  note         = {Machine review of arXiv:2507.23452}
}

abstract

We consider the large deviations of the hydrodynamic rescaling of the zero-range process on $\mathbb{Z}^d$ in any dimension $d\ge 1$. Under mild and canonical hypotheses on the local jump rate, we obtain matching upper and lower bounds, thus resolving the problem opened by \cite{KL99}. On the probabilistic side, we extend the superexponential estimate to any dimension, and prove the superexponential concentration on paths with finite entropy dissipation. In addition, we extend the theory of the parabolic-hyperbolic skeleton equation to the whole space, and remove global convexity/concavity assumptions on the nonlinearity.

Figures

Figures reproduced from arXiv: 2507.23452 by the authors.

**Figure 1.** The geometric nature of the obstruction. Edges {x, y} are highlighted in red, respectively orange, for which dx,y(f) lies in the 1% percentile, respectively top 10%, for f similar to the examples constructed in the proof of Proposition 3.4. No path between x = (0, 0) and y = (20, 0) can avoid the red edges, nor a significant number of orange edges. The novel argument we give for (3.13) is as follows. In infinite vol… view at source ↗

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Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

J.-P. Aubin. Un th\'eor\`eme de compacit\'e. C. R. Acad. Sci. Paris , 256:5042--5044, 1963

work page 1963

[2] [2]

Large deviations for the occupation times of independent particle systems

O Benois. Large deviations for the occupation times of independent particle systems. The Annals of Applied Probability , 6(1):269--296, 1996

work page 1996

[3] [3]

Benois, C

O. Benois, C. Kipnis, and C. Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic Processes and their Applications , 55(1):65--89, January 1995

work page 1995

[4] [4]

Benois, C

O. Benois, C. Kipnis, and C. Landim. Large deviations from the hydrodynamical limit of mean zero asymmetric zero range processes. Stochastic processes and their applications , 55(1):65--89, 1995

work page 1995

[5] [5]

Chen and B

G.-Q. Chen and B. Perthame. Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. Annales de l'Institut Henri Poincar \'e . Analyse Non Lin \'e aire , 20(4):645--668, 2003

work page 2003

[6] [6]

Fehrman and B

B. Fehrman and B. Gess. Non-equilibrium large deviations and parabolic-hyperbolic PDE with irregular drift. Invent. Math. , 234(2):573--636, 2023

work page 2023

[7] [7]

Fehrman and B

B. Fehrman and B. Gess. Well-posedness of the D ean-- K awasaki and the nonlinear D awson-- W atanabe equation with correlated noise. Arch. Rational. Mech. Anal. , 248(20), 2024

work page 2024

[8] [8]

The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion, 2023

Benjamin Gess and Daniel Heydecker. The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion, 2023

work page 2023