Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions

Seho Park

REVIEW 2 major objections 5 minor 29 references

Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions

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Pith's one-line read The paper proves that periodic homogenization of a convex infinite-dimensional Hamilton–Jacobi equation with mean-only initial data converges at the optimal rate O(ε), with an example showing this rate is sharp.

desk verdict A solid extension of the finite-dimensional optimal-rate result to an infinite-dimensional setting, with a clear main theorem and honest proof outline; the catch is that two load-bearing estimates are imported from the author's own preprint or only sketched, so the paper is not yet fully self-contained. read the letter →

arxiv 2608.11449 v1 pith:LP5HYJYV submitted 2026-08-11 math.AP

Seho Park This is my paper · ORCID

classification math.AP MSC 35B4037J5049L2535B2735F2135R15

keywords periodichomogenizationinfinite-dimensionalHamilton-JacobiequationseffectiveHamiltonianoptimalconvergencerateactionmetricrearrangementinvarianceviscositysolutionsmeanconfiguration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that a convex, periodic, rearrangement-invariant Hamilton–Jacobi equation posed on the Hilbert space $V=L^2(I;\mathbb{R}^d)$ -- configurations of indistinguishable particles on a $d$-dimensional torus -- converges, when the initial data depend only on the mean configuration, to a finite-dimensional effective solution at the optimal rate $C\varepsilon$, uniformly in $x$ and $t$. Previously, only a nonconvex rate of $O(\varepsilon^{1/3})$ was known in this infinite-dimensional setting. The proof replaces the full infinite-dimensional comparison by a comparison of mean-restricted value functions through a mean-endpoint action metric, whose large-scale limit is identified with the effective Lagrangian via convex duality and the cell problem. A lower-bound example shows the linear rate is sharp, so the result is optimal.

What carries the argument

The central object is the mean-endpoint action metric $$m(t,a,q)=\inf\left\{\int_0^t L(\gamma(s),\dot\gamma(s))\,ds:\gamma\in AC([0,t];V),\ m(\gamma(0))=a,\ m(\gamma(t))=q\right\},$$ where $L$ is the convex dual of $H$ in the momentum variable and only the means of the endpoints are fixed. The main estimates are dyadic almost-subadditivity and almost-superadditivity: $|m(2t,2a,2q)-2m(t,a,q)|\le C_R$ for $|q-a|\le Rt$. Subadditivity is obtained by cutting, rearranging, and gluing near-minimizing curves using the compact quotient of $V$ by periodicity $\Lambda$ and particle rearrangements $G$; superadditivity uses a topological splitting lemma applied to the finite-dimensional mean-time projection $\xi(s)=(m(\gamma(s)),s)$, which splits the mean-time displacement $(2y,2t)$ into two families of pieces with displacement $(y,t)$, then lifts and reglues the pieces in $V$ with uniformly bounded connectors. These bounds make the rescaled metric converge to a homogenized metric satisfying $$\bar m(t,a,q)=t\bar L\left(\frac{q-a}{t}\right),$$ where $\bar L$ is the effective Lagrangian; identifying $\bar m$ with $t\bar L((q-a)/t)$ through convex duality and the cell problem converts the metric comparison into the linear-in-$\varepsilon$ error for the value functions.

What would settle it

For a Hamiltonian of the form $H(x,p)=\frac{1}{2}\|p\|^2+\int_I W(x(i))\,di$ with smooth $1$-periodic $W$ and $u_0\equiv0$, compute the true $\varepsilon$-dependence of $\sup_{x\in V,t\ge0}|u^\varepsilon(x,t)|$. If any such $W$ yields an error growing faster than linearly in $\varepsilon$, Theorem 1.1 is false; the paper's own example $W(r)=-4\cos^2(\pi r)$ gives the matching lower bound $\varepsilon/6$, so either a proof of the linear upper bound for all such $W$ or one counterexample with a different $W$ would settle the claim.

Watch

Extended reading notes

Core claim

Under assumptions (H1)--(H5) (periodicity, rearrangement invariance, Lipschitz continuity, coercivity, convexity in momentum) and (I1)--(I2) (initial data depending only on the mean $Mx=m(x)\chi_I$), the paper proves $$\|u^\varepsilon(x,t)-\bar u(m(x),t)\|_{L^\infty(V\times[0,\infty))}\le C\varepsilon,$$ with $C$ depending only on $H$ and $\|Du_0\|_{L^\infty(V)}$, where $\bar u$ solves the effective finite-dimensional Hamilton--Jacobi equation with effective Hamiltonian $\bar H$. The argument reduces the full problem to comparing $u^\varepsilon$ with $\bar u$ at mean configurations, using the mean-reduction estimate $|u^\varepsilon(x,t)-u^\varepsilon(Mx,t)|\le C\varepsilon$, and then to showing that the mean-endpoint action metric $m(t,a,q)$ is $O(1)$-close to its homogenized limit. The sharpness example ($H(x,p)=\frac{1}{2}\|p\|^2+\int_I W(x(i))\,di$ with $W(r)=-4\cos^2(\pi r)$, $u_0\equiv0$) gives $u^\varepsilon(0\chi_I,1)\ge\varepsilon/6$, proving the rate cannot be improved.

Load-bearing premise

The paper imports, from its companion work, the estimate $|u^\varepsilon(x,t)-u^\varepsilon(Mx,t)|\le C\varepsilon$ holding uniformly in $t$, and this mean-reduction estimate has exactly the same order as the theorem; if it were only available at a slower rate or with a time-dependent constant, the optimal-rate proof would not go through.

Editorial extensions

If this is right

For mean-dependent initial data, the infinite particle system is quantitatively governed by a finite-dimensional effective equation with a uniform $O(\varepsilon)$ error over all time.
The large-scale limit of the mean-endpoint action metric is exactly the effective Lagrangian cost, so optimal-control problems on the infinite-dimensional space reduce, up to an $O(1)$ action error, to finite-dimensional mean dynamics.
Because the sharpness example has $u^\varepsilon(0\chi_I,1)\ge\varepsilon/6$, no general improvement beyond $O(\varepsilon)$ is possible without adding hypotheses on the Hamiltonian or the initial data.
The curve-surgery template -- cut a near-minimizer through its finite-dimensional mean-time projection, then glue the lifted pieces using the compact quotient by periodicity and rearrangement -- works because the quotient $V/\sim$ is compact; the same scheme should transfer to other infinite-dimensional variational problems with a compact symmetry quotient.

Reading between the lines

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

The proof does not handle initial data depending on the full configuration: the $O(\varepsilon)$ mean-reduction step is specific to mean-only data, so extending the rate to general data would need a different mechanism.
Because the gluing relies on compactness of the quotient by periodicity and rearrangement, the method likely fails for particle systems without rearrangement symmetry; there the curve-surgery connectors would not be uniformly bounded, so a slower rate may be the best available.
The exact constant in Theorem 1.1 is not identified; quantifying it in terms of the diameter of the quotient and the coercivity and Lipschitz constants of $H$ would make the error bound directly usable.
A natural testable extension is to replace the mean by finitely many low-order moments; the analogue would be a moment-endpoint metric, and whether a compact quotient exists for those observables would determine whether the $O(\varepsilon)$ rate persists.

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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

You should know this paper is the real thing, with one important caveat. Park proves O(ε) convergence for periodic homogenization of convex Hamilton–Jacobi equations in L² space of particle configurations, under rearrangement invariance and mean-dependent initial data. That is a genuine extension of Tran–Yu's finite-dimensional result, and the main device—cutting the mean-time projection and gluing pieces back in the full Hilbert space via the compact rearrangement-periodic quotient—is new and fits the problem well. The dyadic almost-subadditivity and almost-superadditivity estimates in Section 3 are carefully argued, and the sharpness example (u^ε ≥ ε/6 at a specific point) checks out. If the main theorem is correct, it closes the optimal-rate question for this class, so this is a meaningful contribution to the homogenization/weak-KAM subfield.

The soft spots are real but not fatal. The first is Lemma 2.1, which asserts |u^ε(x,t)−u^ε(Mx,t)| ≤ Cε uniformly in t. This is imported from the author's preprint [27], and it is load-bearing: it is used to pass from the full solution to the mean-restricted value function and back, and its error order matches the theorem's conclusion. The present paper does not reproduce the proof. The stress-test note is legitimate: if [27] only gives O(ε^{1/3}) for this comparison, the optimal-rate proof would not go through. The author is candid about the dependency, but you cannot verify it from the text. The second issue is Remark 1: the general almost-subadditivity estimates (3.10) and (3.11) are only sketched, yet they are needed for the Fekete-lemma construction in Proposition 4.1 and for the convexity argument in Proposition 4.4. That is a genuine gap in presentation, even if the estimates are probably true by repeating the earlier arguments.

There is nothing circular or fabricated here. The metric identification with the effective Lagrangian is done through the cell problem and comparison, not by assuming the target rate. The citation pattern is honest—earlier work [27], the Wasserstein paper [28], and Tran–Yu [1] are all positioned correctly. The paper is a bit too reliant on [27] for comfort, and the sketched Remark 1 means a referee would have to either trust the author or ask for details.

Bottom line: this deserves a serious referee. I would send it to peer review and ask the author to either prove the mean-reduction estimate in the present paper or supply a precise statement from [27] with the full argument, and to expand Remark 1. If those pieces hold up, the paper is publishable and will be cited.

Referee Report

2 major / 5 minor

Summary. The paper proves an O(ε) convergence rate for periodic homogenization of convex Hamilton–Jacobi equations on the Hilbert space V = L²(I; R^d), where the Hamiltonian is periodic in the configuration variable and invariant under measure-preserving rearrangements, and the initial data depend only on the mean configuration. The proof reduces the problem to a mean-endpoint action metric, establishes dyadic almost-subadditivity and almost-superadditivity estimates by a curve-surgery argument that combines Burago's finite-dimensional cutting lemma applied to the mean-time path with the compactness of the rearrangement-periodicity quotient, constructs a homogenized metric, identifies it with the Legendre transform of the effective Hamiltonian via convex duality, and concludes by comparing the Lax–Oleinik representations. It ends with an explicit example showing that the O(ε) rate cannot be improved.

Significance. If the two missing ingredients identified below are supplied, this is a substantial contribution: it extends the finite-dimensional optimal-rate theorem of Tran–Yu to an infinite-dimensional indistinguishable-particle setting, with a clean two-scale reduction and an explicit sharpness example. The paper's detailed dyadic estimates, careful connector cost bounds, the use of the compact quotient, and the explicit ε/6 lower bound in the optimality example are strengths. The proof is not parameter-fitted; constants are explicit and depend only on H and ||Du0||∞. The result would be the first optimal convergence rate in this infinite-dimensional periodic setting under convexity.

major comments (2)

[§2.1, Lemma 2.1] The O(ε) mean-reduction estimate |u^ε(x,t) - u^ε(Mx,t)| ≤ Cε is imported from the author's preprint [27] and is not proved in this manuscript. This estimate has exactly the same order as Theorem 1.1 and is used in the essential reduction (2.6), in Lemma 2.2, and in the final step of the proof of Theorem 1.1. The paper's own abstract states that [27] establishes only qualitative homogenization with an O(ε^{1/3}) full convergence rate; the O(ε) mean-reduction component therefore cannot be inferred from the cited qualitative result. Without a proof, or a precise statement of the relevant proposition in [27] together with its proof, the main theorem is conditional on an unverified input.
[§3.1, Remark 1] The general almost-subadditivity estimates (3.10) and (3.11) are asserted without proof, with the sentence 'These estimates follow by repeating...' and no details. These estimates are load-bearing: (3.10) is used as (4.3) in Proposition 4.1 to apply Fekete's lemma and obtain the limit defining the homogenized metric, and (3.11) is used in Proposition 4.4, Step 4, to prove convexity of q ↦ m̄(1,0,q). The reader cannot verify the endpoint-connector and mean-correction details for arbitrary scaling factors ρ,σ or for three arbitrary endpoints. Please supply complete proofs of both estimates, or restate them as theorems with full arguments.

minor comments (5)

[§2.2 and §4] The same symbol m is used for the mean functional m(x), the microscopic mean-endpoint metric m(t,a,q), and the homogenized metric (often written m̄); this overloading makes Section 2.2 and Lemma 4.3 hard to follow. Please use distinct notation for the three objects.
[Figure 3.1] Figure 3.1 is referenced in the text but no figure appears in the manuscript; either include the figure or remove the reference.
[§2.3, Lemma 2.3] In Lemma 2.3, the statement that the number of intervals is bounded by (m+1)/2 should explicitly identify m = d+1 for the mean-time path and should briefly explain the topological origin of the bound, since the constant matters in the subsequent time-saving argument.
[§5.1, Example 1] In the proof of the optimality example, the sentence 'Since H is even and convex, it attains its minimum at p=0' would be clearer with the one-line justification that convexity gives H(0) ≤ (H(p)+H(-p))/2 = H(p) for every p.
[References] Because several load-bearing results are cited from the author's unpublished preprint [27], the manuscript should state explicitly which propositions of [27] are used and, ideally, include their statements in an appendix for the reader.

Circularity Check

1 steps flagged · score 4.0 of 10

The O(ε) rate is partly inherited from the author's own prior preprint [27] via Lemma 2.1, but the metric identification and sharpness example retain independent content.

self citation load bearing [Section 2.1, Lemma 2.1; used in (2.6) and in the proof of Theorem 1.1 (Section 5)]
"Lemma 2.1(Uniform Lipschitz estimate and mean reduction).There existsC >0, independent ofε, such that |uε(x,t)−u ε(y,s)|≤C(∥x−y∥ L2 +|t−s|) for allx,y∈Vands,t≥0. Moreover, |uε(x,t)−u ε(Mx,t)|≤Cε for all(x,t)∈V×[0,∞). ... See Propositions 2.5 and 2.8 in [27] for details."

The O(ε) mean-reduction estimate is the load-bearing bridge of the proof: (2.5)-(2.6) convert the target comparison |uε(x,t)−ū(m(x),t)| into a comparison of Uε with ū, and the final step of Theorem 1.1 again invokes (2.6). This estimate has exactly the same O(ε) order as the theorem and is imported verbatim from the author's own preprint [27]; the manuscript gives no derivation, and the abstract describes [27] as providing only an O(ε^{1/3}) full convergence rate. Thus the optimal rate is inherited from a same-author citation rather than established by the new curve-surgery estimates, unless [27] independently proves the O(ε) mean reduction.

full rationale

The main derivation is not definitionally circular: the homogenized metric m̄ is constructed from the microscopic mean-endpoint action metric via almost-subadditivity and almost-superadditivity estimates (Section 3), then identified with the effective Lagrangian through convex duality and the cell-problem corrector in Proposition 4.4. That identification does not assume the target O(ε) rate, and Lemma 4.3 converts the O(1) metric error into the O(ε) variational error without fitting. The sharpness example is also self-contained, giving a direct lower bound uε(0χ_I,1) ≥ ε/6 from the microscopic Lagrangian and a pathwise argument, independent of the upper-rate proof. No fitted parameters, renamings, or ansatz-imported uniqueness theorems appear. The circularity-relevant burden is Lemma 2.1: its O(ε) mean-reduction estimate is the same-order prerequisite for the theorem and is cited from the author's own preprint [27] rather than re-proved here. Because this estimate is a supporting input rather than the definition of the target quantity, and because the finite-dimensional metric argument, dyadic estimates, and cell-problem identification are carried out in this paper, the central claim retains independent content. The sketched general almost-subadditivity estimates (3.10)-(3.11) in Remark 1 are a completeness gap, not a circularity. Score 4 reflects one load-bearing self-citation with substantial independent derivation.

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

All substantive assumptions are mathematical; no data parameters are fitted. The main external inputs are [27]'s qualitative estimates and corrector, the compactness of S_d, and standard convex-analysis and metric lemmas.

assumptions (6)

standard math Burago's curve-cutting lemma: a continuous path in R^m has finitely many disjoint intervals whose total displacement is half the total displacement, with at most (m+1)/2 intervals.
Quoted from [7] and applied to the mean-time path in Proposition 3.3.
domain assumption The author's earlier preprint [27] provides the uniform Lipschitz estimate, the O(epsilon) mean reduction |u^epsilon(x,t)-u^epsilon(Mx,t)| <= C epsilon, and the existence of a bounded corrector for the cell problem.
Used in Lemma 2.1 and in Step 3 of Proposition 4.4; these facts are not re-derived in this paper.
domain assumption The quotient S_d = V/~ is compact, so any two configurations are within bounded L^2-distance up to lattice translation and rearrangement.
Cited to [23]; this gives uniform connector bounds in Lemma 3.1 and in the gluing arguments.
domain assumption Well-posedness and the optimal-control/Lagrangian representation for viscosity solutions of the infinite-dimensional Hamilton-Jacobi equation hold.
Taken from [16,29,22]; used in equations (2.3)-(2.4) and in Section 4.2.
standard math Fekete's lemma for continuous almost-subadditive functions yields the large-scale limit of the mean-endpoint metric.
Used in Proposition 4.1 without proof.
standard math Fenchel-Moreau theorem and Legendre duality connect H_m with the homogenized metric.
Used in Proposition 4.4 to identify the homogenized metric with the effective Lagrangian.

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Pith. "Pith review of Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions." pith.science (2026). https://pith.science/paper/LP5HYJYV

@misc{pith2026260811449,
  author       = {Pith},
  title        = {Pith review of: Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LP5HYJYV}},
  note         = {Machine review of arXiv:2608.11449}
}

abstract

We prove the optimal convergence rate $O(\varepsilon)$ for periodic homogenization of convex Hamilton-Jacobi equations arising from infinite systems of indistinguishable particles on the torus, under the assumption that the initial data depend only on the mean configuration. This extends the finite-dimensional result [1], which is based on the large-time behavior of the Lagrangian action metric and a curve-surgery argument. Here, these tools cannot be applied directly because minimizing curves live in an infinite-dimensional Hilbert space, where local compactness and finite-dimensional topology are unavailable. We overcome this difficulty by cutting the finite-dimensional mean-time projection of a minimizing curve and gluing the lifted pieces in the Hilbert space using the compact quotient induced by periodicity and rearrangement invariance. We conclude with an example showing that this rate is sharp.

Figures

Figures reproduced from arXiv: 2608.11449 by the authors.

**Figure 3.1.** summarizes the curve-cutting and gluing construction underlying the almost-superadditivity estimate [PITH_FULL_IMAGE:figures/full_fig_p018_3_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [27]

Periodic homogenization of Hamilton–Jacobi equations for infinite systems of indis- tinguishable particles.arXiv preprint arXiv:2605.22198(2026)

Park, S. Periodic homogenization of Hamilton–Jacobi equations for infinite systems of indis- tinguishable particles.arXiv preprint arXiv:2605.22198(2026)

arXiv 2026

[2] [28]

Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space

Ding, Z., Ekren, I., Han, Y. & Zitridis, A. Quantitative homogenization of convex Hamilton– Jacobi equations in the Wasserstein space.arXiv preprint arXiv:2606.22103(2026)

work page Pith review arXiv 2026

[3] [1]

Tran, H. V. & Yu, Y. Optimal convergence rate for periodic homogenization of convex Hamilton–Jacobi equations.Indiana Univ. Math. J.74, 555–573 (2025)

work page 2025

[4] [2]

& Varadhan, S

Lions, P.-L., Papanicolaou, G. & Varadhan, S. R. Homogenization of Hamilton–Jacobi equa- tions.Unpublished preprint(1987)

work page 1987

[5] [3]

Evans, L. C. Periodic homogenisation of certain fully nonlinear partial differential equations. Proceedings of the Royal Society of Edinburgh Section A: Mathematics120, 245–265 (1992)

1992

[6] [4]

& Ishii, H

Capuzzo-Dolcetta, I. & Ishii, H. On the rate of convergence in homogenization of Hamilton– Jacobi equations.Indiana University Mathematics Journal50, 1113–1129 (2001)

work page 2001

[7] [5]

Mitake, H., Tran, H. V. & Yu, Y. Rate of convergence in periodic homogenization of Hamilton– Jacobi equations: the convex setting.Archive for Rational Mechanics and Analysis233, 901–934 (2019)

work page 2019

[8] [6]

A near-optimal rate of periodic homogenization for convex Hamilton–Jacobi equations.Archive for Rational Mechanics and Analysis245, 809–817 (2022)

Cooperman, W. A near-optimal rate of periodic homogenization for convex Hamilton–Jacobi equations.Archive for Rational Mechanics and Analysis245, 809–817 (2022)

work page 2022