Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations

Louis-Pierre Chaintron (ENS-PSL); Samuel Daudin (UPCit\'e)

REVIEW 2 major objections 4 minor 1 cited by

Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proves that the viscous approximation error for quadratic Hamilton–Jacobi equations is O(ε log ε) and no better, closing the gap between the classical √ε upper bound and known lower bounds.

desk verdict Important-looking result, but the central Proposition 2.3(ii) is not proven—the O(ε) second-moment bound is false, so the ε log ε lower bound needs a real repair. read the letter →

arxiv 2502.09103 v1 pith:7Q2T36BM submitted 2025-02-13 math.AP math.OCmath.PR

Louis-Pierre Chaintron (ENS-PSL) , Samuel Daudin (UPCité) This is my paper · ORCID

classification math.APmath.OCmath.PR MSC 35F2135B2535K5549L2060H30

keywords vanishingviscosityHamilton-JacobiequationsquadraticHamiltonianoptimalconvergencerateepsilonlogsemiconcavityFokker-Planckentropyestimatesstochasticcontrol

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 settles the optimal convergence rate in the vanishing viscosity limit for Hamilton–Jacobi equations whose Hamiltonian is purely quadratic. Earlier work gave O(√ε) and many suspected that was optimal; the paper proves the true sup-norm rate is O(ε log ε) for globally Lipschitz semiconcave terminal data (functions with a uniform upper bound on second-order differences), and that this rate cannot be sharpened. The argument produces a universal dimensional constant d/2 in a pointwise lower bound. The proof combines sup-convolution with entropy estimates for the Fokker–Planck flow of a carefully chosen stochastic differential equation. A concrete example with terminal data −|Pk(x)| gives a matching expansion and proves optimality in every dimension.

What carries the argument

The engine is the law $µ^{{ε,δ,x}}$_s of the SDE dY_s = −∇$ψ^{{ε,δ}}$_s(Y_s)ds + √ε dB_s, where $ψ^{{ε,δ}}$ is the half-sum of the viscous solution φ^ε and the sup-convolution $φ^{{0,δ}}$ of the inviscid solution. This law satisfies a Fokker–Planck equation, and Proposition 2.3 bounds integrals of $Δφ^{{0,δ}}$ against µ using relative-entropy contraction and the semiconcavity of φ^ε; that integrated Laplacian estimate is what converts the Gaussian entropy factor into ε log ε. Sup-convolution regularizes $φ^{0}$ while preserving its role as an approximate sub-solution, and Itô's formula along Y bridges the PDE inequality (11) to a pointwise comparison of the two value functions.

What would settle it

Take d=2, f≡0, g(x)=−|x_1|, T=1, and t=0. Proposition 3.1 predicts that (φ^ε_0(0)−$φ^{0}$_0(0))/(ε log ε) tends to 1/2 as ε→0^+. Computing this ratio numerically from solutions of (1) and (3) would settle the claim: if the ratio fails to converge to that finite constant, either the upper or the lower bound on the rate is wrong.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: if f and g are globally Lipschitz and semiconcave, then for all ε∈(0,1], sup_{[0,T]×R^d}|φ^ε_t(x)−$φ^{0}$_t(x)| ≤ −C_opt ε log ε, and pointwise φ^ε_t(x)−$φ^{0}$_t(x) ≥ (d/2)ε log ε − C(d,T,L_f,L_g,λ_f,λ_g)ε. When f≡0, the same rate holds for merely Lipschitz g, because the equation itself generates semiconcavity. Proposition 3.1 exhibits explicit data g_k=−|P_k(x)|, f≡0, with the expansion $φ^{{k,ε}}$_t(0)−$φ^{{k,0}}$_t(0) = (k−1)/2 ε log ε + O(ε). Together these statements show that the O(ε log ε) rate is optimal: no vanishing-viscosity scheme for this class can converge faster than ε log ε in sup norm, and the previously standard √ε rate is off by a logarithmic power.

Load-bearing premise

The proof's key estimate is first established for smooth approximations of a probability density, and a single sentence is used to pass to the true density; that passage is not automatic because the quantity being estimated can jump under the kind of approximation used, so the whole ε log ε rate rests on this step being valid.

Editorial extensions

If this is right

Any numerical or analytical approximation of the inviscid solution by adding viscosity ε can expect no better than O(ε log ε) sup-norm accuracy for Lipschitz semiconcave data.
The pointwise lower bound φ^ε−φ^0 ≥ (d/2)ε log ε − Cε shows the signed difference can be as negative as order ε log ε, so the sup-norm rate cannot be improved to o(ε log ε) in any dimension.
For f≡0, the semiconcavity assumption on the terminal condition is unnecessary: the equation generates semiconcavity through the Cole–Hopf transform, extending the optimal rate to merely Lipschitz data.
Through the mean-field control analogy drawn in the paper, the result implies that convergence rates in mean-field control can beat N^{−1/2} but cannot beat N^{−1} log N in the corresponding regular case.
The explicit projection-norm example fixes the asymptotic constant (k−1)/2 for that class of data, demonstrating that the ε log ε rate is actually attained rather than being an artifact of the method.

Reading between the lines

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

The paper anticipates that the argument adapts to HJB equations with a drift b and a uniformly elliptic diffusion coefficient σ; if so, the same ε log ε rate should hold whenever the diffusion appears only through the trace structure of the Laplacian.
The gap between the lower-bound constant d/2 and the example's constant (d−1)/2 suggests the sharp asymptotic constant is not yet pinned down; a refined lower bound with coefficient (d−1)/2 would strengthen Theorem 1.2.
The ε log ε factor also appears in entropic optimal transport, and the same entropy-versus-Laplacian mechanism may provide a quantitative bridge between vanishing viscosity and Schrödinger-bridge convergence; the paper notes the analogy but does not develop it.
A direct numerical check of the explicit expansion—for example computing (φ^ε−φ^0)/(ε log ε) for g=−|x_1| in dimension two—would give a practical test of both the rate and the dimensional constants.

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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

Short version: the paper identifies the right answer but the main proof has a hole. The O(ε log ε) rate for quadratic Hamiltonians and the lower-bound example are worth attention, but Proposition 2.3(ii)—the estimate the whole argument leans on—is not proven. I would not trust the theorem as it stands.

What is genuinely good: the explicit example in Section 3 is correct and gives the first lower bound in every dimension, with the clean (k−1)/2 ε log ε expansion. The paper also honestly acknowledges the prior d=1 result [Qia+24] and the incoming independent work [CG25]. The idea of integrated Laplacian estimates against the Fokker–Planck flow is a nice technique, even if the execution currently fails.

The soft spot is load-bearing. In the proof of Prop 2.3(ii), the authors claim that the second moment M_s = ∫|y−x|² dμ_s satisfies a differential inequality whose comparison solution is O(ε), namely M_T ≤ 4dε e^{4L²T²}. That implication is false. Take f=0 and g(x)=−v·x. Then φε and φ0 are both linear with gradient −v, so the SDE is Y_s = x − v(s−t) + √ε B_s and M_T = |v|²(T−t)² + εd(T−t), which is O(1) as ε→0. The differential inequality itself may hold as an upper bound, but its solutions grow like L²T², not like ε. This matters because the proof lower-bounds the entropy term through −M_T/(2ε); with M_T=O(1), that term is O(1/ε), and after multiplication by ε the final error becomes O(1) instead of the required O(ε). So the lower bound in Theorem 1.2 is not established as written.

The fix is probably available: compare μ_T to a Gaussian centered at the deterministic flow point, with variance ε(T−t), rather than to a Gaussian centered at x. That would restore the log ε term without needing M_T=O(ε). But that is not what is written. The approximation argument for passing from smooth densities to L² H¹ densities is also sketched too quickly, but the false Gronwall step is the bigger issue.

The abstract also overreaches: it states the rate for a globally Lipschitz terminal condition, while Theorem 1.2 requires semiconcavity or f≡0. That should be fixed even after the proof is repaired.

Who this is for: people working on vanishing viscosity, Hamilton–Jacobi equations, and stochastic control. The paper deserves a serious referee because the question is important and the explicit example and overall strategy are valuable. But the referee should demand a corrected proof of Prop 2.3(ii) before publication. If the flow-centered Gaussian repair works, this will be a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper studies the vanishing-viscosity limit for Hamilton-Jacobi equations with a purely quadratic Hamiltonian and Lipschitz terminal data. It claims that when f and g are globally Lipschitz and semiconcave, the convergence rate is O(epsilon log epsilon), with a matching lower bound whose leading term is (d/2) epsilon log epsilon, and that when f=0 the semiconcavity assumption on g can be dropped. Section 3 gives an explicit example showing the rate cannot be improved. The proof combines sup-convolution, stochastic control, entropy estimates for a Fokker-Planck flow, and semiconcavity.

Significance. If the proof is completed, the paper would settle a long-standing question by improving the classical O(sqrt epsilon) rate to the sharp O(epsilon log epsilon) rate for quadratic Hamiltonians and identify a universal leading constant d/2. The explicit example in Section 3 is a genuine strength. However, the key estimate Proposition 2.3(ii) is not proven as written: it relies on a false differential inequality and an insufficiently justified approximation step. These are load-bearing for the main theorem, so the paper cannot be accepted in its current form.

major comments (2)

[Section 2.1, proof of Proposition 2.3(ii)] The proof contains a false differential-inequality estimate. With M_s = integral |y-x|^2 d mu_s(y), the displayed inequality M'_s <= 2L sqrt(M_s) + 2d epsilon does not imply M_T <= 4d epsilon e^{4L^2 T^2}. For a constant drift field, e.g. f=0 and g(x) = -v dot x, the process is Y_s = x - v(s-t) + sqrt(epsilon) B_s, so M_T = |v|^2 (T-t)^2 + d epsilon (T-t), which is O(1) as epsilon goes to 0. This contradicts the claimed O(epsilon) bound. Since the proof uses M_T = O(epsilon) to lower-bound the entropy of mu_T, the estimate as written yields only an O(1) error after multiplication by epsilon, which does not produce the required epsilon log epsilon lower bound in Theorem 2.5. A correct conclusion should compare mu_T to a Gaussian centered at the deterministic flow point z_T, for which E|Y_T - z_T|^2 = O(epsilon), or use a direct Aronson lower bound.
[Section 2.1, approximation step in Proposition 2.3] The passage from smooth C^{1,2} densities to the actual Fokker-Planck density is only justified by a one-sentence 'standard approximation argument' via weak convergence in L^2 H^1. The functional integral log mu d mu is not continuous under weak H^1 convergence, and the Ito formula for log mu requires justification when mu lacks C^{1,2} regularity. Since Proposition 2.3(ii) is the key estimate, this limiting step must be written out in detail, for example by Aronson's Gaussian bounds together with a localization/cutoff argument.

minor comments (4)

[Section 2.2, Proposition 2.9] In the statement of Proposition 2.9 the expression '-d epsilon / 2 * epsilon log epsilon' appears to be a typo; it should read '-d/2 * epsilon log epsilon'.
[End of proof of Theorem 1.2 in Section 2.2] In the final estimate the term 'epsilon * d/2 * log(eta/T)' uses an undefined eta; from the context it should be epsilon, giving 'epsilon * d/2 * log(epsilon/T)'.
[Theorem 2.5 statement] There is a typo in the quantifier 'for all (t,x) in [0,T] in R^d'; it should be 'for all (t,x) in [0,T] times R^d'.
[Equation (14)] The notation 'epsilon^d/2 log(2 pi epsilon (T-t))' is ambiguous; it should be written as '(epsilon d)/2 log(2 pi epsilon (T-t))'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(ε log ε) rate emerges from dimension-dependent entropy estimates and an independent explicit example.

full rationale

The paper's central derivation does not reduce to its own inputs. Theorem 2.5 obtains the d/2 ε log ε lower bound by combining the sup-convolution properties of Lemma 2.2, the semiconcavity upper bound of Lemma 2.1, the algebraic inequality (11) obtained by subtracting the PDE inequalities for φε and φ0,δ, and the entropy/Fokker–Planck estimate of Proposition 2.3. The proof of Proposition 2.3 uses Girsanov's theorem, contraction of relative entropy, Itô's formula, and external Aronson/Gaussian estimates; no target rate is assumed in these estimates, and the leading constant d/2 comes from the Gaussian entropy in dimension d. The optimality example in Proposition 3.1 is an explicit asymptotic computation from the Cole–Hopf formula (14) and the Hopf–Lax formula, with no dependence on the main theorem, so the claim that the rate cannot be sharpened is independent rather than a renaming. Self-citations in the paper (e.g., [Cha23], [DDJ24], [Car+23a], [Cec+25]) are confined to contextual or prospective remarks in Sections 1.3 and 1.4 and are not load-bearing for the proof. The possible gap in the 'standard approximation argument' in Proposition 2.3(ii) is a correctness concern about an unverified limiting procedure, not a circularity, since that step does not assume the ε log ε rate. No fitted parameter is renamed as a prediction and no uniqueness or ansatz is imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The proof introduces no fitted constants; the regularization scales δ=τ=ε are chosen by the argument, not by data. All background facts are standard theorems cited in the text. The explicit example uses the Cole-Hopf solution, needing no extra postulates.

assumptions (7)

standard math Viscosity solution existence, uniqueness and comparison for Hamilton-Jacobi equations (1) and (3) on R^d
Invoked at the start (Section 1) and used in Lemma 2.1 and Proposition 2.9 via comparison.
standard math Girsanov theorem and contraction of relative entropy (Dembo, Theorem D.13)
Used in Proposition 2.3(i) to bound the entropy of the Fokker-Planck flow.
standard math Itô formula for functions with generalized derivatives (Krylov, Chapter 2.10)
Used in the proof of Theorem 2.5 to compute along the SDE (9) with φε - φ0,δ, which is C^{1,1} in space.
standard math Regularity of Green functions for Fokker-Planck equations (Aronson 1968)
Used in the approximation argument in Proposition 2.3(ii).
standard math Sup-convolution estimates for Lipschitz and semiconcave functions (Lasry-Lions)
Lemma 2.2 properties, including the sub-solution property (8).
domain assumption Semiconcavity propagation for Hamilton-Jacobi equations
Lemma 2.1 and Section 1: φε_t is λ-semiconcave when f and g are semiconcave; this is a standard result (Cannarsa-Sinestrari).
standard math Cole-Hopf transform representation for the viscous solution when f=0
Used in Lemma 2.8 and the explicit example of Section 3; derived directly in equation (14).

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Pith. "Pith review of Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations." pith.science (2026). https://pith.science/paper/7Q2T36BM

@misc{pith2026250209103,
  author       = {Pith},
  title        = {Pith review of: Optimal rate of convergence in the vanishing viscosity for quadratic Hamilton-Jacobi equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7Q2T36BM}},
  note         = {Machine review of arXiv:2502.09103}
}

abstract

The purpose of this note is to provide an optimal rate of convergence in the vanishing viscosity regime for first-order Hamilton-Jacobi equations with purely quadratic Hamiltonian. We show that for a globally Lipschitz-continuous terminal condition the rate is of order O($\epsilon$ log $\epsilon$), and we provide an example to show that this rate cannot be sharpened. This improves on the previously known rate of convergence O( $\sqrt$ $\epsilon$), which was widely believed to be optimal. Our proof combines techniques involving regularization by sup-convolution with entropy estimates for the flow of a suitable version of the adjoint linearized equation. The key technical point is an integrated estimate of the Laplacian of the solution against this flow. Moreover, we exploit the semiconcavity generated by the equation.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Convergence Rates for Vanishing Viscosity Approximations of Possibly Degenerate Viscous Hamilton--Jacobi Equations
math.AP 2026-07 accept novelty 6.0 of 10

Vanishing viscosity approximations of possibly degenerate viscous Hamilton–Jacobi equations on the torus converge pointwise at rate O(ε|log ε|) and in averaged form at rate O(ε).

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Works this paper leans on

9 extracted references · 4 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

On some existence theore ms for semi-linear elliptic equations

[AC78] H. Amann and M. G. Crandall. “On some existence theore ms for semi-linear elliptic equations”. In: Indiana University Mathematics Journal 27.5 (1978), pp. 779–790. [Aro68] D. G. Aronson. “Non-negative solutions of linear pa rabolic equations”. In: Annali della Scuola Normale Superiore di Pisa-Scienze Fisiche e Ma tematiche 22.4 (1968), pp. 607–694....

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[FS86] W. H. Fleming and P. E. Souganidis. “Asymptotic serie s and the method of vanishing viscosity”. In: Indiana University Mathematics Journal 35.2 (1986), pp. 425–447. [FH73] A. Friedman and E. Hopf. “The Cauchy problem for ﬁrst o rder partial diﬀerential equations”. In: Indiana University Mathematics Journal 23.1 (1973), pp. 27–40. [Gof24] A. Goﬃ. “R...

arXiv 1986

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First order quasilinear equations i n several independent variables

1964, pp. 743–746. [Kru70] S. N. Kružkov. “First order quasilinear equations i n several independent variables”. In: Mathematics of the USSR-Sbornik 10.2 (1970), p

1970

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12 [Car+23a] P. Cardaliaguet, S. Daudin, J. Jackson, and P. E. S ouganidis. “An algebraic con- vergence rate for the optimal control of McKean-Vlasov dyna mics”. In: SIAM J. Control Optim. 61.6 (2023), pp. 3341–3369. [Car+23b] P. Cardaliaguet, J. Jackson, N. Mimikos-Stamato poulos, and P. E. Souganidis. “Sharp convergence rates for mean ﬁeld control in th...

arXiv 2023

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[Fle64a] W. H. Fleming. “The Cauchy problem for degenerate p arabolic equations”. In: Jour- nal of Mathematics and Mechanics (1964), pp. 987–1008. [Fle64b] W. H. Fleming. “The convergence problem for diﬀere ntial games, II”. In: Advances in game theory 52 (1964), pp. 195–210. [Fle71] W. H. Fleming. “Stochastic control for small noise i ntensities”. In: SI...

work page 1964

[6] [2008]

A higher-dimensi onal problem in the theory of quasilinear equations

[KS64] N. N. Kuznetsov and A. A. Shishkin. “A higher-dimensi onal problem in the theory of quasilinear equations”. In: Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki 4.14 (1964), pp. 192–205. [LL86] J.-M. Lasry and P.-L. Lions. “A remark on regularizat ion in Hilbert spaces”. In: Israel Journal of Mathematics 55 (1986), pp. 257–266. [Lax05] P...

work page 1964

[7] [2009]

Fractal ﬁrst-order partia l diﬀerential equations

[DI06] J. Droniou and C. Imbert. “Fractal ﬁrst-order partia l diﬀerential equations”. In: Archive for Rational Mechanics and Analysis 182 (2006), pp. 299–331. [Eva80] L. C. Evans. “On solving certain nonlinear partial d iﬀerential equations by accretive operator methods”. In: Israel Journal of Mathematics 36 (1980), pp. 225–247. [Eva10] L. C. Evans. “Adjo...

2006

[8] [2021]

Adjoint methods for static Hamilton–Ja cobi equations

14 [Tra11] H. V. Tran. “Adjoint methods for static Hamilton–Ja cobi equations”. In: Calculus of Variations and Partial Diﬀerential Equations 41 (2011), pp. 301–319. [Var66] S. S. Varadhan. “Asymptotic probabilities and diﬀe rential equations”. In: Commu- nications on Pure and Applied Mathematics 19.3 (1966), pp. 261–286. 15

work page 2011