{"id":"ee32fd18-d778-4359-afad-a834aab440e2","arxiv_id":"2502.09103","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The vanishing-viscosity error for quadratic Hamilton-Jacobi equations is of order ε log ε (not √ε), sharp in every dimension, with leading constant d/2.","lead":"This paper determines the exact speed at which a slightly smoothed version of a Hamilton-Jacobi equation approaches the original one as the smoothing vanishes, finding an error of order ε log ε rather than the previously believed √ε. It proves this rate cannot be improved and computes the leading constant in terms of the space dimension.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Prop 2.3(ii) proof's second-moment bound is false for constant drifts; the central ε log ε lower bound is not proven as written.","rationale":"The reader's verdict is CONDITIONAL, and I agree that the paper should be conditionally accepted pending a fix. However, the load-bearing concern is not the approximation argument flagged by the reader. The proof of Proposition 2.3(ii) contains a concrete false estimate: the claimed O(ε) bound on the second moment M_T is wrong for constant drift fields, which are well within the theorem's hypotheses (e.g., f=0, g linear). This is not a mere missing justification; it is an incorrect inequality that invalidates the derivation of the integrated Laplacian lower bound. The theorem itself may still be true, and the gap is likely repairable by centering the Gaussian reference at the deterministic flow, but the current manuscript does not prove the central ε log ε lower bound. Since the verdict remains conditional (major revision), I leave the verdict unchanged. Credit is due for the clear structure, the correct explicit example in Section 3, and the honest statement of limitations in the abstract; the issue is confined to the proof of the key estimate.","tokens_in":14064,"tokens_out":35803,"duration_ms":338935,"concrete_test":"Take d=1, T=1, t=0, x=0, f≡0, g(y)=-y. Then φ^ε = φ^0 = -y - (1-s)/2, so ψ = φ^ε, ∇ψ ≡ -1, and the SDE (9) has solution Y_s = s + √ε B_s. Compute M_1 = 1 + ε. The claimed bound 4dε e^{4L²T²} with L=1 is 4ε e^4, which is < 1 for ε < e^{-4}/4 ≈ 0.0046. Hence the bound fails for all sufficiently small ε. Then trace this failure through Theorem 2.5: replacing the constant 2d e^{4L²T²} in Proposition 2.3(ii) by the true O(1) contribution from M_T/(2ε)·ε leaves an O(1) negative term in the final lower bound, which does not imply the stated d/2 ε log ε - C ε lower bound. A repaired proof must estimate ∫ log μ_T dμ_T against a Gaussian centered at the deterministic flow point, not at x.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2.3(ii) is the key estimate, but its proof contains a false bound. The authors claim that for M_s = ∫|y-x|² dμ_s(y), the differential inequality M'_s ≤ 2L√M_s + 2dε implies M_T ≤ 4dε e^{4L²T²}. This implication is incorrect. For a constant drift field, obtained e.g. with f=0 and g(x) = -v·x, the process 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, contradicting the claimed O(ε) bound. The comparison solution of the inequality grows like L²s², not like ε. This matters because the proof lower-bounds ∫ log μ_T dμ_T by -d/2 log(2πε) - M_T/(2ε) using H(μ_T|γ_ε) ≥ 0; without M_T = O(ε), this lower bound is -O(1/ε). When used in Theorem 2.5, this term is multiplied by ε, producing an O(1) negative constant rather than the O(ε) term required for the ε log ε lower bound. The intended estimate could be rescued by comparing μ_T to a Gaussian centered at the deterministic flow point, or by a direct Aronson lower bound, but as written the proof does not establish Proposition 2.3(ii).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1499,"tokens_out":2912,"duration_ms":199698,"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":[{"comment":"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":"Section 2.1, proof of Proposition 2.3(ii)"},{"comment":"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.","section":"Section 2.1, approximation step in Proposition 2.3"}],"minor_comments":[{"comment":"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'.","section":"Section 2.2, Proposition 2.9"},{"comment":"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)'.","section":"End of proof of Theorem 1.2 in Section 2.2"},{"comment":"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'.","section":"Theorem 2.5 statement"},{"comment":"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))'.","section":"Equation (14)"}],"recommendation":"major_revision","confidential_remarks":"The central claim is plausible and the method is promising, but the proof of the key estimate contains a concrete false bound. The error appears repairable by the Gaussian-centering modification described in the report, so I do not recommend rejection; the revision must contain a fully corrected proof of Proposition 2.3(ii) and a rigorous approximation argument."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":14921,"tokens_out":14555,"would_cite":false,"duration_ms":147406,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35F21","35B25","35K55","49L20","60H30"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["vanishing viscosity","Hamilton-Jacobi equations","quadratic Hamiltonian","optimal convergence rate","epsilon log epsilon rate","semiconcavity","Fokker-Planck entropy estimates","stochastic control"],"falsifier":"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.","tokens_in":13736,"feed_emoji":"","tokens_out":8401,"duration_ms":82240,"temperature":0.7,"pith_summary":"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.","feed_headline":"Vanishing viscosity error rate is ε log ε, not √ε","feed_subtitle":"Sharp bounds for quadratic Hamilton–Jacobi equations close the √ε gap with a dimensional constant.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the classical O(√ε) upper bound that the paper improves.","marker":"[Fle64b]"},{"why":"Establishes existence, gradient estimates, and the viscosity-solution convergence framework used throughout.","marker":"[Lio82; Lio84]"},{"why":"Provides the semiconcavity propagation property on which the upper-bound lemma relies.","marker":"[CS04]"},{"why":"Supplies the sup-convolution regularization used to smooth the inviscid solution.","marker":"[LL86]"},{"why":"Gives the H¹ regularity and approximation of Green functions used in Proposition 2.3.","marker":"[Aro68]"},{"why":"Provides the relative-entropy contraction inequality that bounds the log-density term.","marker":"[Dem09]"},{"why":"Contains the one-dimensional sharp ε log ε example against which the multidimensional result is compared.","marker":"[Qia+24]"},{"why":"Gives the stochastic control representation of the value functions that carries the probabilistic part of the proof.","marker":"[FS06]"},{"why":"Justifies Itô's formula for functions with generalized derivatives applied in the main estimate.","marker":"[Kry08]"}],"fun_headline_variants":["Vanishing viscosity: optimal ε log ε error rate","Quadratic HJ: error rate sharp at ε log ε","ε log ε beats √ε in Hamilton–Jacobi vanishing viscosity","Optimal viscosity rate for quadratic HJ is ε log ε"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Vanishing viscosity: optimal ε log ε error rate","Quadratic HJ: error rate sharp at ε log ε","ε log ε beats √ε in Hamilton–Jacobi vanishing viscosity","Optimal viscosity rate for quadratic HJ is ε log ε"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000521,"raw_usage":{"total_tokens":2509,"prompt_tokens":917,"completion_tokens":1592,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":533,"completion_tokens_details":{"reasoning_tokens":1524}},"tokens_in":533,"tokens_out":1592,"duration_ms":12591,"temperature":1.0,"reasoning_tokens":1524,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T22:38:29.426567+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}