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

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

REVIEW 1 major objections 5 minor 3 cited by

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

T0 review · 1 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Vanishing viscosity converges at $\epsilon \log \epsilon$, not $\sqrt{\epsilon}$.

desk verdict Strong result with a real coefficient error: the lower bound in Theorem 1.2 is false for small theta, though the O(epsilon log epsilon) rate is correct. read the letter →

arxiv 2506.13255 v1 pith:ZVOICZDY submitted 2025-06-16 math.AP

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

classification math.AP MSC 35F2135B2549L25

keywords vanishingviscosityHamilton-JacobiequationsoptimalconvergencerateuniformlyconvexHamiltoniansemiconcavityFokker-Planckequationentropyestimatessolutions

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

Adding a small viscosity term to a first-order Hamilton–Jacobi equation smooths its solutions; the paper asks how fast those smoothed solutions return to the inviscid limit as the viscosity $\epsilon$ vanishes. Under the standard assumptions that the Hamiltonian is uniformly convex and the terminal condition is Lipschitz and semiconcave, it proves that the convergence rate is of order $\epsilon \log \epsilon$, improving the previously known $\sqrt{\epsilon}$ rate that was widely believed optimal. It also proves a matching lower bound with a universal factor $d$ (the space dimension), and constructs an explicit terminal datum for which the expansion has leading term $\frac{k-1}{2}\epsilon \log \epsilon$, showing the rate cannot be sharpened. The interest is that convexity genuinely accelerates convergence, and the logarithmic loss is the true cost of singularities in the limiting solution.

What carries the argument

The proof is carried by two objects: the sup-convolution regularisation $\phi^{0,\delta}(x)=\sup_y(\phi^0(y)-|x-y|^2/(2\delta))$, which is $-1/\delta$-semiconvex and therefore has Laplacian bounded below by $-d/\delta$, and the Fokker–Planck flow $\mu^{\epsilon,\delta}$ driven by the averaged gradient field built from $\nabla\phi^\epsilon$ and $\nabla\phi^{0,\delta}$. Integrating the equation for $\phi^\epsilon-\phi^{0,\delta}$ against this flow turns the error into an integral of $-\frac{\epsilon}{2}\Delta\phi^{0,\delta}$ against $\mu$. The key estimate (Proposition 2.4-(ii)) bounds this integral from below by the entropy of $\mu$ at the exit time, and the entropy itself is at most $-\frac{d}{2}\log(2\pi\epsilon\tau)+\frac{\tau}{2\epsilon}L^2$; that logarithm is the source of the $\epsilon\log\epsilon$ term. Choosing the regularisation scale $\delta=\tau=\epsilon$ balances the $-1/\delta$-semiconvexity bound, the $O(\delta)$ regularisation error, and the entropy gain, producing $d\epsilon\log\epsilon$ as the leading-order error.

What would settle it

For the explicit datum $g(y)=-|y|$ in dimension $d=2$, quadratic Hamiltonian $H(p)=|p|^2/2$, and $x=0$, Proposition 3.4 predicts $\phi^\epsilon_t(0)-\phi^0_t(0)=\frac12\epsilon\log\epsilon+O(\epsilon)$. Evaluating the explicit Cole–Hopf integral (14) at small $\epsilon$ and computing the ratio $(\phi^\epsilon-\phi^0)/(\epsilon\log\epsilon)$ as $\epsilon\to0$ should give $1/2$; any other limit would refute the claimed optimal rate and constant.

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Extended reading notes

Core claim

The central result (Theorem 1.2) is a two-sided estimate: for every $\epsilon\in(0,1/2]$, $\sup_{t,x}|\phi^\epsilon_t(x)-\phi^0_t(x)| \le -C_{\log}\epsilon\log\epsilon$, and $\phi^\epsilon_t(x)-\phi^0_t(x)\ge d\epsilon\log\epsilon-C\epsilon$ for all $(t,x)$. Here $\phi^\epsilon$ solves the viscous Hamilton–Jacobi–Bellman equation with viscosity $\epsilon/2$ and $\phi^0$ solves the first-order equation; the constant $d$ is the spatial dimension. The lower bound carries the content: because $\log\epsilon<0$, the leading term is positive and has a universal dimensional coefficient, so convergence is slower than linear by a logarithm. Section 3.2 proves optimality: for terminal data $g_k(y)=-|P_k(y)|$ in the quadratic case, the explicit Cole–Hopf and Hopf–Lax formulas give $\phi^{k,\epsilon}_t(0)-\phi^{k,0}_t(0)=\frac{k-1}{2}\epsilon\log\epsilon+O(\epsilon)$, which for $k\ge2$ has exactly order $\epsilon\log\epsilon$ and prevents any sharper global rate. The paper also extends the lower bound to merely Lipschitz terminal data when the Hamiltonian is purely quadratic, using the semiconcavity that the equation itself generates.

Load-bearing premise

The whole proof depends on the terminal datum being semiconcave — curvature bounded above by a constant — and on that bound being inherited by every solution $\phi^\epsilon$ uniformly in $\epsilon$; if this propagation fails, the entropy estimate no longer controls the Laplacian term and the argument collapses.

Editorial extensions

If this is right

The global convergence rate for uniformly convex Hamilton–Jacobi equations in every dimension is $\epsilon|\log\epsilon|$, so the $\sqrt{\epsilon}$ rate known from the early differential-game approach is not optimal under uniform convexity.
The lower bound $\phi^\epsilon_t(x)-\phi^0_t(x)\ge d\epsilon\log\epsilon-C\epsilon$ identifies the logarithmic term as the leading, universal cost of the singularities of the inviscid solution; inside the strong-regularity region the paper's comparison argument recovers the faster linear rate.
The explicit datum $g_k(y)=-|P_k(y)|$ shows the rate is sharp in dimensions $d\ge2$, with prefactor $(k-1)/2$ at the origin; no global bound better than order $\epsilon\log\epsilon$ can hold.
For the purely quadratic Hamiltonian with only Lipschitz terminal data, the same rate remains valid because the equation itself generates $(T-t)^{-1}$-semiconcavity, so the semiconcavity assumption on $g$ can be dropped in that case.
In the mean-field control analogue, the paper's result implies that the convergence of value functions as $N\to\infty$ is faster than $N^{-1/2}$ but no faster than $N^{-1}\log N$, giving a quantitative limitation in the non-smooth regime.

Reading between the lines

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

Beyond the paper, the appearance of the dimension $d$ in the leading term suggests that the prefactor counts the dimension of the set of minimisers in the Hopf–Lax representation: in the sharp example the minimising set is a $(d-k)$-flat and the prefactor is $(k-1)/2$, so a conjectural general formula would express the leading constant through the geometry of the argmin set.
A natural stress test is to keep the equation quadratic but replace the identity diffusion by a non-degenerate matrix with bounded derivatives; the entropy mechanism should still yield $\epsilon\log\epsilon$ with the trace of the diffusion matrix replacing $d$, although the paper does not carry out this extension.
Because the proof uses uniform convexity only through the lower bound $\theta I\le\nabla^2_{pp}H$, a merely convex Hamiltonian such as $H(p)=|p|$ should lie on the other side of the divide and revert to the slower $\sqrt{\epsilon}$ rate; a numerical experiment on the explicit flat-Hamiltonian formulas would test whether uniform convexity is the true threshold.

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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

The paper is a real step forward, but the lower bound in Theorem 1.2 is mis-stated. For the uniformly convex setting they prove the right rate, O(epsilon log epsilon), which is a longstanding open improvement over O(sqrt(epsilon)). The entropy method is nice: regularize by sup-convolution and integrate against a Fokker-Planck flow to trade the sqrt(epsilon) bound for epsilon log epsilon. The optimality example is also clean.

The problem is the theta dependence. Proposition 2.4(ii) has a factor 1/theta in the key estimate, and the proof of Theorem 2.6 drops it: the displayed inequality writes epsilon d/2 log(2 pi epsilon tau) instead of epsilon d/(2 theta) log(...). Restoring the factor and choosing delta = tau = epsilon gives a lower bound proportional to (d/theta) epsilon log epsilon, not d epsilon log epsilon. This is not a harmless typo. For theta < 1, the claimed bound with constant d is actually false. Take H(p) = (theta/2)|p|^2 and g(x) = -|x| in R^2. A direct Laplace expansion of the Cole-Hopf formula gives phi^epsilon - phi^0 = (1/(2 theta)) epsilon log epsilon + O(epsilon) at the origin. For theta = 0.1 this is 5 epsilon log epsilon, which is far more negative than the 2 epsilon log epsilon the theorem promises, so the inequality fails for small epsilon. The stress-test note is right about the dropped factor but too charitable in saying it is not a refutation; the stated lower bound is false as written.

The fix is straightforward: the lower bound should have a theta-dependent prefactor, likely d/theta (or something like (d-1)/(2 theta) in the quadratic example), with the constant C depending on theta. The upper bound and the O(epsilon log epsilon) rate are unaffected. The authors also need to check the rescaling claim in the stress-test note, since psi = theta phi keeps the viscosity coefficient epsilon, not theta epsilon, so it does not recover a universal d.

Apart from this, the paper is well written, the literature is handled honestly, and the concurrent torus result [CG25b] is acknowledged. The proof outline is clear and the technical core (Proposition 2.4) appears sound after restoring the theta.

Recommendation: send to peer review, but only after the lower bound statement is corrected and the theta dependence is worked out. A serious referee will catch this immediately; the paper should not be accepted as is.

Referee Report

1 major / 5 minor

Summary. The paper studies the vanishing-viscosity approximation of uniformly convex Hamilton-Jacobi equations with Lipschitz, semiconcave terminal data. It claims an optimal L∞ rate of order ε log ε, improving on the classical √ε rate, and it provides an explicit example showing that the rate cannot be improved in dimension d ≥ 2. The proof combines the semiconcavity of the viscous solutions with a sup-convolution regularization of the first-order solution and entropy estimates for the Fokker-Planck flow associated with the difference of the equations. A separate section treats the purely quadratic case, where the Cole-Hopf formula yields the same rate for merely Lipschitz terminal data.

Significance. If the proof is completed in the form stated, the paper settles a long-standing open question on the optimal vanishing-viscosity rate in all dimensions for uniformly convex Hamiltonians. The entropy-estimate method in Proposition 2.4 is a genuine technical contribution, and the explicit example in Section 3.2 provides a concrete witness that the rate cannot be sharpened. I found no circularity: the entropy bound is derived from Girsanov and standard estimates, not assumed. The simultaneous independent work [CG25b] is acknowledged, and Section 3 usefully extends the result to non-semiconcave data in the quadratic case.

major comments (1)

[Theorem 2.6 and Proposition 2.4(ii)] The transition from Proposition 2.4(ii) to Theorem 2.6 drops the factor 1/θ. Proposition 2.4(ii) states (1/2)∫∫Δφ^{0,δ} dμ ds ≥ d/(2θ) log(2πετ) − τL²/(2εθ) − C. Multiplying by ε and inserting this into Theorem 2.6 gives εd/(2θ) log(2πετ) − τL²/(2θ) − Cε, but the displayed estimate in Theorem 2.6 has εd/2 log(2πετ) − τL²/2 − Cε. With δ=τ=ε, the corrected term yields a leading coefficient d/θ, not d. Since log ε < 0, the advertised lower bound φ^ε−φ^0 ≥ dε log ε − Cε follows from the stated estimates only when θ ≥ 1. For θ < 1, the coefficient d/θ gives a more negative right-hand side, so the claimed inequality is not a consequence of the proof. The order ε log ε is still consistent with the corrected coefficient, but the specific constant d in Theorem 1.2 is not established for all θ>0 as written. This gap needs to be repaired, either by an additional argument that recovers the constant d for θ<1 or by modifying the statement of the lower bound.

minor comments (5)

[Proposition 2.4(ii), proof] The sentence 'Since ∆ϕ^{ε,δ}_t ≥ −λd using the semiconcavity (6)' is not correct as written: semiconcavity gives an upper bound on the Hessian, not a lower bound. The surrounding algebra needs an upper bound on Δϕ^ε_s, namely Δϕ^ε_s ≤ dλ, in order to control the term −(θ/2)∫Δϕ^ε_s dμ. I assume this is a typographical slip, but the sign and the superscript should be corrected.
[Section 2.2] There is a repeated typo 'semiconvavity' where 'semiconcavity' is meant, including near the display following Assumption (A.1).
[Abstract and Section 1.1] The abstract contains a line-break in 'conve rgence' and Section 1.1 has 'direclty'; a proofreading pass is needed.
[Proposition 3.4] The expansion for k=1 has no ε log ε term, consistent with the restriction of the optimality example to d≥2, but the statement 'for 1≤k≤d' could briefly note this so the reader is not misled.
[References] Reference [CG25b] is listed as 'Incoming article'; if a preprint or journal version is available, it should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the epsilon log epsilon rate is derived from independent entropy and semiconcavity estimates, not assumed.

full rationale

The paper's derivation is a forward argument from Assumption (A.1) to Theorem 1.2. The upper bound is obtained from a standard semiconcavity comparison (Lemma 2.2), while the lower bound combines sup-convolution regularization (Lemma 2.3), an entropy/Fisher-information estimate for an auxiliary Fokker-Planck flow (Proposition 2.4), a terminal-time comparison (Lemma 2.5), and a final two-scale optimization with delta = tau = epsilon (Theorem 2.6). The logarithmic factor arises from the Gaussian entropy bound -(d/2) log(2 pi epsilon tau) in Proposition 2.4(i), not from any quantity that already contains the target rate. The self-citations [Cha23] and [Car+23a] support auxiliary, independently checkable facts: a linear algebra trace inequality and the propagation of semiconcavity under the equation. Neither is fitted to the claimed rate, and neither assumes the theorem being proved. The optimality example in Section 3.2 uses the explicit Cole-Hopf and Hopf-Lax formulas, so it is not a renaming or restatement of the main theorem. A possible coefficient inconsistency in the printed proof of Theorem 2.6 involving the theta factor from Proposition 2.4(ii) is a correctness concern rather than a circularity concern, because it affects the internal derivation but does not show that any asserted output was used as an input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data fitting is involved; the auxiliary parameters delta and tau are proof scales sent to zero, not free physical parameters. The proof imports standard facts about viscosity solutions, Fokker-Planck flows, Girsanov transforms, and entropy. No new entities are postulated.

assumptions (5)

domain assumption Well-posedness, comparison and C^{1,2} regularity for (1), with uniform gradient bound |grad phi^epsilon| <= L and semiconcavity grad^2 phi^epsilon <= lambda I (Eq. (6)).
Invoked throughout Section 2; cited to [Lio82; Car+23a].
standard math Properties of sup-convolution: phi^{0,delta} is L-Lipschitz, lambda-semiconcave, -1/delta-semiconvex, and an almost-subsolution of (3) (Lemma 2.3).
Used to split the comparison into a short-time semiconvexity term and a long-time entropy term.
domain assumption The Fokker-Planck equation (9) for the fundamental solution mu^{epsilon,delta} has a positive smooth density, Gaussian bounds, and the entropy production identity (10) with finite Fisher information.
Basis of Proposition 2.4; cited to [Aro68; Bog+22; BRS16].
standard math Girsanov theorem and contraction of relative entropy give the pathwise bound H(mu|w) <= tau L^2/(2 epsilon) in Proposition 2.4(i).
Used to estimate the entropy of the flow at time t+tau.
standard math Uniform convexity (A.1)(iv) together with the linear algebra lemma [Cha23, Lemma A.1] bounds Hessian contractions by theta Delta + lambda(Tr H_pp - d theta).
Converts the divergence of b^{epsilon,delta} into control on Delta phi^{0,delta}.

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

@misc{pith2026250613255,
  author       = {Pith},
  title        = {Pith review of: Optimal rate of convergence in the vanishing viscosity for uniformly convex Hamilton-Jacobi equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVOICZDY}},
  note         = {Machine review of arXiv:2506.13255}
}

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 uniformly convex Hamiltonian. We prove that for a globally Lipschitz-continuous and semiconcave 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 regularisation 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 to handle less regular data in the quadratic case.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

12 extracted references · 11 canonical work pages · cited by 3 Pith papers

[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....

work page 1978
[3]

Construction of signed distance functions through an elliptic equation

[Has+24] T. Hasebe, J. Masamune, T. Oka, K. Sakai, and T. Yama da. “Construction of signed distance functions with an elliptic equation”. In: arXiv preprint arXiv:2401.17665 (2024). [Hop50] E. Hopf. “The partial diﬀerential equation ut + uux = µ xx”. In: Communications on Pure and Applied Mathematics 3 (1950), pp. 201–230. [Kru64] S. Kruzhkov. “The Cauchy...

work page Pith review arXiv 2024
[7]

Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations

[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. 16 [Gof24] A. Goﬃ....

work page Pith review arXiv 1986
[155]

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

work page 1970
[217]

On the rate of convergence of ﬁnite-di ﬀerence approximations for Bell- mans equations with variable coeﬃcients

[Kry00] N. V. Krylov. “On the rate of convergence of ﬁnite-di ﬀerence approximations for Bell- mans equations with variable coeﬃcients.” In: Probability Theory & Related Fields 117.1 (2000). [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 Matematich...

work page 2000
[526]

The convergence rate of vanishing viscosity approximations for mean field games

17 [TTZ25] W. Tang, H. V. Tran, and Y. P. Zhang. “Policy Iteratio n for the Deterministic Control Problems—A Viscosity Approach”. In: SIAM Journal on Control and Op- timization 63.1 (2025), pp. 375–401. [TZ23] W. Tang and Y. P. Zhang. “The convergence rate of vanis hing viscosity approxima- tions for mean ﬁeld games”. In: arXiv preprint arXiv:2303.14560 (...

work page Pith review arXiv 2025
[1997]

Error bounds for monotone ap proximation schemes for parabolic Hamilton-Jacobi-Bellman equations

[BJ07] G. Barles and E. Jakobsen. “Error bounds for monotone ap proximation schemes for parabolic Hamilton-Jacobi-Bellman equations”. In: Mathematics of Computation 76.260 (2007), pp. 1861–1893. [Bog+22] V. I. Bogachev, N. V. Krylov, M. Röckner, and S. V. Shap oshnikov. Fokker–Planck– Kolmogorov Equations. Vol

work page 2007
[2004]

An algebraic con- vergence rate for the optimal control of McKean-Vlasov dyna mics

[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 the r...

arXiv 2023

Show all 12 references

[2006]

The Cauchy problem for degenerate p arabolic equations

[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. ...

1964
[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: ...

2006
[2021]

Adjoint methods for static Hamilton–Ja cobi equations

[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 App...

2011
[2022]

Di stances between transition probabilities of diﬀusions and applications to nonlinear F okker–Planck–Kolmogorov equations

[BRS16] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov. “Di stances between transition probabilities of diﬀusions and applications to nonlinear F okker–Planck–Kolmogorov equations”. In: Journal of Functional Analysis 271.5 (2016), pp. 1262–1300. [Bre+12] A. Bressan, F. Hua...

2016 arXiv

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....

work page 1978

[2] [3]

Construction of signed distance functions through an elliptic equation

[Has+24] T. Hasebe, J. Masamune, T. Oka, K. Sakai, and T. Yama da. “Construction of signed distance functions with an elliptic equation”. In: arXiv preprint arXiv:2401.17665 (2024). [Hop50] E. Hopf. “The partial diﬀerential equation ut + uux = µ xx”. In: Communications on Pure and Applied Mathematics 3 (1950), pp. 201–230. [Kru64] S. Kruzhkov. “The Cauchy...

work page Pith review arXiv 2024

[3] [7]

Remarks on the rate of convergence of the vanishing viscosity process of Hamilton-Jacobi equations

[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. 16 [Gof24] A. Goﬃ....

work page Pith review arXiv 1986

[4] [155]

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

work page 1970

[5] [217]

On the rate of convergence of ﬁnite-di ﬀerence approximations for Bell- mans equations with variable coeﬃcients

[Kry00] N. V. Krylov. “On the rate of convergence of ﬁnite-di ﬀerence approximations for Bell- mans equations with variable coeﬃcients.” In: Probability Theory & Related Fields 117.1 (2000). [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 Matematich...

work page 2000

[6] [526]

The convergence rate of vanishing viscosity approximations for mean field games

17 [TTZ25] W. Tang, H. V. Tran, and Y. P. Zhang. “Policy Iteratio n for the Deterministic Control Problems—A Viscosity Approach”. In: SIAM Journal on Control and Op- timization 63.1 (2025), pp. 375–401. [TZ23] W. Tang and Y. P. Zhang. “The convergence rate of vanis hing viscosity approxima- tions for mean ﬁeld games”. In: arXiv preprint arXiv:2303.14560 (...

work page Pith review arXiv 2025

[7] [1997]

Error bounds for monotone ap proximation schemes for parabolic Hamilton-Jacobi-Bellman equations

[BJ07] G. Barles and E. Jakobsen. “Error bounds for monotone ap proximation schemes for parabolic Hamilton-Jacobi-Bellman equations”. In: Mathematics of Computation 76.260 (2007), pp. 1861–1893. [Bog+22] V. I. Bogachev, N. V. Krylov, M. Röckner, and S. V. Shap oshnikov. Fokker–Planck– Kolmogorov Equations. Vol

work page 2007

[8] [2004]

An algebraic con- vergence rate for the optimal control of McKean-Vlasov dyna mics

[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 the r...

arXiv 2023