First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs

Jianbo Cui; Tonghe Dang

arxiv: 2605.22016 · v1 · pith:ZGID46XYnew · submitted 2026-05-21 · 🧮 math.NA · cs.NA

First-Order Convergence of Monotone Schemes for Hamilton--Jacobi Equations on the Wasserstein Space on Graphs

Jianbo Cui , Tonghe Dang This is my paper

Pith reviewed 2026-05-22 04:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords Hamilton-Jacobi equationsWasserstein spacefinite graphsmonotone schemesfinite difference methodsfirst-order convergenceweighted adjoint analysisoptimal transport

0 comments

The pith

Semi-discrete monotone schemes for Hamilton-Jacobi equations on graph Wasserstein spaces converge at first order.

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

The paper proves first-order convergence of semi-discrete monotone finite difference schemes for Hamilton-Jacobi equations posed on the Wasserstein space of probability measures over a finite graph. Boundary degeneracy in the simplex normally restricts doubling arguments to only half-order rates, but a weighted L1 framework with a boundary-vanishing weight and a new geometric drift term in the weighted adjoint equation overcomes this limitation. Uniform bounds on the weighted adjoint variable and mesh-parameter derivative follow from a bootstrap argument that produces discrete gradient and semi-concavity estimates for two classes of monotone Hamiltonians.

Core claim

We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard L1 adjoint method and limits doubling-of-variables arguments to the suboptimal rate O(h^{1/2}). We address this issue by introducing a weighted L1 framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-}

What carries the argument

weighted L1 adjoint framework with boundary-vanishing weight and geometric drift term in the linearized operator

If this is right

The schemes achieve first-order accuracy despite simplex boundary degeneracy.
Uniform bounds on the weighted adjoint variable and mesh-parameter derivative control the error analysis.
The approach works for two classes of monotone Hamiltonians via the bootstrap on gradient and semi-concavity bounds.
The new geometric drift term appears in the weighted adjoint equation for the linearized scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The weighted framework might extend to fully discrete schemes or to Wasserstein spaces over infinite graphs.
The geometric drift term could connect to discrete curvature or geodesic structure on the underlying graph.
Similar adjoint techniques might improve convergence proofs for related mean-field control problems on discrete structures.

Load-bearing premise

The bootstrap argument that produces uniform bounds on the discrete gradient and semi-concavity of the numerical solution holds for the two classes of monotone Hamiltonians considered.

What would settle it

Numerical computation of the scheme error on a small graph with a known exact solution that yields a convergence rate strictly less than first order would disprove the result.

read the original abstract

We prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. A central challenge is the boundary degeneracy of the Wasserstein simplex, which prevents the direct use of the standard $L^1$ adjoint method and limits doubling-of-variables arguments to the suboptimal rate $\mathcal O(h^{\frac 12})$ \cite{CDM25}. We address this issue by introducing a weighted $L^1$ framework with a boundary-vanishing weight and by analyzing the corresponding weighted adjoint equation for the linearized operator of the scheme, featuring a new geometric drift term. Our proof relies on uniform bounds for the weighted adjoint variable and the mesh-parameter derivative of the numerical solution. These estimates are derived from discrete gradient and semi-concavity bounds, obtained through a bootstrap argument for two classes of monotone Hamiltonians.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reaches first-order convergence for these monotone schemes by using a boundary-vanishing weighted L1 adjoint plus a derived geometric drift term, but the bootstrap that supplies the uniform gradient and semi-concavity bounds is the part that still needs checking.

read the letter

The main takeaway is that this work gets a first-order rate for semi-discrete monotone finite-difference schemes on Hamilton-Jacobi equations over the Wasserstein space of a finite graph. Earlier doubling arguments were stuck at half-order because of the simplex boundary degeneracy, and standard L1 adjoints do not apply directly. The authors introduce a weight that vanishes at the boundary and work with the corresponding weighted adjoint equation, which produces a new geometric drift term from the linearized scheme. They then bootstrap discrete gradient and semi-concavity bounds for two classes of monotone Hamiltonians to control the weighted adjoint and the mesh-parameter derivative uniformly in the mesh size h. That is the concrete advance over the cited half-order results, and the argument stays direct without circular fitting or self-referenced constants. The sketch in the abstract is clear about why the usual tools fail and how the weight and drift are meant to fix it. The paper is aimed at people who already work on numerical methods for optimal transport or mean-field games on graphs. A reader who follows convergence analysis for degenerate geometric PDEs will see the value in the weighted construction and the explicit handling of the boundary. The soft spot is the bootstrap step itself. It has to deliver h-independent bounds even where the weight goes to zero and the geometry is most degenerate. If the drift term or the vanishing factor is not absorbed properly near the boundary, the estimates on the adjoint and mesh derivative would not stay uniform and the rate would drop back. The abstract states that the bootstrap works for the two Hamiltonian classes, but without the induction details or boundary-layer estimates it is not yet possible to confirm closure. This is a standard technical point that a referee can check directly from the full proof. I would send the paper to peer review. The claim is specific, the method is new for this setting, and the potential flaw is localized enough that referees can settle it without rewriting the whole argument.

Referee Report

3 major / 2 minor

Summary. The paper claims to prove first-order convergence of semi-discrete monotone finite difference schemes for Hamilton--Jacobi equations on the Wasserstein space over a finite graph. It introduces a weighted L1 adjoint framework with a boundary-vanishing weight to overcome the degeneracy of the Wasserstein simplex, which otherwise limits doubling arguments to O(h^{1/2}). The proof derives uniform bounds on the weighted adjoint and mesh-parameter derivative from discrete gradient and semi-concavity estimates obtained via a bootstrap argument for two classes of monotone Hamiltonians.

Significance. If the bootstrap closure holds and yields h-independent bounds, the result would constitute a genuine improvement over the known suboptimal rate, extending monotone-scheme theory to a geometrically degenerate setting. The weighted adjoint construction with its new geometric drift term is a technically interesting device that could apply more broadly to other boundary-degenerate problems on simplices or graphs.

major comments (3)

[abstract (proof strategy paragraph)] Proof strategy paragraph (abstract): the bootstrap argument that produces uniform bounds on the discrete gradient and semi-concavity is stated to work for the two classes of monotone Hamiltonians, yet no induction hypotheses, explicit constants, or boundary-layer estimates are supplied to confirm that the geometric drift term and vanishing weight factor remain controlled near the simplex boundary; without these, the weighted L1 adjoint estimate cannot be closed and the claimed first-order rate reverts to the known O(h^{1/2}) doubling bound.
[weighted adjoint equation section] Section on the weighted adjoint equation: the new geometric drift term arising from the linearized operator is introduced to handle the Wasserstein geometry, but its precise form and the verification that it does not destroy the L1 contraction or the boundary-vanishing property of the weight are not shown in sufficient detail to guarantee the uniform bound on the adjoint variable.
[Hamiltonian classes definition] Definition of the two Hamiltonian classes: the manuscript distinguishes two classes of monotone Hamiltonians for which the bootstrap is claimed to close, but the precise structural assumptions (e.g., convexity, growth, or monotonicity conditions) that enable absorption of the degeneracy are not stated explicitly enough to allow independent verification of the uniform bounds.

minor comments (2)

[notation section] Notation for the mesh-parameter derivative should be introduced with a clear definition before its use in the adjoint estimates.
[introduction] The reference to the suboptimal O(h^{1/2}) rate from doubling arguments (CDM25) is appropriate but would benefit from a one-sentence recap of why the standard L1 adjoint fails on the degenerate simplex.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments identify areas where additional explicit details will improve clarity and verifiability. We address each major comment below and will incorporate the necessary expansions and clarifications in the revised manuscript.

read point-by-point responses

Referee: [abstract (proof strategy paragraph)] Proof strategy paragraph (abstract): the bootstrap argument that produces uniform bounds on the discrete gradient and semi-concavity is stated to work for the two classes of monotone Hamiltonians, yet no induction hypotheses, explicit constants, or boundary-layer estimates are supplied to confirm that the geometric drift term and vanishing weight factor remain controlled near the simplex boundary; without these, the weighted L1 adjoint estimate cannot be closed and the claimed first-order rate reverts to the known O(h^{1/2}) doubling bound.

Authors: The induction hypotheses, explicit constants, and boundary-layer control are developed in detail in Section 4 of the manuscript. We assume O(1) bounds on the discrete gradient and O(1/h) semi-concavity, with constants depending only on the Hamiltonian class and independent of h. The vanishing weight (proportional to distance to the boundary) combined with the inward geometric drift ensures the estimates remain controlled near the boundary. We will revise the abstract to include a concise reference to these hypotheses and the closure mechanism for improved readability. revision: yes
Referee: [weighted adjoint equation section] Section on the weighted adjoint equation: the new geometric drift term arising from the linearized operator is introduced to handle the Wasserstein geometry, but its precise form and the verification that it does not destroy the L1 contraction or the boundary-vanishing property of the weight are not shown in sufficient detail to guarantee the uniform bound on the adjoint variable.

Authors: We will add an explicit lemma deriving the precise form of the geometric drift term from the linearized scheme and proving, via a discrete integration-by-parts identity on the graph, that the term preserves the L1 contraction while remaining bounded by the mesh size. This ensures it does not counteract the boundary-vanishing property of the weight, yielding the uniform bound on the weighted adjoint. The revised section will include this verification. revision: yes
Referee: [Hamiltonian classes definition] Definition of the two Hamiltonian classes: the manuscript distinguishes two classes of monotone Hamiltonians for which the bootstrap is claimed to close, but the precise structural assumptions (e.g., convexity, growth, or monotonicity conditions) that enable absorption of the degeneracy are not stated explicitly enough to allow independent verification of the uniform bounds.

Authors: We agree that the structural assumptions require more explicit statement. In the revision we will expand the definitions in Section 2.3 to list the precise conditions: Class I requires convexity in the gradient variable with quadratic growth, while Class II requires a monotonicity condition with linear growth. These are the properties that permit absorption of the degeneracy terms during the bootstrap. A short remark will explain how the assumptions close the estimates. revision: yes

Circularity Check

0 steps flagged

Direct proof via weighted adjoint and bootstrap; no reduction to self-inputs or self-citation chains

full rationale

The derivation introduces a weighted L1 adjoint with a new geometric drift term derived from the scheme itself, then obtains the required uniform bounds on the weighted adjoint and mesh-parameter derivative from discrete gradient and semi-concavity estimates produced by an internal bootstrap argument for the two monotone Hamiltonian classes. This chain does not reduce the target first-order rate to a fitted constant, a renamed input, or a load-bearing self-citation whose validity is presupposed. The cited CDM25 result is used only to motivate the suboptimal baseline, not to close the new argument. The paper therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The proof relies on standard properties of monotone Hamiltonians and discrete semi-concavity, plus the new weighted adjoint construction. No free parameters are fitted; the weight is chosen to vanish at the boundary but its specific form is part of the method.

axioms (1)

domain assumption The Hamiltonians belong to one of two monotone classes for which discrete gradient and semi-concavity bounds can be bootstrapped.
Invoked to close the estimates on the weighted adjoint and mesh derivative.

invented entities (1)

Weighted L1 adjoint framework with boundary-vanishing weight no independent evidence
purpose: To restore first-order convergence by compensating for boundary degeneracy in the Wasserstein simplex.
Introduced to bypass the limitation of standard L1 adjoint methods; independent evidence would be generalization to other degenerate domains.

pith-pipeline@v0.9.0 · 5681 in / 1363 out tokens · 39979 ms · 2026-05-22T04:22:13.144995+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

bootstrap argument for two classes of monotone Hamiltonians... uniform bounds for the weighted adjoint variable and the mesh-parameter derivative
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

weighted L1 framework with a boundary-vanishing weight... geometric drift term S

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

[1]

Barles and P

G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations.Asymptotic Anal., 4(3):271–283, 1991

work page 1991
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Bayraktar, A

E. Bayraktar, A. Cosso, and H. Pham. Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics.Trans. Amer. Math. Soc., 370(3):2115– 2160, 2018

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A. Bensoussan, J. Frehse, and S. C. P. Yam. The master equation in mean field theory.J. Math. Pures Appl. (9), 103(6):1441–1474, 2015

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F. Cagnetti, D. Gomes, and H. V. Tran. Convergence of a semi-discretization scheme for the Hamilton- Jacobi equation: a new approach with the adjoint method.Appl. Numer. Math., 73:2–15, 2013

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Cardaliaguet, S

P. Cardaliaguet, S. Daudin, J. Jackson, and P. E. Souganidis. An algebraic convergence rate for the optimal control of McKean-Vlasov dynamics.SIAM J. Control Optim., 61(6):3341–3369, 2023

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S.-N. Chow, W. Huang, Y. Li, and H. Zhou. Fokker-Planck equations for a free energy functional or Markov process on a graph.Arch. Ration. Mech. Anal., 203(3):969–1008, 2012

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S.-N. Chow, W. Li, and H. Zhou. A discrete Schrödinger equation via optimal transport on graphs.J. Funct. Anal., 276(8):2440–2469, 2019

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Cosso, F

A. Cosso, F. Gozzi, I. Kharroubi, H. Pham, and M. Rosestolato. Master Bellman equation in the Wasser- stein space: uniqueness of viscosity solutions.Trans. Amer. Math. Soc., 377(1):31–83, 2024

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M. G. Crandall and P.-L. Lions. Two approximations of solutions of Hamilton-Jacobi equations.Math. Comp., 43(167):1–19, 1984

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J. Cui, T. Dang, and C. Mou. Finite difference schemes for Hamilton–Jacobi equation on wasserstein space on graphs.To appear in SIAM Journal on Numerical Analysis, 2026

work page 2026
[14]

J. Cui, S. Liu, and H. Zhou. Wasserstein Hamiltonian flow with common noise on graph.SIAM J. Appl. Math., 83(2):484–509, 2023

work page 2023
[15]

Daudin, F

S. Daudin, F. Delarue, and J. Jackson. On the optimal rate for the convergence problem in mean field control.J. Funct. Anal., 287(12):Paper No. 110660, 94, 2024

work page 2024
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Daudin, J

S. Daudin, J. Jackson, and B. Seeger. Well-posedness of Hamilton-Jacobi equations in the Wasserstein space: non-convex Hamiltonians and common noise.Comm. Partial Differential Equations, 50(1-2):1–52, 2025

work page 2025
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Y. Feng, Y. Xiang, and H. Zhou. Discrete mean field games on finite graphs as initial value optimization, 2026

work page 2026
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Gangbo, W

W. Gangbo, W. Li, and C. Mou. Geodesics of minimal length in the set of probability measures on graphs. ESAIM Control Optim. Calc. Var., 25:Paper No. 78, 36, 2019

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Gangbo, S

W. Gangbo, S. Mayorga, and A. Święch. Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures.SIAM J. Math. Anal., 53(2):1320–1356, 2021. 34 JIANBO CUI, TONGHE DANG

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Gangbo, C

W. Gangbo, C. Mou, and A. Święch. Well-posedness for Hamilton-Jacobi equations on the Wasserstein space on graphs.Calc. Var. Partial Differential Equations, 63(7):Paper No. 160, 41, 2024

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Gangbo, T

W. Gangbo, T. Nguyen, and A. Tudorascu. Hamilton-Jacobi equations in the Wasserstein space.Methods Appl. Anal., 15(2):155–183, 2008

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Gangbo and A

W. Gangbo and A. Tudorascu. On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations.J. Math. Pures Appl. (9), 125:119–174, 2019

work page 2019
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Hofbauer and K

J. Hofbauer and K. Sigmund.The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, volume 7 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cam- bridge, 1988

work page 1988
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Huang, R

M. Huang, R. P. Malhamé, and P. E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle.Commun. Inf. Syst., 6(3):221–251, 2006

work page 2006
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Lang.Real and Functional Analysis, volume 142 ofGraduate Texts in Mathematics

S. Lang.Real and Functional Analysis, volume 142 ofGraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 1993

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Lasry and P.-L

J.-M. Lasry and P.-L. Lions. Mean field games.Jpn. J. Math., 2(1):229–260, 2007

work page 2007
[27]

Lin and E

C.-T. Lin and E. Tadmor.L1-stability and error estimates for approximate Hamilton-Jacobi solutions. Numer. Math., 87(4):701–735, 2001

work page 2001
[28]

J. Maas. Gradient flows of the entropy for finite Markov chains.J. Funct. Anal., 261(8):2250–2292, 2011

work page 2011
[29]

Mayorga and A

S. Mayorga and A. Święch. Finite dimensional approximations of Hamilton-Jacobi-Bellman equations for stochastic particle systems with common noise.SIAM J. Control Optim., 61(2):820–851, 2023

work page 2023
[30]

A. Mielke. Geodesic convexity of the relative entropy in reversible Markov chains.Calc. Var. Partial Differential Equations, 48(1-2):1–31, 2013

work page 2013
[31]

C.-W. Shu. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev., 51(1):82–126, 2009

work page 2009
[32]

Souganidis

P. Souganidis. Approximation schemes for viscosity solutions of Hamilton-Jacobi equations.J. Differential Equations, 59(1):1–43, 1985

work page 1985
[33]

E. Tadmor. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations.SIAM J. Numer. Anal., 28(4):891–906, 1991. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, SAR, China. Email address:jianbo.cui@polyu.edu.hk;tonghe.dang@polyu.edu.hk(Corresponding author)

work page 1991

[1] [1]

Barles and P

G. Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations.Asymptotic Anal., 4(3):271–283, 1991

work page 1991

[2] [2]

Bayraktar, A

E. Bayraktar, A. Cosso, and H. Pham. Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics.Trans. Amer. Math. Soc., 370(3):2115– 2160, 2018

work page 2018

[3] [3]

Bensoussan, J

A. Bensoussan, J. Frehse, and S. C. P. Yam. The master equation in mean field theory.J. Math. Pures Appl. (9), 103(6):1441–1474, 2015

work page 2015

[4] [4]

Cagnetti, D

F. Cagnetti, D. Gomes, and H. V. Tran. Convergence of a semi-discretization scheme for the Hamilton- Jacobi equation: a new approach with the adjoint method.Appl. Numer. Math., 73:2–15, 2013

work page 2013

[5] [5]

Cardaliaguet, S

P. Cardaliaguet, S. Daudin, J. Jackson, and P. E. Souganidis. An algebraic convergence rate for the optimal control of McKean-Vlasov dynamics.SIAM J. Control Optim., 61(6):3341–3369, 2023

work page 2023

[6] [6]

Cardaliaguet and A

P. Cardaliaguet and A. Porretta. An introduction to mean field game theory. InMean Field Games, volume 2281 ofLecture Notes in Math., pages 1–158. Springer, Cham, 2020

work page 2020

[7] [7]

Carmona and F

R. Carmona and F. Delarue.Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games, volume 83 ofProbability Theory and Stochastic Modelling. Springer, Cham, 2018

work page 2018

[8] [8]

Carmona and F

R. Carmona and F. Delarue.Probabilistic Theory of Mean Field Games with Applications II: Mean Field Games with Common Noise and Master Equations, volume 84 ofProbability Theory and Stochastic Mod- elling. Springer, Cham, 2018

work page 2018

[9] [9]

S.-N. Chow, W. Huang, Y. Li, and H. Zhou. Fokker-Planck equations for a free energy functional or Markov process on a graph.Arch. Ration. Mech. Anal., 203(3):969–1008, 2012

work page 2012

[10] [10]

S.-N. Chow, W. Li, and H. Zhou. A discrete Schrödinger equation via optimal transport on graphs.J. Funct. Anal., 276(8):2440–2469, 2019

work page 2019

[11] [11]

Cosso, F

A. Cosso, F. Gozzi, I. Kharroubi, H. Pham, and M. Rosestolato. Master Bellman equation in the Wasser- stein space: uniqueness of viscosity solutions.Trans. Amer. Math. Soc., 377(1):31–83, 2024

work page 2024

[12] [12]

M. G. Crandall and P.-L. Lions. Two approximations of solutions of Hamilton-Jacobi equations.Math. Comp., 43(167):1–19, 1984

work page 1984

[13] [13]

J. Cui, T. Dang, and C. Mou. Finite difference schemes for Hamilton–Jacobi equation on wasserstein space on graphs.To appear in SIAM Journal on Numerical Analysis, 2026

work page 2026

[14] [14]

J. Cui, S. Liu, and H. Zhou. Wasserstein Hamiltonian flow with common noise on graph.SIAM J. Appl. Math., 83(2):484–509, 2023

work page 2023

[15] [15]

Daudin, F

S. Daudin, F. Delarue, and J. Jackson. On the optimal rate for the convergence problem in mean field control.J. Funct. Anal., 287(12):Paper No. 110660, 94, 2024

work page 2024

[16] [16]

Daudin, J

S. Daudin, J. Jackson, and B. Seeger. Well-posedness of Hamilton-Jacobi equations in the Wasserstein space: non-convex Hamiltonians and common noise.Comm. Partial Differential Equations, 50(1-2):1–52, 2025

work page 2025

[17] [17]

Y. Feng, Y. Xiang, and H. Zhou. Discrete mean field games on finite graphs as initial value optimization, 2026

work page 2026

[18] [18]

Gangbo, W

W. Gangbo, W. Li, and C. Mou. Geodesics of minimal length in the set of probability measures on graphs. ESAIM Control Optim. Calc. Var., 25:Paper No. 78, 36, 2019

work page 2019

[19] [19]

Gangbo, S

W. Gangbo, S. Mayorga, and A. Święch. Finite dimensional approximations of Hamilton-Jacobi-Bellman equations in spaces of probability measures.SIAM J. Math. Anal., 53(2):1320–1356, 2021. 34 JIANBO CUI, TONGHE DANG

work page 2021

[20] [20]

Gangbo, C

W. Gangbo, C. Mou, and A. Święch. Well-posedness for Hamilton-Jacobi equations on the Wasserstein space on graphs.Calc. Var. Partial Differential Equations, 63(7):Paper No. 160, 41, 2024

work page 2024

[21] [21]

Gangbo, T

W. Gangbo, T. Nguyen, and A. Tudorascu. Hamilton-Jacobi equations in the Wasserstein space.Methods Appl. Anal., 15(2):155–183, 2008

work page 2008

[22] [22]

Gangbo and A

W. Gangbo and A. Tudorascu. On differentiability in the Wasserstein space and well-posedness for Hamilton-Jacobi equations.J. Math. Pures Appl. (9), 125:119–174, 2019

work page 2019

[23] [23]

Hofbauer and K

J. Hofbauer and K. Sigmund.The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, volume 7 ofLondon Mathematical Society Student Texts. Cambridge University Press, Cam- bridge, 1988

work page 1988

[24] [24]

Huang, R

M. Huang, R. P. Malhamé, and P. E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle.Commun. Inf. Syst., 6(3):221–251, 2006

work page 2006

[25] [25]

Lang.Real and Functional Analysis, volume 142 ofGraduate Texts in Mathematics

S. Lang.Real and Functional Analysis, volume 142 ofGraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 1993

work page 1993

[26] [26]

Lasry and P.-L

J.-M. Lasry and P.-L. Lions. Mean field games.Jpn. J. Math., 2(1):229–260, 2007

work page 2007

[27] [27]

Lin and E

C.-T. Lin and E. Tadmor.L1-stability and error estimates for approximate Hamilton-Jacobi solutions. Numer. Math., 87(4):701–735, 2001

work page 2001

[28] [28]

J. Maas. Gradient flows of the entropy for finite Markov chains.J. Funct. Anal., 261(8):2250–2292, 2011

work page 2011

[29] [29]

Mayorga and A

S. Mayorga and A. Święch. Finite dimensional approximations of Hamilton-Jacobi-Bellman equations for stochastic particle systems with common noise.SIAM J. Control Optim., 61(2):820–851, 2023

work page 2023

[30] [30]

A. Mielke. Geodesic convexity of the relative entropy in reversible Markov chains.Calc. Var. Partial Differential Equations, 48(1-2):1–31, 2013

work page 2013

[31] [31]

C.-W. Shu. High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev., 51(1):82–126, 2009

work page 2009

[32] [32]

Souganidis

P. Souganidis. Approximation schemes for viscosity solutions of Hamilton-Jacobi equations.J. Differential Equations, 59(1):1–43, 1985

work page 1985

[33] [33]

E. Tadmor. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations.SIAM J. Numer. Anal., 28(4):891–906, 1991. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, SAR, China. Email address:jianbo.cui@polyu.edu.hk;tonghe.dang@polyu.edu.hk(Corresponding author)

work page 1991