On Mean Field Monotonicity Conditions from Control Theoretical Perspective

Alain Bensoussan; Shanjian Tang; Sheung Chi Phillip Yam; Ziyu Huang

REVIEW 3 major objections 4 minor 1 cited by

On Mean Field Monotonicity Conditions from Control Theoretical Perspective

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Under a new two-part monotonicity condition for the running cost, the FBSDE systems that characterize mean field game equilibria and mean field type control optima are globally well-posed, yielding unique solutions.

desk verdict New monotonicity condition and a solid β-monotonicity proof, but the global well-posedness rests on an unproved lemma that the references don't cover. read the letter →

arxiv 2412.05189 v1 pith:LVDRSIJS submitted 2024-12-06 math.OC

Alain Bensoussan , Ziyu Huang , Shanjian Tang , Sheung Chi Phillip Yam This is my paper · ORCID

classification math.OC MSC 60H3060H1093E2035R15

keywords MeanfieldgamestypecontrolproblemsForward-backwardstochasticdifferentialequationsMonotonicityconditionsβ-MonotonicitySmalleffectDisplacementquasi-monotonicityGenericdriftfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish global well-posedness for mean field games and mean field type control from a control-theoretic monotonicity condition on the running cost. Its central move is to split the cost as $f = f_0 + f_1$, with $f_1$ strongly convex in the state and control with a small mean field effect, and $f_0$ satisfying a displacement quasi-monotonicity condition; this makes the associated forward-backward stochastic differential equations satisfy the paper's β-monotonicity with zero slack term, so a general Hilbert-space FBSDE theorem gives a unique global solution. For mean field type control, the same β-monotonicity is shown to be exactly a convexity condition on the cost. The paper also extends the argument to nonlinear drift coefficients. If the results hold, the standard displacement monotonicity and small-mean-field-effect conditions become two special cases of one unified condition.

What carries the argument

β-monotonicity (Condition 2.1) is the engine: an inequality (2.2) on the FBSDE coefficients $B,A,F$ that bounds their joint pairing by $-\Lambda_\beta \mathbb{E}|\beta'-\beta|^2 + \Gamma_\beta(\|X'-X\|_2^2+\|P'-P\|_2^2+\|Q'-Q\|_2^2)$, together with monotonicity of $G$ and β-Lipschitz bounds. The paper chooses $\beta(s,X,P,Q)(\omega)=\hat v(s,X(\omega),L(X),P(\omega),Q(\omega))$, the optimal feedback control. The key identity (3.9) rewrites the left side of (2.2) for the MFG coefficients (3.4) as the negative of the pairing of $(D_x f, D_v f)$ with $(X'-X, \hat v'-\hat v)$; Condition 3.3 is engineered so this pairing is controlled by convexity of $f_1$, displacement quasi-monotonicity of $f_0$, and Young's inequality, yielding $\Gamma_\beta = 0$. Lemma 2.1, the well-posedness theorem for such abstract FBSDEs, then carries the conclusion.

What would settle it

Compute the a priori estimate for the continuation method on a β-dependent instance of FBSDEs (2.1) that satisfies Condition 2.1 with $\Gamma_\beta = 0$. If such an instance can be produced for which the estimate (3.37) fails or for which (2.1) lacks a unique adapted solution, Lemma 2.1 is false and the paper's global well-posedness claims collapse. Concretely, one can take the linear-quadratic cost $f(s,x,m,v)=|x|^2+|v|^2+v^\top \int y\,m(dy)$ of Remark 3.3, solve the associated Riccati equations explicitly, and check whether the solution exists on the whole interval $[0,T]$ whenever (3.20) holds.

Watch

Extended reading notes

Core claim

The central claim is that the FBSDE selection equations for the mean field game (1.2), and for the mean field type control problem (1.4), are globally well-posed whenever their coefficients satisfy Condition 2.1 with β equal to the optimal control map $\hat v$ defined by $D_v L = 0$. The new sufficient condition, Condition 3.3, writes the running cost as $f_0(s,x,m,v)+f_1(s,x,m,v)$: $f_1$ is strongly convex in $(x,v)$ with small dependence on the measure as quantified by (3.12) and (3.13), while $f_0$ is convex in $v$ and satisfies the displacement quasi-monotonicity inequality (3.18), with constants balanced by (3.20). The proof computes the monotonicity bracket (3.9) and shows it is bounded above by $-\lambda_v \mathbb{E}|\hat v'-\hat v|^2$, i.e. Condition 2.1(i)(a) holds with $\Gamma_\beta = 0$ and $\Lambda_\beta = \lambda_v$; Lemma 2.1 then supplies the unique solution of (1.2), and Theorem 3.1 converts it into the unique solution of the mean field game. For the MFTC problem, Theorem 4.2 shows that the convexity assumption (B3) is precisely the required β-monotonicity for the FBSDEs (1.4). The same scheme is pushed through for nonlinear drift functionals in Theorems 5.2 and 5.5.

Load-bearing premise

The load-bearing premise is Lemma 2.1, which asserts global well-posedness of the abstract FBSDEs (2.1) under Condition 2.1; the paper states this lemma without proof, referring to earlier results that do not cover the general β-dependent term, so the global well-posedness theorems depend on an unproved assertion.

Editorial extensions

If this is right

Under Condition 3.3, the mean field game (1.1) has a unique global solution via Corollary 3.6, so the equilibrium can be computed by solving the associated FBSDEs and applying the maximum principle.
Under the convexity assumption (B3), the mean field type control problem (1.3) has a unique optimal control, with β-monotonicity of the associated FBSDEs serving as the well-posedness mechanism.
Nonlinear drift functionals are covered by Theorems 5.2 and 5.5 under a cone condition and explicit parameter thresholds, widening the class of solvable mean field problems beyond linear drift settings.
Classical displacement monotonicity and the strong-convexity-with-small-mean-field-effect condition appear as special cases of Condition 3.3, so earlier well-posedness results follow from one β-monotonicity verification.
The same Hilbert-space FBSDE lemma remains available for later applications, such as Jacobian and Hessian flows of the decoupling fields, by choosing other β maps.

Reading between the lines

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

If Lemma 2.1 receives a full proof for the β-dependent term, the framework would apply to any choice of feedback map β, not only the optimal-control map analyzed here, making the Hilbert-space FBSDE route a general tool for mean field well-posedness.
The continuation argument sketched for almost anti-monotone terminal costs suggests that the convexity of $g$ can be relaxed substantially; completing that argument would unify displacement-monotone and anti-monotone regimes in one framework.
Testing the linear-quadratic example from Remark 3.3 with explicit Riccati solutions could reveal whether the parameter balance in (3.20) is sharp; a sharper balance would improve the thresholds in the generic-drift theorems.
The MFTC equivalence between convexity and β-monotonicity indicates that future convexity-preserving transformations of mean field type control costs can be reinterpreted as searching for a β map that absorbs nonconvex components.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

Quick take: the paper introduces a genuinely new sufficient condition, displacement quasi-monotonicity, and proves it implies β-monotonicity for MFG FBSDEs. That part is real work and checks out. The global well-posedness conclusion, however, rests on Lemma 2.1, an unproved Hilbert-space FBSDE theorem whose β-dependent statement is not covered by the cited references. That gap is load-bearing.

What is new and good: Condition 3.3 splits the running cost into a strongly convex part and a displacement quasi-monotonic part, and the proof of Theorem 3.5 is detailed; the Young's inequality step uses exactly the parameter condition (3.20). The paper also shows Condition 3.3 reduces to known displacement monotonicity and small mean field effect, so the extension claim is honest. The generic drift and MFTC sections are natural extensions and give the paper breadth. Self-citations are frequent, but the key estimate is derived from scratch, so the new-result claim is credible.

The soft spot is Lemma 2.1. Its proof is omitted and the four cited results do not contain the general term in (2.2), where the right side is controlled by |β(X',P',Q')−β(X,P,Q)|² rather than by state and control differences. Condition 2.1(ii) also lacks separate P,Q Lipschitz terms, so the continuation method would need to propagate P,Q regularity through β. Whether that works is exactly what the omitted argument must show. If the lemma fails, Theorems 3.5, 5.2 and 5.5 do not follow. I found no flaw in the β-monotonicity estimate itself; this is a completeness gap, not a contradiction.

Minor: Theorem 3.1's proof is also omitted but likely standard, and the anti-monotonicity sketch is labeled future work, so it shouldn't count against the paper.

Who this is for: researchers working on probabilistic MFG well-posedness and FBSDEs with monotone functionals. If the lemma gap is filled, the paper is a solid contribution. It deserves a serious referee. I would send it to review and ask the authors to supply the proof of Lemma 2.1 or a precise reference that covers the β-dependent statement, and to explain how the continuation method handles the missing P,Q regularity.

Recommendation: engage with it, but don't rely on the main theorems until the gap is addressed.

Referee Report

3 major / 4 minor

Summary. The paper studies well-posedness of mean field game (MFG) and mean field type control (MFTC) problems through a stochastic control and FBSDE approach. It introduces a β-monotonicity condition for general FBSDEs on Hilbert space, proposes a new monotonicity condition for MFGs in which the running cost is split as f = f0 + f1, and proves that certain convexity/small-mean-field-effect assumptions on f1 together with a displacement quasi-monotonicity assumption on f0 imply β-monotonicity with Γβ = 0 (Theorem 3.5). From this, global well-posedness of the associated FBSDEs is claimed via Lemma 2.1, yielding a unique global MFG solution (Corollary 3.6). The paper also treats the MFTC case, showing that convexity of the cost functional corresponds to β-monotonicity, and extends both settings to nonlinear drift functionals (Theorems 5.2 and 5.5).

Significance. If the results are correct, the paper provides a useful unifying control-theoretic framework: Condition 3.3 includes classical displacement monotonicity and the previously used strong convexity plus small mean field effect as special cases, and it offers a genuinely new quasi-monotonicity condition. The treatment of degenerate and state- and control-dependent diffusions, and the extension to generic nonlinear drifts, are valuable and go beyond several existing analytic approaches. The main caveat is that the global well-posedness conclusions all depend on Lemma 2.1, whose proof is omitted and whose precise hypotheses are not exactly covered by the cited references; the paper's positive contribution would be fully established if that lemma is supplied with a complete proof.

major comments (3)

[§2, Lemma 2.1] Lemma 2.1 is the load-bearing well-posedness result for the whole paper: Theorems 3.2, 3.5, 5.2 and 5.5 all invoke it to convert β-monotonicity into existence and uniqueness of the FBSDEs. However, its proof is omitted and described only as similar to [40, Theorem 2.3], [2, Theorem 1], [6, Lemma 4.1] and [7, Lemma 2.2]. Those results do not contain the present general β-dependent term in (2.2), where the right-hand side is controlled by |β(X',P',Q')−β(X,P,Q)|² rather than by ||X'−X||²+||P'−P||²+||Q'−Q||², and Condition 2.1(ii) has no separate P,Q Lipschitz terms for B and A. A failure of this lemma would invalidate the claimed global well-posedness in Corollary 3.6 and in Theorems 5.2 and 5.5. Please provide a complete proof of Lemma 2.1, or a precise statement with conditions that are verifiably satisfied by the examples in Sections 3 and 5.
[§3.1, Theorem 3.1 and Corollary 3.6] Corollary 3.6 states that (3.3) gives a solution of the MFG, but this relies on Theorem 3.1, whose proof is also omitted and only described as similar to [7, Lemma 2.1]. Since Theorem 3.1 converts a solution of the FBSDEs (1.2) into a solution of the MFG (1.1), the sufficiency of the maximum principle is part of the central claim. Please either include the proof or give a precise reference with the exact statement needed here.
[§5, Theorems 5.2 and 5.5] The nonlinear-drift results in Section 5 depend on the same unproved Lemma 2.1 and also on lengthy estimates involving the cone property. In Theorem 5.2, the final line asserts Condition 2.1(i)(a) with Λβ = λv − 2L²Lv_b/λb, but the displayed chain only shows a bound after Young's inequality; the parameter inequality (5.10) is stated in a very compressed form. Please spell out the final Young-inequality step and verify that the constants in (5.10) are exactly those needed to absorb the cross term into the negative ||ΔV||² and ||ΔX||² terms. The same request applies to the corresponding step in Theorem 5.5.

minor comments (4)

[§3.2, Theorems 3.3–3.5] There are repeated typographical errors: “with with the choice of β” appears in Theorems 3.3, 3.4 and 3.5, and the proof of Theorem 3.5 contains “we adopt the use the notations”. These should be corrected.
[§2, Condition 2.1(ii)] In the second displayed inequality of Condition 2.1(ii), the first term appears to be missing a square: it reads ||F(s,X',P',Q')−F(s,X,P,Q)||₂ rather than ||...||₂². Also, the arguments of β are written inconsistently, e.g. “β(X′,s;P′,Q′)” in that line versus “β(s,X′,P′,Q′)” elsewhere.
[§3.2, after Condition 3.2] The paper explicitly notes that the constant 1/8 in Condition 3.2 is not optimal and that the paper does “not drill down into the details”. This is acceptable, but it would help the reader to state clearly that the condition is sufficient and that no attempt is made at sharp constants.
[§5.2] In the derivation of D²_pH, the text says “we shall explain its well-posedness without exploding to infinity in the following”, but no such explanation actually follows. Either provide the missing justification or delete the promise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new Condition 3.3 to beta-monotonicity implication is proved directly; the omitted proof of Lemma 2.1 is a completeness gap, not a circularity.

full rationale

The paper's central new claim is that Condition 3.3 (split running cost into strongly convex f1 and displacement quasi-monotone f0) implies the beta-monotonicity Condition 2.1(i)(a) with beta equal to the optimal control v-hat and Gamma_beta = 0. This is proved by direct estimation in Theorem 3.5 (inequalities (3.22)-(3.28)), with no fitted parameter and no assumption of the desired well-posedness conclusion. The MFTC analogue, Theorem 4.2, similarly verifies Condition 2.1 directly from the convexity assumption (B3). The generic-drift results in Section 5 are further direct verifications of the same beta-monotonicity inequalities. The only load-bearing ingredient whose proof is not given is Lemma 2.1, the general Hilbert-space FBSDE well-posedness result under beta-monotonicity; the text says the proof is 'similar to [40, Theorem 2.3], [2, Theorem 1], [6, Lemma 4.1] and [7, Lemma 2.2]' and is omitted. Since [6] and [7] are by the same authors, this is a self-citation, and the exact beta-dependent term in (2.2) may not be literally covered by the external references. However, this is a completeness or correctness risk, not circularity: Lemma 2.1 is a parameter-free general statement whose assumptions do not include any of the paper's target results, and the paper's new contribution is the verification of those assumptions, which is carried out independently. Therefore no circular step is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The novel mathematical object is the displacement quasi-monotonicity condition, which is a proof device rather than a postulated entity.

free parameters (1)

Constant 1/8 in Condition 3.2 = 1/8
Introduced ad hoc to close the Young's inequality in the proof of Theorem 3.4; the paper notes it is not optimal.

assumptions (4)

domain assumption Lemma 2.1: well-posedness of Hilbert-space FBSDEs (2.1) under Condition 2.1
Stated without proof; cited as similar to prior continuation-method results, but the exact β-dependent statement is not proved in this paper.
domain assumption Theorem 3.1: sufficient maximum principle for MFG (1.1)
Proof omitted, referenced to [7, Lemma 2.1].
domain assumption Theorem 4.1: sufficient maximum principle for MFTC (1.3)
Proof omitted, referenced to [9].
standard math Linear functional derivative and Itô calculus are used throughout
Background framework from [14,22].

how reviews work

0 comments

Cite this review

Pith. "Pith review of On Mean Field Monotonicity Conditions from Control Theoretical Perspective." pith.science (2026). https://pith.science/paper/LVDRSIJS

@misc{pith2026241205189,
  author       = {Pith},
  title        = {Pith review of: On Mean Field Monotonicity Conditions from Control Theoretical Perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LVDRSIJS}},
  note         = {Machine review of arXiv:2412.05189}
}

abstract

In this article, from the viewpoint of control theory, we discuss the relationships among the commonly used monotonicity conditions that ensure the well-posedness of the solutions arising from problems of mean field games (MFGs) and mean field type control (MFTC). We first introduce the well-posedness of general forward-backward stochastic differential equations (FBSDEs) defined on some suitably chosen Hilbert spaces under the $\beta$-monotonicity. We then propose a monotonicity condition for the MFG, namely partitioning the running cost functional into two parts, so that both parts still depend on the control and the state distribution, yet one satisfies a strong convexity and a small mean field effect condition, while the other has a newly introduced displacement quasi-monotonicity. To the best of our knowledge, the latter quasi type condition has not yet been discussed in the contemporary literature, and it can be considered as a bit more general monotonicity condition than those commonly used. Besides, for the MFG, we show that convexity and small mean field effect condition for the first part of running cost functional and the quasi-monotonicity condition for the second part together imply the $\beta$-monotonicity and thus the well-posedness for the associated FBSDEs. For the MFTC problem, we show that the $\beta$-monotonicity for the corresponding FBSDEs is simply the convexity assumption on the cost functional. Finally, we consider a more general setting where the drift functional is allowed to be non-linear for both MFG and MFTC problems.

Discussion (0). Continue with ORCID to comment.

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

Control on Hilbert Space and Mean Field Control: the Common Noise Case
math.OC 2025-02 conditional novelty 6.0 of 10

Under convexity and monotonicity assumptions, the common-noise mean-field control value function satisfies a Bellman equation whose gradient yields the master equation.

Reference graph

Works this paper leans on

43 extracted references · 42 canonical work pages · cited by 1 Pith paper

[1]

Ahuja , Wellposedness of mean ﬁeld games with common noise under a w eak monotonicity condition

S. Ahuja , Wellposedness of mean ﬁeld games with common noise under a w eak monotonicity condition. SIAM J. Control Optim. 54 (2016), no. 1, 30–48

work page 2016
[2]

Ahuja, W

S. Ahuja, W. Ren and T. W. Yang , Forward-backward stochastic diﬀerential equations with monotone functionals and mean ﬁeld games with common noise. Stochastic Process. Appl. 129 (2019), no. 10, 3859–3892

work page 2019
[3]

Ambrosio, N

L. Ambrosio, N. Gigli and G. Savar ´e, Gradient ﬂows in metric spaces and in the space of probabilit y measures. second edition, Lectures in Mathematics ETH Z¨ urich, Birkh ¨ auser, Basel, 2008

work page 2008
[4]

Bensoussan, J

A. Bensoussan, J. Frehse and S. C. P. Yam , Mean ﬁeld games and mean ﬁeld type control theory . Springer- Briefs in Mathematics, Springer, New York, 2013

work page 2013
[5]

Bensoussan, P

A. Bensoussan, P. J. Graber and S. C. P. Yam , Control on Hilbert spaces and application to some mean ﬁeld type control problems. Ann. Appl. Probab. 34 (2024), no. 4, 4085–4136

work page 2024
[6]

Degenerate Mean Field Type Control with Linear and Unbounded Diffusion, and their Associated Equations

A. Bensoussan, Z. HUang, S. Tang and S. C. P. Yam , Degenerate mean ﬁeld type control with linear and unbounded diﬀusion, and their associated equations. 2023, arxiv: 2311.09138

work page Pith review arXiv 2023
[7]

Bensoussan, Z

A. Bensoussan, Z. HUang, S. Tang and S. C. P. Yam , A class of degenerate mean ﬁeld games, associated FBSDEs and master equations. 2024, arxiv: 2410.12404

work page arXiv 2024
[8]

Bensoussan, Z

A. Bensoussan, Z. HUang and S. C. P. Yam , Control theory on Wasserstein space: a new approach to optimality conditions. Ann. Math. Sci. Appl. 8 (2023), no. 3, 565–628

work page 2023

Show all 43 references

[9]

Bensoussan, Z

A. Bensoussan, Z. HUang and S. C. P. Yam , Maximum principle for mean ﬁeld type control problems with general volatility functions. Int. Game Theory Rev. 26 (2024), no. 2, Paper No. 2440003, 31 pp

2024
[10]

Bensoussan, H

A. Bensoussan, H. M. Tai and S. C. P. Yam , Mean ﬁeld type control problems, some Hilbert-space-valu ed FBSDEs, and related equations. 2023, arxiv: 2305.04019

2023 arXiv
[11]

Bensoussan, H

A. Bensoussan, H. M. Tai, T. K. Wong and S. C. P. Yam , A control theoretical approach to mean ﬁeld games and associated master equations. 2024, arxiv: 2402.0 1639

2024
[12]

Bensoussan, T

A. Bensoussan, T. K. Wong, S. C. P. Yam and H. Yuan , Global well-posedness of ﬁrst-order mean ﬁeld games and master equations with nonlinear dynamics. 2023, a rxiv: 2311.11896. 34

2023 arXiv
[13]

Bensoussan, T

A. Bensoussan, T. K. Wong, S. C. P. Yam and H. Yuan , A theory of ﬁrst order mean ﬁeld type control problems and their equations. 2023, arxiv: 2305.11848

2023 arXiv
[14]

Bensoussan and S

A. Bensoussan and S. C. P. Yam , Control problem on space of random variables and master equ ation. ESAIM Control Optim. Calc. Var. 25 (2019), Paper No. 10, 36 pp

2019
[15]

Briani and P

A. Briani and P. Cardaliaguet , Stable solutions in potential mean ﬁeld game systems. NoDEA Nonlinear Diﬀerential Equations Appl. 25 (2018), no. 1, Paper No. 1, 26 pp

2018
[16]

Buckdahn, J

R. Buckdahn, J. Li, S. Peng and C. Rainer , Mean-ﬁeld stochastic diﬀerential equations and associat ed PDEs. Ann. Probab. 45 (2017), no. 2, 824–878

2017
[17]

Cardaliaguet , Notes on mean ﬁeld games, from P

P. Cardaliaguet , Notes on mean ﬁeld games, from P. L. Lions’ lectures at colle ge de France, 2013, www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf

2013
[18]

Cardaliaguet, M

P. Cardaliaguet, M. Cirant and A. Porretta , Splitting methods and short time existence for the master equations in mean ﬁeld games. J. Eur. Math. Soc. (JEMS) 25 (2023), no. 5, 1823–1918

2023
[19]

Cardaliaguet, F

P. Cardaliaguet, F. Delarue, J. M. Lasry and P. L. Lions , The master equation and the convergence problem in mean ﬁeld games . Annals of Mathematics Studies, 201, Princeton Univ. Press , Princeton, NJ, 2019

2019
[20]

Cardaliaguet, P

P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon , Second order mean ﬁeld games with degen- erate diﬀusion and local coupling. NoDEA Nonlinear Diﬀerential Equations Appl. 22 (2015), no. 5, 1287–1317

2015
[21]

R. A. Carmona and F. Delarue , Forward-backward stochastic diﬀerential equations and c ontrolled McKean- Vlasov dynamics. Ann. Probab. 43 (2015), no. 5, 2647–2700

2015
[22]

R. A. Carmona and F. Delarue , Probabilistic theory of mean ﬁeld games with applications. I–II. Probability Theory and Stochastic Modelling, 83–84, Springer, Cham, 20 18
[23]

J. F. Chassagneux, D. Crisan and F. Delarue , A probabilistic approach to classical solutions of the mas ter equation for large population equilibria. Mem. Amer. Math. Soc. 280 (2022), no. 1379, v+123 pp

2022
[24]

M. F. Djete, D. Possama ¨ı and X. Tan, McKean-Vlasov optimal control: the dynamic programming p rinciple. Ann. Probab. 50 (2022), no. 2, 791–833

2022
[25]

Gangbo and A

W. Gangbo and A. R. M ´esz´aros, Global well-posedness of master equations for determinis tic displacement convex potential mean ﬁeld games. Comm. Pure Appl. Math. 75 (2022), no. 12, 2685–2801

2022
[26]

Gangbo, A

W. Gangbo, A. R. M ´esz´aros, C. Mou and J. Zhang , Mean ﬁeld games master equations with nonseparable Hamiltonians and displacement monotonicity. Ann. Probab. 50 (2022), no. 6, 2178–2217

2022
[27]

D. A. Gomes, E. A. Pimentel and V. Voskanyan , Regularity theory for mean-ﬁeld game systems . Springer- Briefs in Mathematics, Springer, Cham, 2016

2016
[28]

P. J. Graber and A. R. M ´esz´aros, On monotonicity conditions for mean ﬁeld games. J. Funct. Anal. 285 (2023), no. 9, Paper No. 110095, 45 pp

2023
[29]

Hu and S

Y. Hu and S. Peng , Solution of forward-backward stochastic diﬀerential equ ations. Probab. Theory Related Fields 103 (1995), no. 2, 273–283

1995
[30]

Huang, P

M. Huang, P. E. Caines and R. P. Malham ´e, Large-population cost-coupled LQG problems with nonunif orm agents: individual-mass behavior and decentralized ǫ-Nash equilibria. IEEE Trans. Automat. Control 52 (2007), no. 9, 1560–1571

2007
[31]

Huang, R

M. Huang, R. P. Malham ´e and P. E. Caines , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence p rinciple. Commun. Inf. Syst. 6 (2006), no. 3, 221–251

2006
[32]

Huang and S

Z. Huang and S. Tang , Fully coupled nonlocal quasilinear forward-backward par abolic equations arising from mean ﬁeld games. 2021, arxiv:2105.13213

2021 arXiv
[33]

Huang and S

Z. Huang and S. Tang , Mean ﬁeld games with common noises and conditional distrib ution dependent FBSDEs. Chinese Ann. Math. Ser. B 43 (2022), no. 4, 523–548

2022
[34]

Huang and S

Z. Huang and S. Tang , State-density ﬂows of non-degenerate density-dependent mean ﬁeld SDEs and associ- ated PDEs. Trans. Amer. Math. Soc. 376 (2023), no. 3, 1553–1599

2023
[35]

J. M. Lasry and P. L. Lions , Jeux ` a champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 619–625

2006
[36]

J. M. Lasry and P. L. Lions , Jeux ` a champ moyen. II. Horizon ﬁni et contrˆ ole optimal. C. R. Math. Acad. Sci. Paris 343 (2006), no. 10, 679–684. 35

2006
[37]

J. M. Lasry and P. L. Lions , Mean ﬁeld games. Jpn. J. Math. 2(2007), no. 1, 229–260

2007
[38]

Mou and J

C. Mou and J. Zhang , Mean ﬁeld game master equations with anti-monotonicity co nditions. J. Eur. Math. Soc. (2024)

2024
[39]

Orrieri, A

C. Orrieri, A. Porretta and G. Savar ´e, A variational approach to the mean ﬁeld planning problem. J. Funct. Anal. 277 (2019), no. 6, 1868–1957

2019
[40]

Peng and Z

S. Peng and Z. Wu , Fully coupled forward-backward stochastic diﬀerential e quations and applications to optimal control. SIAM J. Control Optim. 37 (1999), no. 3, 825–843

1999
[41]

Pham and X

H. Pham and X. Wei , Dynamic programming for optimal control of stochastic McK ean-Vlasov dynamics. SIAM J. Control Optim. 55 (2017), no. 2, 1069–1101

2017
[42]

Porretta , Weak solutions to Fokker-Planck equations and mean ﬁeld ga mes

A. Porretta , Weak solutions to Fokker-Planck equations and mean ﬁeld ga mes. Arch. Ration. Mech. Anal. 216 (2015), no. 1, 1–62

2015
[43]

Yong and X

J. Yong and X. Y. Zhou , Stochastic controls. Hamiltonian systems and HJB equation s. Applications of Math- ematics (New York), 43, Springer, New York, 1999. 36

1999

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Ahuja , Wellposedness of mean ﬁeld games with common noise under a w eak monotonicity condition

S. Ahuja , Wellposedness of mean ﬁeld games with common noise under a w eak monotonicity condition. SIAM J. Control Optim. 54 (2016), no. 1, 30–48

work page 2016

[2] [2]

Ahuja, W

S. Ahuja, W. Ren and T. W. Yang , Forward-backward stochastic diﬀerential equations with monotone functionals and mean ﬁeld games with common noise. Stochastic Process. Appl. 129 (2019), no. 10, 3859–3892

work page 2019

[3] [3]

Ambrosio, N

L. Ambrosio, N. Gigli and G. Savar ´e, Gradient ﬂows in metric spaces and in the space of probabilit y measures. second edition, Lectures in Mathematics ETH Z¨ urich, Birkh ¨ auser, Basel, 2008

work page 2008

[4] [4]

Bensoussan, J

A. Bensoussan, J. Frehse and S. C. P. Yam , Mean ﬁeld games and mean ﬁeld type control theory . Springer- Briefs in Mathematics, Springer, New York, 2013

work page 2013

[5] [5]

Bensoussan, P

A. Bensoussan, P. J. Graber and S. C. P. Yam , Control on Hilbert spaces and application to some mean ﬁeld type control problems. Ann. Appl. Probab. 34 (2024), no. 4, 4085–4136

work page 2024

[6] [6]

Degenerate Mean Field Type Control with Linear and Unbounded Diffusion, and their Associated Equations

A. Bensoussan, Z. HUang, S. Tang and S. C. P. Yam , Degenerate mean ﬁeld type control with linear and unbounded diﬀusion, and their associated equations. 2023, arxiv: 2311.09138

work page Pith review arXiv 2023

[7] [7]

Bensoussan, Z

A. Bensoussan, Z. HUang, S. Tang and S. C. P. Yam , A class of degenerate mean ﬁeld games, associated FBSDEs and master equations. 2024, arxiv: 2410.12404

work page arXiv 2024

[8] [8]

Bensoussan, Z

A. Bensoussan, Z. HUang and S. C. P. Yam , Control theory on Wasserstein space: a new approach to optimality conditions. Ann. Math. Sci. Appl. 8 (2023), no. 3, 565–628

work page 2023