A study of common noise in mean field games

Charles Bertucci (CMAP); Charles Meynard (CMAP)

REVIEW 2 major objections 6 minor 1 cited by

A study of common noise in mean field games

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that mean-field games with endogenous common noise admit unique global-in-time Lipschitz solutions under joint monotonicity conditions, and that additive common noise can be included by a shift transformation.

desk verdict Genuinely new results on MFG master equations with endogenous common noise, but the L2 well-posedness proof rests on a comparison principle that is not rigorously established. read the letter →

arxiv 2412.12741 v1 pith:2XJ2M472 submitted 2024-12-17 math.AP

Charles Meynard (CMAP) , Charles Bertucci (CMAP) This is my paper · ORCID

classification math.AP MSC 35Q8949N8091A16

keywords meanfieldgamesmasterequationcommonnoiseendogenousLipschitzsolutionsflatmonotonicitydisplacementglobalwell-posedness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that mean field games remain solvable over arbitrary time horizons when the common noise is endogenous: a random variable observed by all players whose drift may depend on the distribution of players or on the value function. It proves that the associated master equation has a unique global-in-time Lipschitz solution, provided a joint monotonicity condition couples the noise drift to the game data. The result is obtained in two standard regimes of monotonicity, flat and L2/displacement, and it extends to additive common noise through a change of variables that removes the second-order derivative in the measure argument. A careful reader would care because this is a systematic treatment of noise that feeds back into the game, and because the paper identifies the structural assumption that prevents finite-time blow-up.

What carries the argument

The engine is the notion of Lipschitz solution, a solution of the differentiated master equation obtained as a fixed point of a stochastic characteristic flow, requiring only Lipschitz regularity of the gradient. Blow-up control is carried by a penalized monotonicity functional: for two copies $(\theta,\mu)$ and $(\tilde\theta,\nu)$, the paper studies $$\tfrac{1}{2}(\$\theta$-\tilde\$\theta$)^\top A(\$\theta$-\tilde\$\theta$)+\langle U(\cdot,\$\theta$,\mu)-U(\cdot,\tilde\$\theta$,\nu),\mu-\nu\rangle$$ and proves through a comparison principle and a weakly singular integral inequality that this quantity stays nonnegative with a quantitative lower bound. Nonnegativity yields Lipschitz estimates in the noise variable and in the Wasserstein metric, preventing the fixed-point norm from blowing up. For additive common noise, the key object is the shift transformation $U(t,x,\theta,m)=V(t,x-\theta,\theta,(\mathrm{id}-\theta)_\# m)$, which turns the second-order measure term into a finite-dimensional Laplacian.

What would settle it

The sharp place to refute the central claim is the boundary of the joint monotonicity hypotheses: take smooth data satisfying all regularity assumptions and with the inequalities (3.3)--(3.4) or (4.20)--(4.21) holding at equality, then compute the quantity in (3.5) or (4.18) along the local Lipschitz solution; if this quantity becomes negative before the local existence time, the propagation lemma fails and the proof of Theorem 3.12 or Theorem 4.30 cannot hold.

Watch

Extended reading notes

Core claim

The central claim is that the master equation with an extra finite-dimensional common-noise variable theta, whose drift may depend on the measure or on the value function, is globally well-posed in the class of Lipschitz solutions under appropriate monotonicity hypotheses. In the flat monotone regime, Theorem 3.12 gives a unique global-in-time Lipschitz solution under Hypothesis 3.8; in the L2/displacement regime, Theorem 4.30 gives the same conclusion under Hypotheses 4.22 and 4.26. Corollary 5.4 then transfers these results to the full equation with additive common noise, even when the additive noise is correlated with the noise driving theta. Along the way, the paper shows that propagation of L2-monotonicity is equivalent to the Hilbertian displacement-monotone approach, and that the resulting solution serves as the decoupling field of a mean-field forward-backward stochastic differential equation.

Load-bearing premise

The global existence results rest on joint monotonicity conditions coupling the noise drift with the other coefficients, conditions that the paper itself notes are strong and generally require either the payoff or the noise drift to be strongly monotone.

Editorial extensions

If this is right

For any finite horizon, the master equation with a common noise variable whose drift depends on the distribution or the value function has a unique Lipschitz solution in the flat monotone regime.
The same global wellposedness holds in the L2/displacement monotone regime, and the result also covers mean-field FBSDE decoupling fields and some extended mean field games.
Additive common noise, even correlated with the Brownian motion driving theta, can be absorbed through a change of variables and does not destroy global existence.
When the joint monotonicity hypotheses fail, finite-time blow-up is possible, so the hypotheses are not merely technical additions.
In the L2-monotone setting, the constructed solution is the decoupling field of the associated mean-field forward-backward system, yielding an existence result for those systems.

Reading between the lines

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

The paper does not address N-player convergence; a natural next step is to test whether these global Lipschitz solutions are the limit of symmetric N-player equilibria as the number of players grows.
The penalization device $\theta \mapsto A\theta$ is the simplest case of completing the value function with an auxiliary variable so that the pair is jointly monotone; other choices of auxiliary variable could relax the joint monotonicity conditions and connect to G-monotonicity for FBSDEs.
Because the shift transformation removes the second-order measure derivative, comparison-based weak solution theories for master equations with additive common noise could be developed without second derivatives in the measure argument, a direction the paper only sketches.
The condition in Example 4.28 is invariant under rescaling the noise drift, suggesting that the geometry of the coupling, rather than the strength of the noise, is what guarantees global existence.

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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

Colleague—

The paper is a real step forward: it extends the authors' finite-state work on endogenous common noise in MFG master equations to continuous state space, covering both flat and L2/displacement monotone regimes, and shows how additive common noise folds into the same framework. The main theorems (3.12, 4.30, Cor 5.4) are substantial, the examples are meaningful, and the exposition is clear about what is new. I did not see circular reasoning, fitting, or invented entities.

The soft spot is in the L2 section. Lemma 4.13, which gives the non-negativity of Z_beta and underpins Theorem 4.30, uses a test function with cubic growth in |y| and E_3^3. But Definition 4.10's Htest class only allows measure derivatives of linear growth (q=2). The cubic test is therefore outside the admissible class. Remark 4.12 claims local behaviour near the minimum justifies it, but that remark does not supply the missing uniform bounds needed for the dominated convergence argument in Lemma 4.11, nor a rigorous comparison principle for unbounded test functions along the lines of Crandall-Lions or Ishii. Without that, the contradiction inequality 0 >= alpha/(t-s*)^2 + alpha e^{kappa s*}(kappa-c)(1+...) is not actually derived. This is a load-bearing gap, not a cosmetic one, since all the global L2 Lipschitz bounds rely on it.

The same pattern shows up elsewhere: Lemma 3.6 is deferred to 'a proof similar to Lemma 2.6', Lemma 4.25 says 'no particular difficulty', and Corollary 5.4 says 'no difficulty in extending'. Those are probably fine, but they are exactly where a referee should press. The self-citation footprint is high, but the cited machinery is real and foundational; that is not the issue. The monotonicity assumptions are restrictive—the authors admit this in Remarks 3.9 and 4.27—but that is a limitation, not a flaw.

Who is this for: anyone working on master equations with common noise, or on MFG with distribution-dependent noise. The paper deserves a serious referee; the gap is fixable and the results are worth having. I would send it out, with a clear request for a rigorous treatment of the comparison principle in Lemma 4.13.

Referee Report

2 major / 6 minor

Summary. This paper studies the master equation of mean field games with an additional common noise variable θ that may itself be affected by the distribution of players or by the value function. The main results are global-in-time well-posedness for Lipschitz solutions: Theorem 3.12 under flat (Lasry-Lions) monotonicity with a joint monotonicity condition (Hypothesis 3.8), Theorems 4.15 and 4.30 under L2/displacement-type monotonicity (Hypotheses 4.7 and 4.26), and Corollary 5.4 extending these to the full master equation with additive common noise. The proofs adapt the method of Lipschitz solutions and rely on propagation of monotonicity via maximum-principle/comparison arguments for auxiliary functions Z and Zβ. The paper also discusses the link between the Hilbertian L2 approach and displacement monotonicity, and gives applications to mean field FBSDEs.

Significance. If the results are correct, the paper makes a solid contribution to the study of master equations with endogenous common noise, a setting where finite-time blow-up can occur without structural assumptions. The main theorems are clearly stated and cover two important monotonicity regimes. The paper is honest about the restrictive nature of the joint monotonicity hypotheses (Remarks 3.9 and 4.27) and provides nontrivial examples (Examples 3.10 and 4.28) where they hold. Section 5's reduction of additive common noise to a finite-dimensional variable is elegant and avoids second-order derivatives in the measure variable. The principal weakness is that a central comparison argument in Section 4 is incomplete, which affects the L2-monotonicity part of the paper.

major comments (2)

[Section 4.3.1, Lemma 4.13 and Remark 4.12] The proof of non-negativity of Zβ uses the test function αe^{κs}(1+|θ|^3+E_3^3(γ))+α/(t−s), whose measure derivative has cubic growth. Definition 4.10 restricts test functions to Htest, where |Dmϕ|+|DyDmϕ| ≤ C(1+|y|^{q-1}) with q=2 in the present P2 setting, i.e., linear growth. The assertion in Remark 4.12 that it suffices to check the Htest conditions only in a neighbourhood of the minimum is not justified, since the support of the minimizer γ* is not bounded and the cubic derivative can be arbitrarily large on that support. The contradiction inequality 0 ≥ α/(t−s*)² + αe^{κs*}(κ−c)(1+|θ*|³+E_3³(γ*)) relies exactly on those cubic terms. Since Lemma 4.14, Theorem 4.15 and Theorem 4.30 all depend on the non-negativity of Zβ, this comparison step is load-bearing and must be rigorously established, e.g., by an approximation with truncated moments or by an unbounded viscosity comparison principle in the spirit of [20,21,25].
[Section 4.3.1 (Lemma 4.25) and Section 5 (Corollary 5.4)] The proofs of Lemma 4.25 and Corollary 5.4 are only sketched. Lemma 4.25 states that Z^A_β is a viscosity supersolution and its proof says 'there is no particular difficulty in extending the computations of Lemma 4.11', but the extension involves a doubling of variables in θ and a non-autonomous drift b[W]; the details are needed to verify the correct form of the second-order operator, the cross terms, and the subsequent comparison principle in Lemma 4.29. Similarly, Corollary 5.4 says 'there is no difficulty in extending the arguments of section 4.3' while the equation contains additional second-order terms in the measure argument and cross derivatives; a proof should be supplied. Because the main theorem for the full equation (1.2) depends on these steps, the current exposition is insufficient. These omissions should be addressed in a revision.

minor comments (6)

[Abstract and throughout] There are several typos: 'mast er equation' in the abstract, 'expending' for 'expanding' in Section 1.3, 'Form' for 'For' in Lemma 4.3, and 'exemple'/'fonction' in the footnote of Theorem 4.30.
[Section 2.1.2] The passage from the W1-based Lipschitz solutions of [8] to the Wq-based setting is said to be 'very straightforward'; a few details on the contraction argument in the Wq norm would help the reader.
[Section 3.1.1, Lemma 3.6] The proof delegates to Lemma 2.6 for the derivation of (3.2), but Lemma 2.6 concerns the identification of ∇xU with W, not the two-measure monotonicity estimate; a direct derivation would improve verifiability.
[Section 4.1.2, Remark 4.12] Remark 4.12 is the only justification for using cubic test functions with the Htest framework; it should be expanded into a rigorous lemma, or the comparison proof should be modified.
[References] References [12] and [13] are identical; they should be merged or differentiated.
[Section 3] The notation P(Td) is used without specifying the integrability class; in the flat monotone section with W1, it would be clearer to write P_1(Td).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the global-existence theorems are proved from stated monotonicity data hypotheses; the cited prior Lipschitz-solution framework supplies local existence but not the conclusions.

full rationale

No circular step could be identified. The paper's central results (Theorems 3.12, 4.30, and Corollary 5.4) are proved from explicit monotonicity assumptions on the data (Hypotheses 3.8, 4.26) via propagation lemmas for auxiliary quantities Z, Z_beta, and Z_A_beta; these quantities are not defined in terms of the desired global-existence conclusion, and their non-negativity is proved rather than assumed. The local-in-time Lipschitz-solution framework is imported from the published prior work [8] and adapted from W1 to Wq in Theorem 2.3, and the finite-state method is credited to [7]; these citations carry independent content and are not used to assume the present theorems' conclusions. The monotonicity hypotheses are restrictive, as the paper itself concedes in Remarks 3.9 and 4.27, but restrictiveness is not circularity. The skeptic's concern about Lemma 4.13 is a potential correctness gap: the comparison argument uses a cubic test function that is not obviously in the Htest class of Definition 4.10, and Remark 4.12's justification is brief. A gap in a proof is not, however, a reduction of the result to its inputs by construction. Since no fitted parameter is renamed as a prediction, no equation coincides with an input by definition, and no load-bearing uniqueness theorem is imported solely from the authors' own prior work, the derivation chain is self-contained in the circularity sense.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new ontological entities; theta is a finite-dimensional common noise variable already present in the model. All assumptions are clearly stated monotonicity and regularity conditions on the data, which are necessary for the theorems and are acknowledged by the authors as strong in several places.

assumptions (9)

standard math Standard probability space with Brownian motions (Section 1.5)
The probabilistic representation of Lipschitz solutions relies on this background.
domain assumption Hypothesis 2.2: Lipschitz regularity of U0, DpH, DxH, and b with respect to x, theta, m, p
Gives local existence of Lipschitz solutions via fixed point (Theorem 2.3).
domain assumption Hypothesis 3.5: separated Hamiltonian H = Hbar - f, alpha-H-convex Hbar, flat monotone U0 and f
Key condition for propagation of flat monotonicity in Section 3.1.1.
domain assumption Hypothesis 3.8: joint flat monotonicity coupling f and b through matrix A (equations 3.3 and 3.4)
Load-bearing for Theorem 3.12; the paper acknowledges this is strong (Remark 3.9).
domain assumption Hypothesis 4.1: Lipschitz F, G, W0 and b in W2 metric
Ensures local existence for the general system in Section 4.
domain assumption Hypothesis 4.7: strong joint L2-monotonicity of W0 and of (F,G) (equations 4.5 and 4.6)
Propagated via Z_beta auxiliary function to yield global existence in Theorem 4.15.
domain assumption Hypothesis 4.22: Lipschitz regularity of b with functional dependence measured in W2 with respect to the coupling
Needed to bound b differences in the stability estimates for Theorem 4.30.
domain assumption Hypothesis 4.26: joint L2-monotonicity coupling b with (F,G) through matrix A (equations 4.20 and 4.21)
Central load-bearing condition for Theorem 4.30; the paper provides a concrete condition in Example 4.28.
domain assumption Lipschitz solution framework from [8]
The notion of solution and local well-posedness are imported from the authors' earlier work.

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Pith. "Pith review of A study of common noise in mean field games." pith.science (2026). https://pith.science/paper/2XJ2M472

@misc{pith2026241212741,
  author       = {Pith},
  title        = {Pith review of: A study of common noise in mean field games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XJ2M472}},
  note         = {Machine review of arXiv:2412.12741}
}

read the original abstract

This paper is concerned with the study of mean field games master equations involving an additional variable modelling common noise. We address cases in which the dynamics of this variable can depend on the state of the game, which requires in general additional monotonicity assumptions on the coefficients. We explore the link between such a common noise and more traditional ones, as well as the links between different monotone regimes for the master equation.

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Analysis on spaces of measures
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Works this paper leans on

39 extracted references · 37 canonical work pages · cited by 1 Pith paper

[1]

A User’s Guide to Opt imal Transport

Ambrosio, Luigi and Gigli, Nicola. “A User’s Guide to Opt imal Transport”. Modelling and Optimisation of Flows on Networks: Cetraro, Italy 2009, Editors: Benedet to Piccoli, Michel Rascle . Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 1–155. isbn: 978-3-642-32160-3

work page 2009
[2]

Well-posedn ess of mean-ﬁeld type forward–backward stochastic diﬀerential equations

Bensoussan, A., Yam, S.C.P., and Zhang, Z. “Well-posedn ess of mean-ﬁeld type forward–backward stochastic diﬀerential equations”. Stochastic Processes and their Applications 125.9 (2015), pp. 3327–3354

work page 2015
[3]

The Master equation in mean ﬁeld theory

Bensoussan, Alain, Frehse, Jens, and Yam, Sheung Chi Phi llip. “The Master equation in mean ﬁeld theory”. Journal de Mathématiques Pures et Appliquées 103.6 (2015), pp. 1441–1474

work page 2015
[4]

Monotone solutions for mean ﬁeld games mas ter equations: continuous state space and common noise

Bertucci, C. “Monotone solutions for mean ﬁeld games mas ter equations: continuous state space and common noise”. Communications in Partial Diﬀerential Equations 48.10-12 (2023), pp. 1245–1285

work page 2023
[5]

Monotone solutions for mean ﬁeld games mas ter equations: ﬁnite state space and optimal stopping

Bertucci, C. “Monotone solutions for mean ﬁeld games mas ter equations: ﬁnite state space and optimal stopping”. Journal de l’École polytechnique — Mathématiques 8 (2021), pp. 1099–1132

work page 2021
[6]

A Mean Field Game Approach to Bitcoin Mining

Bertucci., C., Bertucci, L., Lasry, J.-M., and Lions, P. -L. “A Mean Field Game Approach to Bitcoin Mining”. SIAM Journal on Financial Mathematics 15.3 (2024), pp. 960–987

work page 2024
[7]

Noise through on additionn al variable for mean ﬁeld games master equation on ﬁnite state space

Bertucci, C. and Meynard, C. “Noise through on additionn al variable for mean ﬁeld games master equation on ﬁnite state space” (2024). arXiv: 2402.05635

arXiv 2024
[8]

On Lipschitz solutions of mean ﬁeld games master equations

Bertucci, Charles, Lasry, Jean-Michel, and Lions, Pier re-Louis. “On Lipschitz solutions of mean ﬁeld games master equations”. Journal of Functional Analysis 287.5 (2024), p. 110486

work page 2024

Show all 39 references

[9]

A two spaces extension of Cauchy-Lipschitz Theorem

Bertucci, Charles and Lions, Pierre Louis. A two spaces extension of Cauchy-Lipschitz Theorem . 2024. arXiv: 2402.19092

2024 arXiv
[10]

The master equation and the convergence problem in mean ﬁeld games

Cardaliaguet, P., Delarue, F., Lasry, J.-M., and Lions , P.-L. “The master equation and the convergence problem in mean ﬁeld games”. Princeton University Press (2019)

2019
[11]

Mean Field Game of Controls and An Application To Trade Crowding

Cardaliaguet, P. and Lehalle, C.-A. “Mean Field Game of Controls and An Application To Trade Crowding”. Mathematics and Financial Economics 12.3 (2019), pp. 335–363

2019
[13]

Mean ﬁeld games with common noise and degenerate idiosyncratic noise

Cardaliaguet, Pierre, Seeger, Benjamin, and Souganid is, Panagiotis. Mean ﬁeld games with common noise and degenerate idiosyncratic noise . 2022. arXiv: 2207.10209

2022 arXiv
[14]

Mo notone Solutions of the Master Equation for Mean Field Games with Idiosyncratic Noise

Cardaliaguet, Pierre and Souganidis, Panagiotis. “Mo notone Solutions of the Master Equation for Mean Field Games with Idiosyncratic Noise”. SIAM Journal on Mathematical Analysis 54.4 (2022), pp. 4198–4237

2022
[15]

Mean ﬁeld forwar d-backward stochastic diﬀerential equations

Carmona, René and Delarue, François. “Mean ﬁeld forwar d-backward stochastic diﬀerential equations”. Elec- tronic Communications in Probability 18 (2013), pp. 1 –15

2013
[16]

Probabilistic Theory of Mean Field Games with Applications II

Carmona, René and Delarue, François. Probabilistic Theory of Mean Field Games with Applications II. Ed. by Springer. 2018

2018
[17]

Mean ﬁeld games with common noise

Carmona, René, Delarue, François, and Lacker, Daniel. “Mean ﬁeld games with common noise”. The Annals of Probability 44.6 (2016), pp. 3740 –3803

2016
[18]

A probabilistic appro ach to mean ﬁeld games with major and minor players

Carmona, René and Zhu, Xiuneng. “A probabilistic appro ach to mean ﬁeld games with major and minor players”. The Annals of Applied Probability 26.3 (2016), pp. 1535–1580. issn: 10505164

2016
[19]

Weak solutions to the maste r equation of potential mean ﬁeld games

Cecchin, A. and Delarue, F. “Weak solutions to the maste r equation of potential mean ﬁeld games” (2022). arXiv: 2204.04315

2022 arXiv
[20]

Hamilto n-Jacobi equations in inﬁnite dimensions I. Uniqueness of viscosity solutions

Crandall, Michael G and Lions, Pierre-Louis. “Hamilto n-Jacobi equations in inﬁnite dimensions I. Uniqueness of viscosity solutions”. Journal of Functional Analysis 62.3 (1985), pp. 379–396

1985
[21]

Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations

Crandall, Michael G. and Louis Lions, pierre. “Remarks on the existence and uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations”. Illinois Journal of Mathematics 31.4 (1987), pp. 665 –688

1987
[22]

The covariation for Banach space valued processes and applications

Di Girolami, Cristina, Fabbri, Giorgio, and Russo, Fra ncesco. “The covariation for Banach space valued processes and applications”. Metrika 77 (Jan. 2013)

2013
[23]

Mean Fi eld Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity

Gangbo, W., Mészáros, A., Mou, C., and Zhang, J. “Mean Fi eld Games Master Equations with Non-separable Hamiltonians and Displacement Monotonicity”. Annals of Probability 50 (2022)

2022
[24]

On monotonic ity conditions for mean ﬁeld games

Graber, P. Jameson and Mészáros, Alpár R. “On monotonic ity conditions for mean ﬁeld games”. Journal of Functional Analysis 285.9 (2023), p. 110095

2023
[25]

Uniqueness of Unbounded Viscosity Solution of Hamilton-Jacobi Equations

Ishii, H. “Uniqueness of Unbounded Viscosity Solution of Hamilton-Jacobi Equations”. Indiana University Mathematics Journal 33.5 (1984), pp. 721–748

1984
[26]

Convex analysis and minimization algorithms

Jean-Baptiste, Hiriart-Urruty. Convex analysis and minimization algorithms. II, advanced theory and bundle methods. Grundlehren der mathematischen Wissenschaften 306. Berl in New York: Springer, 1993

1993
[27]

Stochastic In tegration

Karatzas, Ioannis and Shreve, Steven E. “Stochastic In tegration”. Brownian Motion and Stochastic Calculus . New York, NY: Springer New York, 1998, pp. 128–238. REFERENCES 37

1998
[28]

On the convergence of closed-loop Nas h equilibria to the mean ﬁeld game limit

Lacker, Daniel. “On the convergence of closed-loop Nas h equilibria to the mean ﬁeld game limit”. Annals of Applied Probability 30 (2020), pp. 1693–1761

2020
[29]

Mean-ﬁeld games with a ma jor player

Lasry, J.-M. and Lions, P.-L. “Mean-ﬁeld games with a ma jor player”. Comptes Rendus Mathematique 356.8 (2018), pp. 886–890

2018
[30]

Mean ﬁel d games

Lasry, Jean-Michel and Lions, Pierre-Louis. “Mean ﬁel d games”. Japanese Journal of Mathematics 2.3 (2007), pp. 229–260

2007
[31]

Lectures at Collège de France

Lions, P.-L. “Lectures at Collège de France” (2007-201 1). url: http://www.college-de-france.fr

2007
[32]

“Viscosity solutions of fully nonlinear s econd-order equations and optimal stochastic control in inﬁnite dimensions

Lions, P.-L. “Viscosity solutions of fully nonlinear s econd-order equations and optimal stochastic control in inﬁnite dimensions. III. Uniqueness of viscosity soluti ons for general second-order equations”. Journal of Functional Analysis 86.1 (1989), pp. 1–18

1989
[33]

Extended mean-ﬁeld ga mes

Lions, P.-L. and Souganidis, P. “Extended mean-ﬁeld ga mes”. Rendiconti Lincei 3 (2020), 611–625

2020
[34]

Bansil, Alpár R

M. Bansil, Alpár R. Mészáros and Mou, C. Global Well-Posedness of Displacement Monotone Degenerat e Mean Field Games Master Equations . 2023. arXiv: 2308.16167

2023 arXiv
[35]

Wellposedness of S econd Order Master Equations for Mean Field Games with Nonsmooth Data

Mou, Chenchen and Zhang, Jianfeng. “Wellposedness of S econd Order Master Equations for Mean Field Games with Nonsmooth Data”. Memoirs of the American Mathematical Society (2019)

2019
[36]

Mean Field Games Systems under Displacement Monotonicity

Mészáros, A. and Mou, C. “Mean Field Games Systems under Displacement Monotonicity”. SIAM Journal on Mathematical Analysis 56.1 (2024), pp. 529–553

2024
[37]

Fully Coupled Forward-Backward Sto chastic Diﬀerential Equations and Applications to Optimal Control

Peng, S. and Wu, Z. “Fully Coupled Forward-Backward Sto chastic Diﬀerential Equations and Applications to Optimal Control”. SIAM Journal on Control and Optimization 37.3 (1999), pp. 825–843

1999
[38]

Convex ity in Hamilton–Jacobi Theory I: Dynamics and Duality

Rockafellar, R. Tyrrell and Wolenski, Peter R. “Convex ity in Hamilton–Jacobi Theory I: Dynamics and Duality”. SIAM Journal on Control and Optimization 39.5 (2000), pp. 1323–1350

2000
[39]

Mean ﬁeld games with common noise via Malliavin calculus

Tangpi, Ludovic and Wang, Shichun. Mean ﬁeld games with common noise via Malliavin calculus . 2024. arXiv: 2405.02244

2024
[40]

Weakly singular Gronwall inequalities a nd applications to fractional diﬀerential equations

Webb, J.R.L. “Weakly singular Gronwall inequalities a nd applications to fractional diﬀerential equations”. Journal of Mathematical Analysis and Applications 471.1 (2019), pp. 692–711

2019

Pith tools

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

[1] [1]

A User’s Guide to Opt imal Transport

Ambrosio, Luigi and Gigli, Nicola. “A User’s Guide to Opt imal Transport”. Modelling and Optimisation of Flows on Networks: Cetraro, Italy 2009, Editors: Benedet to Piccoli, Michel Rascle . Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 1–155. isbn: 978-3-642-32160-3

work page 2009

[2] [2]

Well-posedn ess of mean-ﬁeld type forward–backward stochastic diﬀerential equations

Bensoussan, A., Yam, S.C.P., and Zhang, Z. “Well-posedn ess of mean-ﬁeld type forward–backward stochastic diﬀerential equations”. Stochastic Processes and their Applications 125.9 (2015), pp. 3327–3354

work page 2015

[3] [3]

The Master equation in mean ﬁeld theory

Bensoussan, Alain, Frehse, Jens, and Yam, Sheung Chi Phi llip. “The Master equation in mean ﬁeld theory”. Journal de Mathématiques Pures et Appliquées 103.6 (2015), pp. 1441–1474

work page 2015

[4] [4]

Monotone solutions for mean ﬁeld games mas ter equations: continuous state space and common noise

Bertucci, C. “Monotone solutions for mean ﬁeld games mas ter equations: continuous state space and common noise”. Communications in Partial Diﬀerential Equations 48.10-12 (2023), pp. 1245–1285

work page 2023

[5] [5]

Monotone solutions for mean ﬁeld games mas ter equations: ﬁnite state space and optimal stopping

Bertucci, C. “Monotone solutions for mean ﬁeld games mas ter equations: ﬁnite state space and optimal stopping”. Journal de l’École polytechnique — Mathématiques 8 (2021), pp. 1099–1132

work page 2021

[6] [6]

A Mean Field Game Approach to Bitcoin Mining

Bertucci., C., Bertucci, L., Lasry, J.-M., and Lions, P. -L. “A Mean Field Game Approach to Bitcoin Mining”. SIAM Journal on Financial Mathematics 15.3 (2024), pp. 960–987

work page 2024

[7] [7]

Noise through on additionn al variable for mean ﬁeld games master equation on ﬁnite state space

Bertucci, C. and Meynard, C. “Noise through on additionn al variable for mean ﬁeld games master equation on ﬁnite state space” (2024). arXiv: 2402.05635

arXiv 2024

[8] [8]

On Lipschitz solutions of mean ﬁeld games master equations

Bertucci, Charles, Lasry, Jean-Michel, and Lions, Pier re-Louis. “On Lipschitz solutions of mean ﬁeld games master equations”. Journal of Functional Analysis 287.5 (2024), p. 110486

work page 2024