On the interpretation of the Master Equation

Mean field games

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Mean Field Games and Mean Field Type Control Theory

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Stochastic Hamilton–Jacobi–Bellman Equations

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The Master equation in mean field theory

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The Master Equation for Large Population Equilibriums

Carmona, R. A., & Delarue, F. (2014) · Springer proceedings in mathematics & statistics · doi:10.1007/978-3-319-11292-3_4 · Cited by 118 · see citing works

Dynamic programming for mean-field type control

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THE MASTER EQUATION

van Kampen, N. G. (2007) · Elsevier eBooks · doi:10.1016/b978-044452965-7/50008-8 · Cited by 35 · see citing works

Mean Field Games and Systemic RiskPreprint

Carmona, R. A., Fouque, J., & Sun, L. (2013) · arXiv (Cornell University) · doi:10.48550/arxiv.1308.2172 · Cited by 8 · see citing works

The Master Equation and the Convergence Problem in Mean Field Games

Cardaliaguet, P., Delarue, F., Lasry, J., & Lions, P. (2019) · Princeton University Press eBooks · doi:10.23943/princeton/9780691190716.001.0001 · Cited by 248 · see citing works

The Master Equation and the Convergence Problem in Mean Field Games

Lions, P.-L., Delarue, F., & Lasry, J. (2019) · Princeton University Press eBooks · doi:10.1515/9780691193717 · Cited by 228 · see citing works

Dynamic Programming for Optimal Control of Stochastic McKean--Vlasov Dynamics

Pham, H., & Wei, X. (2017) · SIAM Journal on Control and Optimization · doi:10.1137/16m1071390 · Cited by 168 · see citing works

A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria

Chassagneux, J.-F., Crisan, D., & Delarue, F. (2022) · Memoirs of the American Mathematical Society · doi:10.1090/memo/1379 · Cited by 165 · see citing works

Bellman equation and viscosity solutions for mean-field stochastic control problem

Pham, H., & Wei, X. (2017) · ESAIM Control Optimisation and Calculus of Variations · doi:10.1051/cocv/2017019 · Cited by 126 · see citing works

An Extended Mean Field Game for Storage in Smart Grids

Alasseur, C., Tahar, I. B., & Matoussi, A. (2020) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-019-01619-3 · Cited by 94 · see citing works

From the master equation to mean field game limit theory: a central limit theorem

Delarue, F., Lacker, D., & Ramanan, K. (2019) · Electronic Journal of Probability · doi:10.1214/19-ejp298 · Cited by 81 · see citing works

Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics

Bayraktar, E., Cosso, A., & Pham, H. (2016) · Transactions of the American Mathematical Society · doi:10.1090/tran/7118 · Cited by 75 · see citing works

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

Graber, P. J. (2016) · Applied Mathematics & Optimization · doi:10.1007/s00245-016-9385-x · Cited by 71 · see citing works

McKean–Vlasov optimal control: The dynamic programming principle

Djete, M. F., Possamaï, D., & Tan, X. (2022) · The Annals of Probability · doi:10.1214/21-aop1548 · Cited by 65 · see citing works

From the master equation to mean field game limit theory: Large deviations and concentration of measure

Delarue, F., Lacker, D., & Ramanan, K. (2020) · The Annals of Probability · doi:10.1214/19-aop1359 · Cited by 59 · see citing works

Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space

de Raynal, P.-É. C., & Frikha, N. (2021) · Journal de Mathématiques Pures et Appliquées · doi:10.1016/j.matpur.2021.12.001 · Cited by 59 · see citing works

Numerical method for FBSDEs of McKean–Vlasov type

Chassagneux, J.-F., Crisan, D., & Delarue, F. (2019) · The Annals of Applied Probability · doi:10.1214/18-aap1429 · Cited by 44 · see citing works

Mean–field moral hazard for optimal energy demand response management

Élie, R., Hubert, E., Mastrolia, T., & Possamaï, D. (2020) · Mathematical Finance · doi:10.1111/mafi.12291 · Cited by 43 · see citing works

Quadratic mean field games

Ullmo, D., Swiecicki, I., & Gobron, T. (2019) · Physics Reports · doi:10.1016/j.physrep.2019.01.001 · Cited by 40 · see citing works

Viscosity Solutions for Controlled McKean--Vlasov Jump-Diffusions

Burzoni, M., Ignazio, V., Reppen, M., & Soner, H. M. (2020) · SIAM Journal on Control and Optimization · doi:10.1137/19m1290061 · Cited by 39 · see citing works

Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

Bandini, E., Cosso, A., Fuhrman, M., & Pham, H. (2018) · Stochastic Processes and their Applications · doi:10.1016/j.spa.2018.03.014 · Cited by 35 · see citing works

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

Mou, C., & Zhang, J. (2024) · Memoirs of the American Mathematical Society · doi:10.1090/memo/1515 · Cited by 35 · see citing works

Finite Dimensional Approximations of Hamilton--Jacobi--Bellman Equations in Spaces of Probability Measures

Gangbo, W., Mayorga, S., & Święch, A. (2021) · SIAM Journal on Mathematical Analysis · doi:10.1137/20m1331135 · Cited by 33 · see citing works

Control problem on space of random variables and master equation

Bensoussan, A., & Yam, S. C. P. (2018) · ESAIM Control Optimisation and Calculus of Variations · doi:10.1051/cocv/2018034 · Cited by 31 · see citing works

Mean-Field-Type Games with Jump and Regime Switching

Bensoussan, A., Djehiche, B., Tembiné, H., & Yam, S. C. P. (2019) · Dynamic Games and Applications · doi:10.1007/s13235-019-00306-2 · Cited by 26 · see citing works

Minimal-time mean field games

Mazanti, G., & Santambrogio, F. (2019) · Mathematical Models and Methods in Applied Sciences · doi:10.1142/s0218202519500258 · Cited by 25 · see citing works

Short time solution to the master equation of a first order mean field game

Mayorga, S. (2019) · Journal of Differential Equations · doi:10.1016/j.jde.2019.11.031 · Cited by 25 · see citing works

Itô’s formula for flows of measures on semimartingales

Guo, X., Pham, H., & Wei, X. (2023) · Stochastic Processes and their Applications · doi:10.1016/j.spa.2023.02.004 · Cited by 20 · see citing works

Restoring uniqueness to mean-field games by randomizing the equilibria

Delarue, F. (2019) · Stochastic Partial Differential Equations Analysis and Computations · doi:10.1007/s40072-019-00135-9 · Cited by 17 · see citing works

The turnpike property and the longtime behavior of the Hamilton–Jacobi–Bellman equation for finite-dimensional LQ control problems

Esteve, C., Kouhkouh, H., Pighin, D., & Zuazua, E. (2022) · Mathematics of Control Signals and Systems · doi:10.1007/s00498-022-00325-2 · Cited by 16 · see citing works

Control on Hilbert spaces and application to some mean field type control problems

Bensoussan, A., Graber, P. J., & Yam, S. C. P. (2024) · The Annals of Applied Probability · doi:10.1214/24-aap2060 · Cited by 15 · see citing works

Backward SDEs for control with partial information

Papanicolaou, A. C. (2018) · Mathematical Finance · doi:10.1111/mafi.12174 · Cited by 15 · see citing works

Stochastic Control with Delayed Information and Related Nonlinear Master Equation

Saporito, Y. F., & Zhang, J. (2019) · SIAM Journal on Control and Optimization · doi:10.1137/17m1154746 · Cited by 13 · see citing works

Memory-Limited Partially Observable Stochastic Control and Its Mean-Field Control Approach

Tottori, T., & Kobayashi, T. J. (2022) · Entropy · doi:10.3390/e24111599 · Cited by 11 · see citing works

Forward-Backward Sweep Method for the System of HJB-FP Equations in Memory-Limited Partially Observable Stochastic Control

Tottori, T., & Kobayashi, T. J. (2023) · Entropy · doi:10.3390/e25020208 · Cited by 11 · see citing works

Finite Dimensional Approximations of Hamilton–Jacobi–Bellman Equations for Stochastic Particle Systems with Common Noise

Mayorga, S., & Święch, A. (2023) · SIAM Journal on Control and Optimization · doi:10.1137/22m1489186 · Cited by 10 · see citing works

Sharp semi-concavity in a non-autonomous control problem and $$\pmb {L^p}$$ estimates in an optimal-exit MFG

Dweik, S., & Mazanti, G. (2020) · Nonlinear Differential Equations and Applications NoDEA · doi:10.1007/s00030-019-0612-4 · Cited by 10 · see citing works

Stationary fully nonlinear mean-field games

Andrade, P. D. S., & Pimentel, E. A. (2021) · Journal d Analyse Mathématique · doi:10.1007/s11854-021-0193-0 · Cited by 9 · see citing works

Decentralized Stochastic Control with Finite-Dimensional Memories: A Memory Limitation Approach

Tottori, T., & Kobayashi, T. J. (2023) · Entropy · doi:10.3390/e25050791 · Cited by 7 · see citing works

Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes

Kolokoltsov, V. N., & Troeva, M. S. (2018) · Problemy analiza · doi:10.15393/j3.art.2018.5250 · Cited by 7 · see citing works

The Master Equation in a bounded domain with Neumann conditions

Ricciardi, M. (2021) · Communications in Partial Differential Equations · doi:10.1080/03605302.2021.2008965 · Cited by 7 · see citing works

Recent Developments in Controlled Crowd Dynamics

Banda, M. K., Herty, M., & Trimborn, T. (2020) · Modeling and simulation in science, engineering & technology · doi:10.1007/978-3-030-50450-2_7 · Cited by 7 · see citing works

Matrix-Valued Mean-Field-Type Games: Risk-Sensitive, Adversarial, and Risk-Neutral Linear-Quadratic CasePreprint

Barreiro‐Gomez, J., Duncan, T. E., & Tembiné, H. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.11346 · Cited by 7 · see citing works

Theory for optimal estimation and control under resource limitations and its applications to biological information processing and decision-making

Tottori, T., & Kobayashi, T. J. (2025) · Physical Review Research · doi:10.1103/gvl6-cvby · Cited by 6 · see citing works

Existence of Solutions of the Master Equation in the Smooth Case

Bessi, U. (2016) · SIAM Journal on Mathematical Analysis · doi:10.1137/15m1018782 · Cited by 6 · see citing works

The Convergence Problem in Mean Field Games with Neumann Boundary Conditions

Ricciardi, M. (2023) · SIAM Journal on Mathematical Analysis · doi:10.1137/22m1479075 · Cited by 5 · see citing works

Ergodic control of McKean–Vlasov SDEs and associated Bellman equation

Bao, X., & Tang, S. (2023) · Journal of Mathematical Analysis and Applications · doi:10.1016/j.jmaa.2023.127404 · Cited by 5 · see citing works

Linear Quadratic Extended Mean Field Games and Control Problems

Bensoussan, A., Kim, J., Li, B., & Yam, S. C. P. (2025) · Journal of Optimization Theory and Applications · doi:10.1007/s10957-025-02712-6 · Cited by 4 · see citing works

Numerical approximations of coupled forward–backward SPDEs

Molla, H. U., & Qiu, J. (2021) · Stochastic Analysis and Applications · doi:10.1080/07362994.2021.2011318 · Cited by 4 · see citing works

Remarks on potential mean field games

Graber, P. J. (2025) · Research in the Mathematical Sciences · doi:10.1007/s40687-024-00494-3 · Cited by 4 · see citing works

Mean field type control problems, some Hilbert-space-valued FBSDES, and related equations

Bensoussan, A., Tai, H. M., & Yam, S. C. P. (2025) · ESAIM Control Optimisation and Calculus of Variations · doi:10.1051/cocv/2025022 · Cited by 4 · see citing works

A Hessian-Dependent Functional With Free Boundaries and Applications to Mean-Field Games

Correa, J. C., & Pimentel, E. A. (2024) · Journal of Geometric Analysis · doi:10.1007/s12220-023-01542-0 · Cited by 4 · see citing works

Optimization in a Random Environment

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_1 · Cited by 3 · see citing works

Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs

Gangbo, W., Mou, C., & Święch, A. (2024) · Calculus of Variations and Partial Differential Equations · doi:10.1007/s00526-024-02758-w · Cited by 3 · see citing works

The Master Field and the Master Equation

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_4 · Cited by 3 · see citing works

Quantile-based Mean-Field Games with Common NoisePreprint

Tembiné, H. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1708.06035 · Cited by 3 · see citing works

Stochastic McKean-Vlasov Control Problem with Regime-Switching and Its Applications

Zhang, L., & Li, X. (2025) · Journal of Systems Science and Complexity · doi:10.1007/s11424-025-4283-4 · Cited by 2 · see citing works

Solving MFGs with a Common Noise

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_3 · Cited by 2 · see citing works

On Mean Field Games with Common Noise based on Stable-Like Processes

Kolokoltsov, V. N., & Troeva, M. S. (2019) · Statistics Optimization & Information Computing · doi:10.19139/soic.v7i2.637 · Cited by 2 · see citing works

Multiscale control of Stackelberg games

Herty, M., Steffensen, S., & Thünen, A. (2022) · Mathematics and Computers in Simulation · doi:10.1016/j.matcom.2022.04.028 · Cited by 2 · see citing works

Control in Hilbert Space and First-Order Mean Field Type Problem

Bensoussan, A., Cheung, H., & Yam, S. C. P. (2022) · Springer eBooks · doi:10.1007/978-3-030-98519-6_1 · Cited by 2 · see citing works

MFGs with a Common Noise: Strong and Weak Solutions

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_2 · Cited by 1 · see citing works

State-density flows of non-degenerate density-dependent mean field SDEs and associated PDEs

Huang, Z., & Tang, S. (2022) · Transactions of the American Mathematical Society · doi:10.1090/tran/8809 · Cited by 1 · see citing works

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

Bensoussan, A., Huang, Z., & Yam, S. C. P. (2024) · International Game Theory Review · doi:10.1142/s0219198924400036 · Cited by 1 · see citing works

Stochastic recursive optimal control of McKean-Vlasov type: A viscosity solution approach

Zhang, L. (2024) · Journal of Differential Equations · doi:10.1016/j.jde.2024.07.015 · Cited by 1 · see citing works

Extensions for Volume I

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-58920-6_7 · Cited by 1 · see citing works

Theoretical Analysis of Resource-Induced Phase Transitions in Estimation Strategies

Tottori, T., & Kobayashi, T. J. (2026) · Physical Review Letters · doi:10.1103/5ynb-7k4v

Convergence and Approximations

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_6

Classical Solutions to the Master Equation

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_5

On a mean field optimal control problem

Carrillo, J. A., Pimentel, E. A., & Voskanyan, V. (2020) · Nonlinear Analysis · doi:10.1016/j.na.2020.112039

Derivative over Wasserstein spaces along curves of densitiesPreprint

Buckdahn, R., Li, J., & Liang, H. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2010.01507

TOWARDS A REGULARITY THEORY FOR FULLY NONLINEAR MODELS

Andrade, P. D. S. (2020) · doi:10.17771/pucrio.acad.51015

Extensions for Volume II

Carmona, R. A., & Delarue, F. (2018) · Probability theory and stochastic modelling · doi:10.1007/978-3-319-56436-4_7

Marginal Problem

Mikami, T. (2021) · SpringerBriefs in mathematics · doi:10.1007/978-981-16-1754-6_3

Density constraints in optimal transport, PDEs and mean field games

Mészáros, A. R. (2015) · doi:10.70675/0e5ac0a4ze3fbz48dfza71ez592ca3158255

Some Advances in Mean Field Games Theory

Ricciardi, M. (2020) · doi:10.70675/84c22ff9z6020z46dfz99b4z46cf83aa5e60

Interactions and incitatives : between contract theory and mean-field games

Hubert, E. (2020) · doi:10.70675/75903d35z105ez4a64zae64zb6ce8c27c4a7

Some results on the McKean–Vlasov optimal control and mean field games : Limit theorems, dynamic programming principle and numerical approximations

Djete, M. F. (2020) · doi:10.70675/4c908101z3572z4e03z8915z93320b477485

The Master equation in mean field theory

Bensoussan, A., Frehse, J., & Yam, S. C. P. (2015) · Journal de Mathématiques Pures et Appliquées · doi:10.1016/j.matpur.2014.11.005 · Cited by 138 · see citing works

Equivalence of the master equation and the Langevin equation

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Langevin description of markovian integro-differential master equations

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Langevin equation with multi-Poissonian noise

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Connection between master equation and Lie-admissible formulation

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