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On some mean field games and master equations through the lens of conservation laws (2024)
Journal Article
Graber, P. J., & Mészáros, A. R. (2024). On some mean field games and master equations through the lens of conservation laws. Mathematische Annalen, https://doi.org/10.1007/s00208-024-02859-z

In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of deterministic mean field games and master equations, where the interaction of the agents happens only at the... Read More about On some mean field games and master equations through the lens of conservation laws.

Mean Field Games Systems under Displacement Monotonicity (2024)
Journal Article
Mészáros, A. R., & Mou, C. (2024). Mean Field Games Systems under Displacement Monotonicity. SIAM Journal on Mathematical Analysis, 56(1), 529-553. https://doi.org/10.1137/22m1534353

In this note we prove the uniqueness of solutions to a class of mean field games systems subject to possibly degenerate individual noise. Our results hold true for arbitrary long time horizons and for general nonseparable Hamiltonians that satisfy a... Read More about Mean Field Games Systems under Displacement Monotonicity.

On monotonicity conditions for mean field games (2023)
Journal Article
Graber, P. J., & Mészáros, A. R. (2023). On monotonicity conditions for mean field games. Journal of Functional Analysis, 285(9), Article 110095. https://doi.org/10.1016/j.jfa.2023.110095

In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for unconditional uniqueness that is independent of the length of the time... Read More about On monotonicity conditions for mean field games.

Well-posedness of mean field games master equations involving non-separable local Hamiltonians (2023)
Journal Article
Ambrose, D. M., & Mészáros, A. R. (2023). Well-posedness of mean field games master equations involving non-separable local Hamiltonians. Transactions of the American Mathematical Society, 376(4), 2481-2523. https://doi.org/10.1090/tran/8760

In this paper we construct short time classical solutions to a class of master equations in the presence of non-degenerate individual noise arising in the theory of mean field games. The considered Hamiltonians are non-separable and local functions o... Read More about Well-posedness of mean field games master equations involving non-separable local Hamiltonians.

A variational approach to first order kinetic Mean Field Games with local couplings (2022)
Journal Article
Griffin-Pickering, M., & Mészáros, A. R. (2022). A variational approach to first order kinetic Mean Field Games with local couplings. Communications in Partial Differential Equations, 47(10), 1945-2022. https://doi.org/10.1080/03605302.2022.2101003

First order kinetic mean field games formally describe the Nash equilibria of deterministic differential games where agents control their acceleration, asymptotically in the limit as the number of agents tends to infinity. The known results for the w... Read More about A variational approach to first order kinetic Mean Field Games with local couplings.

Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2022)
Journal Article
Gangbo, W., & Mészáros, A. R. (2022). Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games. Communications on Pure and Applied Mathematics, 75(12), 2685-2801. https://doi.org/10.1002/cpa.22069

This manuscript constructs global in time solutions to master equations for potential mean field games. The study concerns a class of Lagrangians and initial data functions that are displacement convex, and so this property may be in dichotomy with t... Read More about Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games.

Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients (2021)
Journal Article
Kwon, D., & Mészáros, A. R. (2021). Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients. Journal of the London Mathematical Society, 104(2), 688-746. https://doi.org/10.1112/jlms.12444

This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in particular on... Read More about Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients.

Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport (2020)
Journal Article
Jacobs, M., Kim, I., & Mészáros, A. R. (2020). Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport. Archive for Rational Mechanics and Analysis, 239(1), 389-430. https://doi.org/10.1007/s00205-020-01579-3

Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based... Read More about Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport.

The planning problem in mean field games as regularized mass transport (2019)
Journal Article
Graber, P. J., Mészáros, A. R., Silva, F. J., & Tonon, D. (2019). The planning problem in mean field games as regularized mass transport. Calculus of Variations and Partial Differential Equations, 58(3), Article 115. https://doi.org/10.1007/s00526-019-1561-9

In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians... Read More about The planning problem in mean field games as regularized mass transport.