As this text was posted on the arXiv roughly 8 years before it has appeared in print, the reader is holding in their hands (or reading on their electronic device) essentially the first result on the global in time well-posedness of a large class of master equations arising both in mean field games and mean field type control problems.
The theory of mean field games started with the seminal works of J.-M. Lasry and P.-L. Lions [C. R., Math., Acad. Sci. Paris 343, No. 9, 619–625 (2006; Zbl 1153.91009); C. R., Math., Acad. Sci. Paris 343, No. 10, 679–684 (2006; Zbl 1153.91010); Jpn. J. Math. (3) 2, No. 1, 229–260 (2007; Zbl 1156.91321)] and M. Huang et al. [Commun. Inf. Syst. 6, No. 3, 221–252 (2006; Zbl 1136.91349)] around 2007, and ever since this subject has witnessed a huge expansion both from the theoretical viewpoint and the point of view of applications.
Master equations, which encode all features of the underlying games, was first proposed by Lions in his lectures at the Collège de France. These are PDEs of hyperbolic nature set on the Cartesian product of the state space (which is typically a euclidean space) and the space of probability measures supported on the state space. As the master equation is a nonlinear and nonlocal PDE on an infinite dimensional space, its well-posedness theory requires significant new ideas, beyond the finite dimensional setting. As such, there is no debate about notions of classical solutions, while notions of weak solutions are much more subtle to even define.
The present text settles the first global in time existence and uniqueness questions for such master equations, subject to a possibly degenerate noise in the dynamics of the individual agents. In a follow up work by P. Cardaliaguet et al. [The master equation and the convergence problem in mean field games. Princeton, NJ: Princeton University Press (2019; Zbl 1430.91002)] similar results have been established also in the presence of common noise, in the periodic setting (when the state space is the flat torus \(\mathbb{T}^d\)). This work uses PDE analysis, and in contrast to the text under review, it required a non-degenerate idiosyncratic noise with constant intensity.
The text under review relies on a probabilistic approach, which nicely builds on FBSDE systems, often of McKean-Vlasov type. The heart of the analysis is the observation that the candidate for the solution of the master equation (or the derivative of such) can be seen as the decoupling field for the FBSDE. At an initial step the authors establish the solvability of the FBSDE system and its various linearizations for short time. This short time horizon depends on various Lipschitz constants of the data, but most notably on the 2-Wasserstein (\(W_2\))-Lipschitz constant of the final condition with respect to the measure variable. The various linearized FBSDE systems give rise to the representation formulas of the derivatives of the decoupling field, which are shown to exist in a suitable sense, so that the corresponding master equation has a classical solution. As a major technical work, the authors establish chain rules and Ito formulas for functions defined on the space of probability measures.
Then, as the second set of main results of the paper, under various monotonicity and convexity assumptions on the data, the authors are able to patch together short time solutions and obtain global in time classical solutions. The key element here is the propagation of the \(W_2\)-Lipschitz condition, which can be guaranteed in three different scenarios. First, when the extended Hamiltonian of the control problem is convex in both the state and control variables and when the Lasry-Lions monotonicity condition is present. Second, when the extended Hamiltonian is not convex in the position variable, but the cost functionals are bounded in the position and are linear-quadratic in the control variable, the volatility is non-degenerate and the Lasry-Lions condition is in force. Finally, in the case when convexity must hold in the state and control variables and also in the direction of the measure (in which case there is no need of the Lasry-Lions condition).
As a last remark it worth mentioning that the typical assumptions on the data are somewhat weaker than similar assumptions imposed by R. Carmona and F. Delarue [Probabilistic theory of mean field games with applications I. Mean field FBSDEs, control, and games. Cham: Springer (2018; Zbl 1422.91014); Probabilistic theory of mean field games with applications II. Mean field games with common noise and master equations. Cham: Springer (2018; Zbl 1422.91015)]. Notably, the text under review gives short time existence results and propagation of Lipschitz estimates with respect to the \(W_2\)-metric, while in the mentioned reference this is done under the \(W_1\)-metric which is stronger.
All in all, the results obtained in this text via a thorough and careful analysis, are extremely important for the study of classical solutions of master equations. These results served and continue to serve as source of inspiration for many other results on the solvability of similar master equations.