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Register a new userA note on non-coercive first order Mean Field Games with analytic data
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We study first order evolutive Mean Field Games whose operators are non-coercive. This situation occurs, for instance, when some directions are `forbidden to the generic player at some points. Under some regularity assumptions, we establish existence of a weak solution of the system. Mainly, we shall describe the evolution of the populations distribution as the push-forward of the initial distribution through a flow, suitably defined in terms of the underlying optimal control problem.
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We study first order evolutive Mean Field Games where the Hamiltonian is non-coercive. This situation occurs, for instance, when some directions are forbidden to the generic player at some points. We establish the existence of a weak solution of the system via a vanishing viscosity method and, mainly, we prove that the evolution of the populations density is the push-forward of the initial density through the flow characterized almost everywhere by the optimal trajectories of the control problem underlying the Hamilton-Jacobi equation. As preliminary steps, we need that the optimal trajectories for the control problem are unique (at least for a.e. starting points) and that the optimal controls can be expressed in terms of the horizontal gradient of the value function.
We study the asymptotic behavior of solutions to the constrained MFG system as the time horizon $T$ goes to infinity. For this purpose, we analyze first Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM theory, constructing a Mather measure for the associated variational problem. Using these results, we show that a solution to the constrained ergodic mean field games system exists and the ergodic constant is unique. Finally, we prove that any solution of the first-order constrained MFG problem on $[0,T]$ converges to the solution of the ergodic system as $T to +infty$.
We consider Mean Field Games without idiosyncratic but with Brownian type common noise. We introduce a notion of solutions of the associated backward-forward system of stochastic partial differential equations. We show that the solution exists and is unique for monotone coupling functions. This the first general result for solutions of the Mean Field Games system with common and no idiosynctratic noise. We also use the solution to find approximate optimal strategies (Nash equilibria) for N-player differential games with common but no idiosyncratic noise. An important step in the analysis is the study of the well-posedness of a stochastic backward Hamilton-Jacobi equation.
In this paper we study second order master equations arising from mean field games with common noise over arbitrary time duration. A classical solution typically requires the monotonicity condition (or small time duration) and sufficiently smooth data. While keeping the monotonicity condition, our goal is to relax the regularity of the data, which is an open problem in the literature. In particular, we do not require any differentiability in terms of the measures, which prevents us from obtaining classical solutions. We shall propose three weaker notions of solutions, named as {it good solutions}, {it weak solutions}, and {it viscosity solutions}, respectively, and establish the wellposedness of the master equation under all three notions. We emphasize that, due to the game nature, one cannot expect comparison principle even for classical solutions. The key for the global (in time) wellposedness is the uniform a priori estimate for the Lipschitz continuity of the solution in the measures. The monotonicity condition is crucial for this uniform estimate and thus is crucial for the existence of the global solution, but is not needed for the uniqueness. To facilitate our analysis, we construct a smooth mollifier for functions on Wasserstein space, which is new in the literature and is interesting in its own right. As an important application of our results, we prove the convergence of the Nash system, a high dimensional system of PDEs arising from the corresponding $N$-player game, under mild regularity requirements. We shall also prove a propagation of chaos property for the associated optimal trajectories.
In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation for the density of the distribution of states (forward in time) and a Hamilton-Jacobi (HJ) equation for the optimal value of a representative agent (backward in time). The state variable is the pair $(x, v)in R^Ntimes R^N$ where x stands for the position and v stands for the velocity. The dynamics is often referred to as the double integrator. In this case, the Hamiltonian of the system is neither strictly convex nor coercive, hence the available results on MFGs cannot be applied. Moreover, we will assume that the Hamiltonian is unbounded w.r.t. the velocity variable v. We prove the existence of a weak solution of the MFG system via a vanishing viscosity method and we characterize the distribution of states as the image of the initial distribution by the flow associated with the optimal control.
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