A quadratic Mean Field Games model for the Langevin equation

We consider a Mean Field Games model where the dynamics of the agents is given by a controlled Langevin equation and the cost is quadratic. A change of variables, introduced in [9], transforms the Mean Field Games system into a system of two coupled kinetic Fokker-Planck equations. We prove an existence result for the latter system, obtaining consequently existence of a solution for the Mean Field Games system.


Introduction
The Mean Field Games (MFG in short) theory concerns the study of differential games with a large number of rational, indistinguishable agents and the characterization of the corresponding Nash equilibria. In the original model introduced in [11,14], an agent can typically acts on its velocity (or other first order dynamical quantities) via a control variable. Mean Field Games where agents control the acceleration have been recently proposed in [1,3,4].
A prototype of stochastic process involving acceleration is given by the Langevin diffusion process, which can be formally defined as X(t) = −b(X(t),Ẋ(t)) + σḂ(t), (1.1) whereẊ,Ẍ are respectively the first and second time derivatives of the stochastic process X in R d ,Ḃ a white noise process and σ a positive parameter. The solution of (1.1) can be rewritten as a Markov process (X, V ) solving V (t) = −b(X(t), V (t)) + σḂ(t).
The density function of the previous process satisfies the kinetic Fokker-Planck equation The previous equation, in the case b ≡ 0, was first studied by Kolmogorov [12] who provided an explicit formula for its fundamental solution. Then considered by Hörmander [10] as motivating example for the general theory of the hypoelliptic operators (see also [2,13]). We consider a Mean Field Games model where the dynamics of the single agent is given by a controlled Langevin diffusion process, i.e In (1.2), α, the control law, is chosen to maximize the functional where m(s) is the distribution of the agents at time s. Let u the value function associated to the previous control problem. Formally, the couple (u, m) satisfies the MFG system where the first equation is a backward Hamilton-Jacobi-Bellman equation, degenerate in the x-variable and with a quadratic Hamiltonian in the v variable, and the second equation is forward kinetic Fokker-Planck equation. In the standard setting, MFG with quadratic Hamiltonians has been extensively considered in literature both as a reference model for the general theory and also since, thanks to the Hopf-Cole change of variable, the nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a linear equation, allowing to use all the tools developed for this type of problem (see for example [6,7,8,9,14]). We study (1.3) by means of a change of variable introduced in [8,9] for the standard case. By defining the new unknowns φ = e u/σ 2 and ψ = me −u/σ 2 , the system (1.3) is transformed into a system of two kinetic Fokker-Planck equations (1.4) In the previous problem, the coupling between the two equations is only in the source terms. Following [8], we prove existence of a (weak) solution to (1.4) by showing the convergence of an iterative scheme defined, starting from ψ (0) ≡ 0, by solving alternatively the (backward and forward) equations .
(1.6) We show that the resulting sequence (φ (k+ 1 2 ) , ψ (k+1) ), k ∈ N, monotonically converges to the solution of (1.4). Hence, by the inverse change of variable u = ln(φ)/σ 2 and m = φψ, we obtain a solution of the original problem (1.3). The previous iterative procedure also suggests a monotone numerical method for the approximation of (1.4), hence for (1.3). Indeed, by approximating (1.5) and (1.6) by finite differences and solving alternatively the resulting discrete equations, we obtain an approximation of the converging sequence (φ (k+ 1 2 ) , ψ (k+1) ). A corresponding procedure for the standard quadratic MFG system was studied in [8], where the convergence of the method is proved. We plan to study the properties of the previous numerical procedure in a future work.

Well posedness of the kinetic Fokker-Planck system
In this section, we study the existence of a solution to system (1.4). The proof of the result follows the strategy implemented in [8, Section 2] for the case of a standard MFG system with quadratic Hamiltonian and relies on the results for linear kinetic Fokker-Planck equations in [5, Appendix A].
We fix the assumptions we will assume in all the paper. The vector field b : and the coupling cost f : Moreover, the diffusion coefficient σ is strictly positive and the initial and terminal data satisfy We denote with (·, ·) the scalar product in L 2 ([0, T ] × R 2d ) and with ·, · the pairing between ). We define the following functional space and we also set [5,Lemma A.1]) and therefore the initial/terminal conditions for (1.4) are well defined in L 2 sense. We first prove the well posedness of problems (1.5) and (1.6).
Moreover, for any R > 0, there exists δ R ∈ R such that (ii) Let Φ : Y 0 → Y 0 be the map which associates to ψ the unique solution of (2.3). Then, if ψ 2 ≤ ψ 1 , we have Φ(ψ 2 ) ≥ Φ(ψ 1 ). (2.5) By [5,Prop. A.2], φ belongs to Y and it coincides with the unique solution of (2.5) in this space. Moreover, the following estimate holds for some constant C which depends only on e uT /σ 2 L 2 , f L ∞ and σ. Hence F maps B C , the closed ball of radius C of L 2 ([0, T ] × R 2d ), into itself. To show that the map F is continuous on B C , consider a {ϕ n } n∈N , ϕ ∈ L 2 ([0, T ]× R 2d ) such that ϕ n − ϕ L 2 → 0 and set φ n = F (ϕ n ). Then φ n ∈ Y, and, by the estimate (2.6), we get that, up to a subsequence, there existsφ ∈ Y such that φ n →φ, ) and, moreover ϕ n → ϕ almost everywhere. By the definition of weak solution for (2.5), we have that for any w ∈ D([0, T ] × R 2d ), the space of infinite differentiable functions with compact support in [0, T ] × R 2d . Employing weak convergence for left hand side of (2.7) and the Dominated Convergence Theorem for the right hand one, we get for n → ∞ Given the previous properties, we conclude, by Schauder's Theorem, that there exists a fixed-point of the map F in L 2 , hence in Y, and therefore a solution to the nonlinear parabolic equation (2.3).
We set where C R is defined as in (2.4). 14) where C R as in (2.4).
(ii) Let Ψ : Y R → Y 0 be the map which associates to φ ∈ Y R the unique solution of (2.13).