Charafeddine Mouzouni
email: mouzouni@math.univ-lyon1.fr

ON QUASI-STATIONARY MEAN FIELD GAMES MODELS

Keywords: 2010 Mathematics Subject Classification. 35Q91, 49N70, 35B40 Mean field games, quasi-stationnary models, nonlinear coupled PDE systems, long time behavior, self-organization, N-person games, Nash equilibria, myopic equilibrium

We explore a mechanism of decision-making in Mean Field Games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N -players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic Mean Field Games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic.

INTRODUCTION

The Mean Field Games formalism has been introduced some years ago by series of seminal papers by J.-M. Lasry and P.-L. Lions [START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF][START_REF]Jeux à champ moyen. II. Horizon fini et contrôle optimal[END_REF] and M. Huang, R. Malhamé and P. Caines [START_REF] Caines | Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized -Nash equilibria[END_REF][START_REF]Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF]. It describes the evolution of stochastic differential games with a large population of rational players and where the strategies of the agents are not only affected by their own preferences but also by the state of the other players through a global mean field effect. In terms of partial differential equations, these models are typically a system of a transport or Fokker-Plank equation for the distribution of the agents coupled with a Hamilton-Jacobi-Bellman equation.

The motivation of this paper is to study a strategy-choice mechanism that is different from classical Mean Field Games. Our agents are myopic, and choose their actions according to the information available at time t, by fixing the future state of their opponents and trying to get the best possible gain in the future ps ą tq. Players anticipate no evolution of the system, undergo changes in their environment and adapt their strategies. A system of interacting agents can have such irrational behavior in panic situations for instance. In this framework, agents build a strategy at each moment and the global (in time) strategy is the history of all the chosen strategies. This decision-making mechanism intrinsically implies the existence of two time scales: a fast time scale which is linked to the optimization of the expected future cost; and a slow time scale linked to the actual evolution of the system. The coexistence of these two time scales gives rise to equations of quasi-stationary nature. We are also interested in the formation of equilibria for this type of evolution systems, and in the rate at which these systems converge towards these equilibria.

In general the decision-making mechanism in mean field games (MFG for short) involves solving a stochastic control problem, that provides a global in time optimal strategy. In the case where the players aim to minimize a long time average cost, it is well known that the MFG system of partial differential equations is stationary and takes the following form [START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF][START_REF] Feleqi | The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF],

(1.1)

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ´σ∆ū `Hpx, Dūq `λ " F px, mq in Q :" T d ´σ∆ m ´divp mH p px, Dūqq " 0 in Q m ě 0, ă m ą:" ż Q m " 1, ă ū ą" 0
Here σ ą 0, all the functions are Z d -periodic, the unknowns are the constant λ and the functions ū and m, T d is the d-dimensional torus, H is the Hamiltonian and F the coupling, both related to the structure of the cost, and H p is the partial derivative of H with respect to the second variable. The solution of the first equation in (1.1) can be interpreted as the equilibrium value function of a "small" player whose cost depends on the density m of the other players, while the second equation characterizes the distribution of players at the equilibrium. It is well known (see e.g. [START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF]) that there exists a solution p λ, ū, mq in R ˆC2 pQq ˆW 1,p pQq for all 1 ď p ă 8 to (1.1), under a wide range of sufficient conditions. Moreover, uniqueness holds under the following monotonicity condition on F :

(1.2) @m, m 1 P PpQq, ż Q `F px, mq ´F px, m 1 q ˘dpm ´m1 qpxq ě 0.

The interpretation of the above monotonicity condition is that the players dislike congested regions and prefer configurations in which they are scattered. Another well known example of stationary MFG systems, is the case where players aim to minimize a discounted infinite-horizon cost functional. In that case, the MFG system takes the following form (see e.g. [START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF], among others):

(1.3) $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % ´σ∆v `Hpx, Dvq `ρv " F px, μq in Q ´σ∆μ ´divpμH p px, Dvqq " 0 in Q μ ě 0, ă μ ą" 1
where ρ ą 0. It is also well known (see [START_REF] Alvarez | Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF][START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF]) that, under several technical conditions on H and F , there exists a solution pv, μq P C 2 pQq ˆW 1,p pQq for all 1 ď p ă 8 to (1.3). Moreover, if H has a linear growth, i.e. |Hpx, pq| ď Cp1 `|p|q for some constant C ą 0, system (1.1) is obtained as a limit of system (1.3) when ρ Ñ 0. More precisely, we have that (1.4) p ρ ă v ą, v´ă v ą, μq ÝÑ p λ, ū, mq in R ˆC2 pQq ˆL8 pQq as ρ Ñ 0.

In this paper, we consider a situation where the evolution of the players is driven by a system of stochastic differential equations, and where choosing a strategy amounts to choosing a drift vector field that has a suitable regularity; at any time t ě 0, each player seeks to minimize a cost functional which depends on the current state of the system, and on the possible future evolution ps ą tq of the player, which is related to her/his choice of a vector field α t p.q at time t. Thus, choosing the optimal α t p.q amounts to plan optimally the future evolution of the player, assuming no evolution in her/his environment. Players follow their planned evolution and adjust their drift α t p.q according to the observed changes. Further details and explanations about the model will be given in Section 3.

For the choice of α t p.q we consider two different cost structures: a discounted cost functional; and a long-time average cost (see Section 3). As we already pointed out, the scheduling gives rise to two time scales: a slow time scale "t" linked to the evolution of the state of the system; and a fast scale "s" (which does not appear explicitly in the MFG systems) related to the scheduling. Under some assumptions on H and F , we show in Section 3 that at the mean field limit one gets systems of equations of the following form:

(1.5) $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % ´σ1 ∆v `Hpx, Dvq `ρv " F px, µptqq in p0, T q ˆQ B t µ ´σ∆µ ´divpµH p px, Dvqq " 0 in p0, T q ˆQ µp0q " m 0 ě 0 in Q, ă m 0 ą" 1;
and

(1.6) $ ' ' ' ' ' ' ' & ' ' ' ' ' ' ' % ´σ1 ∆u `Hpx, Duq `λptq " F px, mptqq in p0, T q ˆQ B t m ´σ∆m ´divpmH p px, Duqq " 0 in p0, T q ˆQ mp0q " m 0 ě 0 in Q, ă m 0 ą" 1, ă u ą" 0 in p0, T q,
where ρ ą 0 is fixed throughout this paper, σ 1 , T ą 0, m 0 is the initial density of players, and all functions are Z d -periodic. Note that pλ, uq (resp. v) depends on time only through m (resp. µ). The parameters σ 1 and σ are respectively: the noise level related to the prediction process (the assessment of the future evolution), and the noise level associated to the evolution of the players. System (1.6) corresponds to the case of a long time average cost functional, while system (1.5) corresponds to the case of a discounted cost functional. We shall see that for any time t, pλptq, uptqq (resp. vptq) characterizes a local Nash equilibrium related to a long time average cost (resp. a discounted cost). The first equations in (1.6) and (1.5) give the "evolution" of the game value of a "small" player, and expresse the adaptation of players choices to the environment evolution. The evolution of µ and m expresses the actual evolution of the population density. We refer to Section 3 for more detailed explanations.

In contrast to most MFG systems, the uniqueness of solutions to systems (1.6) and (1.5) does not require the monotonicity condition (1.2) nor the convexity of H with respect to the second variable. This fact is essentially related to the forward-forward structure of the systems. We also show that the small-discount approximation (1.4) holds for quasi-stationary models under the same conditions as for the stationary ones. Under the monotonicity condition (1.2), we prove in Section 4 that for a quadratic Hamiltonian, a solution pλ, u, mq to (1.6) converges exponentially fast in some sense to the unique equilibrium p λ, ū, mq of (1.1) as t Ñ `8, provided that m 0 ´m is sufficiently small and σ " σ 1 . An analogous result holds also for systems (1.3)-(1.5) when the discount rate ρ is small enough. This asymptotic behavior is interpreted by the emergence of a self-organizing phenomenon and a phase transition in the system. Note that this entails in particular that our systems can exhibit a large scale structure even if the cohesion between the agents is only maintained by interactions between neighbors. The techniques used to prove this asymptotic results rely on some algebraic properties pointed out in [START_REF] Cardaliaguet | Long time average of mean field games, with nonlocal coupling[END_REF] specific to the quadratic Hamiltonian. On the other hand, one can not use the usual duality arguments to show convergence for general data. Therefore the convergence remains an open problem for more general cases.

Similar asymptotic results were established for the MFG system in [START_REF] Cardaliaguet | Long time average of mean field games, with nonlocal coupling[END_REF][START_REF]Long time average of mean field games[END_REF] for local and non local coupling. Long time convergence of forward-forward MFG models is also discussed in [START_REF] Achdou | Mean field games models of segregation[END_REF][START_REF] Gomes | One-Dimensional Forward-Forward Mean-Field Games[END_REF]. Self-organizing and phase transition in Mean Field Games were addressed in [START_REF] Mehta | Synchronization of Coupled Oscillators is a Game[END_REF][START_REF]Bifurcation Analysis of a Heterogeneous Mean-field Oscillator Game[END_REF][START_REF]Learning in Mean-field Oscillator Games[END_REF], for applications in neuroscience, biology, economics, and engineering. For an overview on collective motions and self-organization phenomena in mean field models, we refer to [START_REF] Degond | Collective dynamics and self-organization: some challenges and an example[END_REF] and the references therein. The derivation of the Mean Field Games system was addressed in [START_REF] Feleqi | The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF] for the ergodic case (long time average cost). More general cases were analyzed in the important recent paper [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] on the master equation and its application to the convergence problem in Mean Field Games. The reader will notice in Section 3 that the analysis of the mean-field limit in our case is very similar to that of the McKean-Vlasov equation. Therefore the proof of convergence is less technical than in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] and is based on the usual coupling arguments (see e.g. [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF][START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF], among others). MFG models with myopic players are briefly addressed in [START_REF] Achdou | Mean field games models of segregation[END_REF] for applications to urban settlements and residential choice. However, the sense given to "myopic players" is different from the one we are considering in this paper: indeed, "myopic players" in [START_REF] Achdou | Mean field games models of segregation[END_REF] corresponds to individuals which compute their cost functional taking only into account their very close neighbors, while in this paper "myopic players" refers to individuals which anticipate nothing and only undergo the evolution of their environement. In [START_REF] Cristiani | Modeling rationality to control self-organization of crowds: an environmental approach[END_REF], the authors introduce a model for the study of crowds dynamics, that is very similar to the one addressed in this paper: in Section 2.2.2, the authors consider a situation where at any time pedestrians build the optimal path to destination, based on the observed state of the system. Although the approaches are different, the two models have many similarities.

Local Nash equilibria for mean field systems of rational agents were also considered in [START_REF] Degond | Large-scale dynamics of Mean-Field Games driven by local Nash equilibria[END_REF][START_REF]Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria[END_REF][START_REF]Evolution of wealth in a non-conservative economy driven by local Nash equilibria[END_REF]. The authors use the "Best Reply Strategy approach" to derive a kinetic equation and provide applications to the evolution of wealth distribution in a conservative [START_REF]Evolution of the distribution of wealth in an economic environment driven by local Nash equilibria[END_REF] and non-conservative [START_REF]Evolution of wealth in a non-conservative economy driven by local Nash equilibria[END_REF] economy. The link between Mean Field Games and the "Best Reply Strategy approach" is analyzed in [START_REF] Degond | Meanfield games and model predictive control[END_REF].

The paper is organized as follows: In Section 2, we give sufficient conditions for the existence and uniqueness of classical solutions for systems (1.5) and (1.6). The proofs rely on continuous dependence estimates for Hamilton-Jacobi-Bellman equations [START_REF] Marchi | Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem[END_REF], the small-discount approximation, and the non-local coupling which provides compactness and regularity. Section 3 is devoted to a detailed derivation of systems (1.5) and (1.6) from N -players stochastic differential games models. In Section 4, we prove the exponential convergence result for system (1.6). Finally, the Appendix recall some elementary facts on the Fokker-planck equation.

Notations and assumptions.

For simplicity, we work in a periodic setting in order to avoid issues related to boundary conditions or conditions at infinity. Therefore we will often consider functions as defined on Q :" T d (the d-dimensional torus). Throughout the paper, d ě 1, and the usual inner product on R d is denoted by x.y or ă x, y ą. We use the notation ă f ą:" ş Q f and denote PpQq the set of probability measures on Q. Recall that PpQq becomes a compact topological space when endowed with the weak ˚convergence thanks to Prokhorov's theorem. Moreover, this topology is metrizable, e.g. by the Kantorowich-Rubinstein distance:

d 1 pπ, π 1 q :" sup "ż Q f pxq dpπ ´π1 qpxq, where f : Q Ñ R is 1-Lipschitz continuous * .
We denote by CpQq the set of Z d -periodic continuous functions on R d , by C m` pQq, m P N, P p0, 1s, the set of Z d -periodic functions having m-th order derivatives which are -H ölder continuous, by pL p pQq, }.} p q, 1 ď p ď 8, the set of p-summable Lebesgue measurable and Z d -periodic functions on Q, by W k,p pQq, k P N, 1 ď p ď 8, the Sobolev space of Z d -periodic functions having a weak derivatives up to order k which are p-summable on Q, and for a given Lipschitz continuous function f , we define }f } Lip :" sup x‰y |f pxq ´f pyq| |x ´y| .

For γ P p0, 1q, we use the notation C 1`γ{2,2`γ for parabolic H ölder spaces, with the norm }.} C 1`γ{2,2`γ , as defined in [START_REF] Ladyzenskaja | Linear and quasilinear equations of parabolic type[END_REF]. For a given T ą 0, we note Q T :" p0, T q ˆQ, and Q T :" r0, T s ˆQ, and C 1`γ{2,2`γ pQ T q the space of Z d -periodic function in C 1`γ{2,2`γ pr0, T s ˆRd q.

Throughout the proofs, we denote by C a generic constant, and we use the notation C η,β to point out the dependence of the constant on parameters η, β. For any vector py j q 1ďjďN we use the notation y ´i " py j q j‰i . For a random variable X defined on some probability space, LpXq denotes the law of X. Throughout the paper, γ P p0, 1q is a fixed parameter.

Acknowledgement. This work was supported by LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and partially supported by project (ANR-16-CE40-0015-01) on Mean Field Games.

The author would like to thank Martino Bardi and Pierre Cardaliaguet for fruitful discussions.

ANALYSIS OF THE QUASI-STATIONARY SYSTEMS

This section is devoted to the analysis of systems (1.5) and (1.6). A detailed derivation of these systems from a N -person differential game will be given in Section 3.

We shall use the following conditions:

(H1) the operator m Ñ F p., mq is defined from PpQq into LippQq :" C 0`1 pQq, and satisfies (H6) m 0 is a probability measure, absolutely continuous with respect to the Lebesgue measure, and its density m 0 belongs to C 2`γ pQq.

The Hamiltonian H satisfies one of the following sets of conditions:

C1. H grows at most linearly in p, i.e., there exists κ H ą 0, such that

|Hpx, pq| ď κ H p1 `|p|q , @x P Q, @p P R d ; C2. H is superlinear in p uniformly in x, i.e., inf xPQ |Hpx, pq|{|p| Ñ `8 as |p| Ñ `8,
and there exists θ P p0, 1q, κ ą 0, such that a.e x P Q and |p| large enough,

(2.2) D x H, p `θ.H 2 ě ´κ|p| 2 .
Condition (C1.) arises naturally in control theory when the controls are chosen in a bounded set, whereas under condition (C2.) the control variable of each player can take any orientation in states space and can be arbitrary large with a large cost. As it is pointed out in [START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF], the condition (2.2) is interpreted as a condition on the oscillations of H and plays no role when d " 1.

A triplet pλ, u, mq is a classical solution to (1.6), if m : r0, T s ˆRd ÝÑ R is continuous, of class C 2 in space, and of class C 1 in time, u : p0, T q ˆRd ÝÑ R is of class C 2 in space, and pλ, u, mq satisfies (1.6) in the classical sense. Similarly, a couple pv, µq is a classical solution to (1.5)

, if µ : r0, T s ˆRd ÝÑ R is continuous, of class C 2 in space, of class C 1 in time, v : p0, T q ˆRd ÝÑ R is of class C 2 in
space and pv, µq satisfies (1.5) in the classical sense.

In this section, we give an existence and uniqueness result of classical solutions for system (1.5) and (1.6) under condition (C1.). In addition, we show that system (1.6) is also well-posed under condition (C2.).

We start by dealing with the case where the Hamiltonian has a linear growth (condition (C1.)). Let us consider the quasi-stationary approximate problem (1.5). We start by analyzing the first equation in (1.5).

Lemma 2.1. Under assumptions (H1), (H2) and (C1.), for any µ P PpQq and ą 0, the problem

(2.3) ´σ1 ∆v `Hpx, Dvq ` v " F px, µq in Q
has a unique solution v rµs P C 2`γ pQq. In addition, there exists constants κ 0 ą 0 and θ P p0, 1s, such that for any µ P PpQq and ą 0, the following estimates hold

(2.4a) } v rµs} 8 ď }F } 8 `κH , (2.4b) }v rµs ´ v rµs } C 2`θ ď κ 0 .
Proof. The proof of existence and uniqueness for equation (

Dκ 1 H ą 0, |Hpx, pq| ď κ 1 H ´1 `|p| 2 ¯, @x P Q, @p P R d ;
which is less restrictive than (C1.).

We now state a continuous dependence estimate due to Marchi [START_REF] Marchi | Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem[END_REF], which plays a crucial role. Lemma 2.3. Assume (H1)-(H3), and (C1.). For any µ, µ 1 P PpQq and ą 0, we have that (2.5a) }v rµs ´v rµ 1 s} 8 ď ´1}F p., µq ´F p., µ 1 q} 8 .

Moreover, for any M ą 0, there exists a constant χ M ą 0, such that for any P p0, M q and µ, µ 1 P PpQq, the following holds

(2.5b) }w rµs ´w rµ 1 s} C 2 ď χ M }F p., µq ´F p., µ 1 q} 8 ,
where w " v ´ v .

Proof. Note that v ˘:" v rµ 1 s ˘ ´1}F p., µq ´F p., µ 1 q} 8 , are respectively a super-and a sub-solution to equation (2.3) with the coupling term F p., µq. Thus, estimate (2.5a) follows thanks to the comparison principle.

The proof of (2.5b) is similar to [START_REF] Marchi | Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem[END_REF]Theorem 2.2]. Nevertheless we give a proof to this result because in this particular framework we do not need to fulfill all the conditions of [START_REF] Marchi | Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem[END_REF]. We shall proceed by contradiction assuming that there exists sequences p n q P p0, M q, pµ n q, pµ 1 n q P PpQq, such that for any n ě 0, (2.6) c n ě n}F p., µ n q ´F p., µ 1 n q} 8 , where c n :" }w n rµ n s ´w n rµ 1 n s} C 2 , and lim n n " 0. Note that the function

W n :" c ´1 n `w n rµ n s ´w n rµ 1 n s satisfies the following equation R n ´σ1 ∆W n `fn .DW n " 0,
where

R n :" n W n `c´1 n n v n rµ n s ´v n rµ 1 n s `c´1 n `F p., µ n q ´F p., µ 1 n q ˘, and
f n pxq :" ż 1 0 H p `x, sDw n rµ n s `p1 ´sqDw n rµ 1 n s ˘ds.
Using (2.6) and (2.5a), one checks that lim n }R n } 8 " 0. In addition, (H3) and (2.4b) entails that }f n } Lip is uniformly bounded. Moreover, invoking standard regularity theory for linear elliptic equations (see e.g. [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]), the sequence pW n q is uniformly bounded in C 2`θ 1 pQq for some θ 1 P p0, 1s. We infer that pf n , W n q converge uniformly to some pf, W q in CpQq ˆC2 pQq which satisfies ´σ1 ∆W `f.DW " 0, }W } C 2 " 1, W " 0.

Since W is periodic, we deduce from the strong maximum principle that W must be constant; this provides the desired contradiction.

Corollary 2.4. Under (H5) and assumptions of Lemma 2.3, for any ą 0 there exists a constant κ ą 0 such that,

(2.7a) }v rµs ´v rµ 1 s} C 2 ď κ d 1 pµ, µ 1 q
for any µ, µ 1 P PpQq.

We shall give now an existence and uniqueness result for system (1.5).

Theorem 2.5. Under conditions (H1)-(H6) and (C1.), there exists a unique classical solution pv, µq in C 1{2 `r0, T s; C 2 pQq ˘ˆC 1,2 pQ T q to the problem (1.5).

Proof. Existence : For a constant δ ą 0 large enough to be chosen below, let C δ be the set of maps µ P Cpr0, T s; PpQqq such that

sup s‰t d 1 pµptq, µpsqq |t ´s| 1{2 ă δ.
Note that C δ is compact thanks to Ascoli's Theorem, and the compactness of pPpQq, d 1 q. We aim to prove our claim using Schauder's fixed point theorem (see e.g. [34, p. 25]). Set for any px, νq P Q ˆPpQq, Gpx; νq :" H p px, Dv ρ rνspxqq.

Note that G and D x G are uniformly bounded thanks to (H3), (H4), and the uniform bound (2.4b). We define an operator

Ψ : C δ Ñ C δ ,
such that, for a given ν P C δ , Ψν :" µ is the solution to the following "McKean-Vlasov" equation (2.8) B t µ ´σ∆µ ´divpµGpx; νptqqq " 0, µp0q " m 0 .

Let us check that Ψ is well defined. Note that the above equation can be written as B t µ ´σ∆µ ´xDµ, Gpx; νptqqy ´µdiv pGpx; νptqqq " 0.

Using In the same way, one checks that functions px, tq Ñ Gpx, νptqq and px, tq Ñ div rGpx, νptqqs are in C γ 1 {2,γ 1 pQ T q, where γ 1 " minpγ, θq, thanks to Lemma 2.1 and (H4). Here γ and θ are the H ölder exponents appearing in (H6) and (2.4b) respectively. We infer that problem (2.8) has a unique solution µ P C 1`γ 1 {2,2`γ 1 pQ T q which satisfies (2.9)

}µ} C 1`γ 1 {2,2`γ 1 ď C }G} C 1 }m 0 } C 2`γ 1 ,
owing to existence and uniqueness theory for parabolic equations in H ölder spaces [23, Theorem IV.5.1 p. 320]. Furthermore, using classical properties of Fokker-Planck equation (see Lemma A.1), it follows that

d 1 pµptq, µpsqq ď C T p1 `}G} 8 q |t ´s| 1{2 .
Therefore µ P C δ for big enough δ, since }G} 8 and C T does not dependent on ν nor on δ.

In particular, the operator Ψ is well defined form C δ into C δ X C 1`γ 1 {2,2`γ 1 pQ T q. Let us check now that Ψ is continuous. Given a sequence ν n Ñ ν in C δ , let µ n :" Ψν n , and µ :" Ψν.

Invoking Ascoli's Theorem, estimate (2.9) and the uniqueness of the solution to (2.8), it holds that lim

n }µ n ´µ} C 1,2 " 0.
The convergence is then easily proved to be in Cpr0, T s, PpQqq. Thus, by Schauder fixed point theorem the map Ψ : C δ Ñ C δ has a fixed point µ P C 1,2 pQ T q and pv ρ rµs, µq is a classical solution to (1.5). In addition, estimate (2.7a) entails that v ρ rµs P C 1{2 `r0, T s; C 2 pQq ˘. Uniqueness : Let pv, µq and pv 1 , µ 1 q be two solutions to the system (1.5), w :" v ´v1 and ν :" µ ´µ1 . One checks that # B t ν ´σ∆ν ´divpνH p px, Dvqq " div `µ1 `Hp px, Dvq ´Hp px, Dv 1 q ˘˘, ν t"0 " 0.

˙.

From (2.7a) and (2.9), we infer that ż

Q |µ 1 Dw| 2 ď C}m 0 } 2 C 2`γ d 1 pµ, µ 1 q 2 ď C}m 0 } 2 C 2`γ }νptq} 2 2 .
Plugging this into (2.10) provides

}νptq} 2 2 ď C ż t 0 }νpsq} 2 2 ds,
which implies that ν " 0, and so w " 0 thanks to (2.7a). The proof is complete.

Let us now deal with system (1.6). We shall start by proving the well-posedness for the first equation in (1.6) and by giving a continuous dependence estimate.

Lemma 2.6. Under conditions (H1)-(H3), (H5) and (C1.), for any measure m P PpQq, there exists a unique solution pλrms, urmsq P R ˆC2 pQq to the problem

(2.11) ´σ1 ∆u `Hpx, Duq `λ " F px, mq in Q, ă u ą" 0.
Moreover, for any m, m 1 P PpQq, the following estimates hold

(2.12a) ˇˇλrms ´λrm 1 s ˇˇď κ F d 1 pm, m 1 q, (2.12b) › › urms ´urm 1 s › › C 2 ď χ 1 κ F d 1 pm, m 1 q. Proof.
It is well known (see e.g. [START_REF] Alvarez | Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF]) that for a given m P PpQq, there exists a unique periodic solution pλrms, urmsq in R ˆCpQq to (2.11). Regularity of the solution, and estimates (2.12a), (2.12b) follow from Lemma 2.3, and small-discount approximation techniques (see e.g. [START_REF] Alvarez | Ergodicity, stabilization and singular perturbations for Bellman-Isaacs equations[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF][START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF]).

Remark 2.7. It is possible to show more regularity for the maps m Ñ λrms, m Ñ urms under additional regularity assumptions on F and H. For instance, if H pp ě κ e I d for some κ e ą 0, and F satisfies

sup m‰m 1 d 1 pm, m 1 q ´1 › › › › δF δm p., m, .q ´δF δm p., m 1 , .q › › › › C 0 ˆC1
ă 8, then ur.s and λr.s are of class C 1 in PpQq. We refer to [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] for the definition of derivatives in PpQq and notations. In addition, we have that δu δm pmqpνq :" wpm, νq and δλ δm pmqpνq :" ςpm, νq for any m in PpQq and any signed measure ν on Q, where pς, wq is the solution to the following problem ´σ1 ∆w `Hp px, Durmsq .Dw `ς " δF δm pmqpνq in Q, and w " 0.

One has also an analogous result for the map v r.s defined in Lemma 2.1. We omit the details and invoke [8, Proposition 3.8] for a similar approach.

We prove now well-posedness for system (1.6).

Theorem 2.8. Under assumptions (H1)-(H6) and (C1.), there exists a unique classical solution pλ, u, mq in C 1{2 pr0, T sq ˆC1{2 `r0, T s; C 2 pQq ˘ˆC 1,2 `QT ˘to system (1.6). This result holds if one replaces condition (C1.) by condition (C2.).

Proof. The proof of existence relies on small-discount approximation techniques. We give here an adaptation of these techniques for the quasi-stationary case. The crucial point in this proof is estimates (2.4a) and (2.4b).

Assume first that H satisfies condition (C1.). Let pv , µ q be the unique classical solution to (1.5) replacing ρ by , and set w :" v ´ v . Invoking (2.4a) and (2.4b), we have

(2.13) $ ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % ´σ1 ∆w `Hpx, Dw q ` w " F px, µ ptqq ´ ă v ą in Q T , B t µ ´σ∆µ ´div pH p px, Dv qµ q " 0 in Q T , µ p0q " m 0 in Q, sup 0ďtďT }w ptq} C 2`θ ď κ 0 , sup 0ďtďT } v rµptqs} 8 ď }F } 8 `κH .
On the other hand, recall that according to [23, Theorem IV.5.1 p. 320] it holds that

(2.14) }µ } C 1`γ 1 {2,2`γ 1 ď C 1 }m 0 } C 2`γ 1 ,
where γ 1 " minpγ, θq, and the constant C 1 ą 0 is independent of thanks to (2.4b). Hence, one can extract a subsequence n Ñ 0 such that for any t P r0, T s (2.15) p n v n ptq , w n ptq, µ n q Ñ pλptq, uptq, mq in R ˆC2 pQq ˆC1,2 pQ T q as n Ñ 8, where pλ, u, mq is a classical solution to (1.6). In addition, for any t, s P r0, T s, estimates (2.12a) and (2.12b) provide

}urmptqs ´urmpsqs} C 2 ď χ 1 κ F d 1 pmptq, mpsqq,
and |λrmptqs ´λrmpsqs| ď κ F d 1 pmptq, mpsqq. Thus, urms P C 1{2 `r0, T s; C 2 pQq ˘and λrms P C 1{2 pr0, T sq. The proof of uniqueness is identical to Theorem 2.5. Hence, the proof of well-posedness under (C1.) is complete.

If we suppose that H satisfies only (C2.), by virtue of (2.1) one can derive the following uniform bound using Bernstein's method (see [START_REF] Lasry | Mean field games[END_REF][START_REF]Jeux à champ moyen. I. Le cas stationnaire[END_REF] and [4, Theorem 2.1]):

(2. [START_REF]Evolution of wealth in a non-conservative economy driven by local Nash equilibria[END_REF])

Dκ B ą 0, @ν P PpQq N , }Durνs} 8 ď κ B .
Thus, by a suitable truncation of H one reduces the problem to the previous case.

Remark 2.9. All the results of this section hold true if one replaces the elliptic parts of the equations with a more general operator L of the following form:

L :" ´Tr `ψpxqD 2 ˘,
where ψ is Z d -periodic, }ψ} Lip ă 8, and there exists κ ψ ą 0 such that ψpxq ě κ ψ I d .

MODELS EXPLANATION & MEAN FIELD LIMIT

We provide in this section a rigorous interpretation for the quasi-stationary systems (1.5) and (1.6) in terms of N -players stochastic differential games. We shall start by writing systems of equations for N players, then we pass to the limit when the number of players goes to infinity assuming that all the players are identical. Throughout this section, we employ the notations introduced in Lemma 2.1 and Lemma 2.6.

3.1. Stochastic differential games models for N -players. We consider a game of Nplayers where at each time agents choose their strategy -assuming no evolution in their environment; -according to an evaluation of their future situation emanating from the choice.

Observing the evolution of the system, players adjust their strategies without anticipating. More precisely, each player observe the state of the system at time t and chooses the best drift vector field α t p.q which optimize her/his future evolution ps ą tq. The player adapts and corrects her/his choice as the system evolves. This situation amounts to resolving at each moment an optimization problem which consists in finding the vector field (strategy) which guarantees the best future cost. Our agents are myopic: they anticipate no evolution and only undergo changes in their environment.

Let us now give a mathematical formalism to our model. Let pW j q 1ďjďN be a family of N independent Brownian motions in R d over some probability space pΩ, F, Pq, and pD i q 1ďiďN be closed subsets of R d . We suppose that the probability space pΩ, F, Pq is rich enough to fulfill the assumptions that will be formulated in this section. Let V :" pV 1 , ..., V N q be a vector of i.i.d random variables with values in R d that are independent of pW j q 1ďjďN and let F t :" σ V j , W j u , 1 ď j ď N, u ď t ( be the information available to the players at time t. We suppose that F t contains the P-negligible sets of F. Consider a system driven by the following stochastic differential equations (3.1) dX i t " α i t pX i t q dt `?2σ i dW i t , X i 0 " V i , i " 1, ..., N. For any t ě 0, the i-th player choses α i t in the set of admissible strategies denoted by A i , that is, the set of Z d -periodic processes α i defined on Ω, indexed by R d with values in D i , such that (3.2) sup ωPΩ }α i pω, .q} Lip ă 8.

The reason of considering condition (3.2) will be clear in (3.4) below. At each time t ě 0, player i faces an optimization problem for choosing α i t p.q P A i which insures the best future cost. We will explain the optimization problem in Section 3.1.1.

These instant choices give rise to a global (in time) strategies pα 1 t , ..., α N t q tě0 which does not necessarily guarantee the well-posedness of equations (3.1) in a suitable sense. Hence we need to introduce the following definitions: Definition 3.1. Let T ą 0 and i " 1, ..., N . We say that the i-th equation of (3.1) is wellposed on r0, T s, if there exists a process X i , unique a.s, with continuous sample paths on r0, T s which satisfies the following properties:

(i) pX i q tPr0,T s is pF t q tPr0,T s -adapted;

(ii) P " X i 0 " V i ‰ " 1; (iii) P " ş t 0 ˇˇα i s pX i s q ˇˇds ă 8 ı " 1 @t P r0, T s; (iv) for any t P r0, T s, the following holds

X i t " V i `ż t 0 α i s pX i s q ds `?2σ i W i t a.s .
System (3.1) is well-posed if all equations are. Definition 3.2. Let T ą 0 and i " 1, ..., N . We say that the global strategy pα i t q tě0 is feasible on r0, T s, if the i-th equation of (3.1) is well-posed on r0, T s.

Note that in contrast to standard optimal control situations, the optimal global strategy is not a solution to a global (in time) optimization problem, but it is the history of all the choices made during the game. The agents plan and correct their plans as the game evolves, and the global strategy is achieved through this process of planning and selfcorrection.

3.1.1. The case of a long time average cost. Consider the case where the i-th player seeks to minimize the following long time average cost:

(3.3) J i `t, V, α 1 t , ..., α N t ˘:" lim inf τ Ñ`8 1 τ E "ż τ t L i pX i s,t , α i t pX i s,t qq `F i pX i s,t ; X ´i t q ds ˇˇF t  ,
where L i : R d ˆDi Ñ R and F i : R d ˆRdpN´1q Ñ R are continuous and Z d -periodic with respect to the first variable. At any time t ě 0, the process pX i s,t q sąt represents the possible future trajectory of player i, related to the chosen strategy (vector field) α i t P A i . In other words, pX i s,t q sąt is what is likely to happen (in the future s ą t) if player i plays α i t at the instant t. Mathematically, we consider that pX i s,t q sąt are driven by the following (fictitious) stochastic differential equations

(3.4) $ & % dX i s,t " α i t pX i s,t q ds `b2σ 1 i dB i s´t,t s ą t, X i t,t " X i t , i " 1, ..., N,
where tpB i .,t q 1ďiďN u tě0 is a family of standard Brownian motions, and for any t ě 0, the process pB i s´t,t q sąt represents the noise related to the future prediction (or guess) of the i-th player. For simplicity, we assume that for any i P t1, ..., N u, t ě 0, and s ą t, (3.5) B i s´t,t is independent from F t . Note that system (3.4) is well-posed in the strong sense, and that the definition of pX i s,t q sąt introduces a fast (instantaneous) scale 's' related to the projection in future, which is different from the real (slow) scale 't'.

The cost functional (3.3) is an evaluation of the future cost of player i, given the information available at time t. In this model we consider that the evaluation horizon is infinite. The cost structure expresses the fact that agents are myopic: they anticipate no future change and act as if the system will remain immutable. As they adjust, they undergo changes and do not anticipate them.

We now give a definition of Nash equilibrium for our game.

Definition 3.3. We say that a vector of global strategies pα 1 t , ..., αN t q tě0 is a Nash equilibrium of the N -person game on r0, T s, for the initial position V " pV 1 , ..., V N q, if for any i " 1, ..., N , pα i t q tě0 is feasible on r0, T s, and αi t " arg max

α i PA i J i `t, V, α1 t , ..., αi´1 t , α i , αi`1 t , .
.., αN t ˘a.s @t P r0, T s.

In other words, a Nash equilibrium on r0, T s is the history of local Nash equilibria, which is feasible on [0,T].

Next we provide a verification result that produces a Nash equilibrium for the Nperson game associated to the cost (3.3). Let us introduce the following notation for empirical measures:

νM Y :" 1 M M ÿ i"1 δ Y i , @Y " pY i q P R M d .
For any i " 1, ..., N , we suppose that F i depends only on x P Q and on the empirical density of the other variables. Namely, for any x P Q and Y " pY 1 , ..., Y N ´1q P R dpN ´1q ,

F i px; Y q :" F i ´x; νN´1 Y ¯.
Set for px, pq P Q ˆRd ,

H i px, pq :" sup αPD i
´p.α ´Li px, αq ( .

Throughout this section, we assume that assumptions of Theorem 2.8 hold for H i and F i , and that the supremum is achieved at a unique point ᾱi in the definition of H i , for all px, pq, so that (3.6) H i p px, pq " ´ᾱ i px, pq :" arg max

αPD i ´p.α ´Li px, αq ( .
We also employ the notations introduced in Lemma 2.6: namely, for any π P PpQq, we denote by pλ i rπs, u i rπsq the unique solution to

´σ1 i ∆u i `Hi px, Du i q `λi " F i px, πq in Q, ă u ą" 0.
Remark 3.4. It is possible to consider a more general form for the drift in system (3.4). For instance, one can replace α by the following (more general) affine form:

f i px, αq :" g i pxq `Gi pxqα,
where G i P LippQq dˆd and g i P LippQq d . Then

H i px, pq " ´p.g i `sup αPD i
´p.G i pxqα ´Li px, αq ( . then the supremum in the definition of H i is attained. Uniqueness of the supremum holds if L i is strictly convex with respect to the second variable.

If L i is Lipschitz in x,
The following result characterizes a Nash equilibrium on r0, T s associated to the cost functional (3.3). Proposition 3.5.

(1) The following system of equations is well-posed on r0, T s,

(3.7) d Xi t " ᾱi ´X i t , Du i " νN´1 X´i t ı p Xi t q ¯dt `?2σ i dW i t , Xi 0 " V i , i " 1, ..., N.
(2) Let for x P Q and t P r0, T s

ᾱi t pxq :" ᾱi ´x, Du i " νN´1 X´i t ı pxq ¯, i " 1, ..., N.
The vector pᾱ 1 t , ..., ᾱN t q tě0 defines a Nash equilibrium on r0, T s for any initial data. (3) The following holds

λ i " νN´1 X´i t ı " lim inf τ Ñ`8 1 τ E "ż τ t L i p X i s,t , ᾱi t p X i s,t qq `F i ´X i s,t , νN´1 X´i t ¯ds ˇˇF t  ,
where p X i s,t q sąt are obtained by solving $ & % d X i s,t " ᾱi t p X i s,t q ds `b2σ 1 i dB i s´t,t , s ą t, X i t,t " Xi t , i " 1, ..., N. Proof. Assertion (1) is a consequence of the regularity results of Lemma 2.6, while assertions (2) and (3) follows by standard verification arguments (see e.g. [START_REF] Bardi | Nonlinear elliptic systems and mean-field games[END_REF]Theorem 3.4], among many others). For any τ ą t ě 0 and i P t1, .., N u one has

u i p X i τ,t q " u i p X i t,t q `ż τ t Du i p X i s,t q.ᾱ i t p X i s,t q ds `ż τ t σ 1 i ∆u i p X i s,t q ds `b2σ 1 i ż τ t Du i p X i s,t q dB i s´t,t ,
where here u i " u i " νN´1

X´i t ı in order to simplify the presentation. Owing to (3.6) one gets

u i p X i τ,t q " u i p X i t,t q `ż τ t `´H i p X i s,t , Du i p X i s,t qq `σ1 i ∆u i p X i s,t q ˘ds `b2σ 1 i ż τ t Du i p X i s,t q dB i s´t,t ´ż τ t L i p X i s,t , ᾱi t p X i s,t qq ds " u i p X i t,t q ´ż τ t L i p X i s,t , ᾱi t p X i s,t qq `F i p X i s,t ; X´i t q ( ds `pτ ´tqλ i `b2σ 1 i ż τ t Du i p X i s,t q dB i s´t,t .
Hence, from (3.5) we infer that

τ ´1E " u i pX i τ,t q | F t ‰ " p1 ´tτ ´1qλ i `τ ´1E " u i pX i t,t q | F t ‰ ´τ ´1E "ż τ t L i p X i s,t , ᾱi t p X i s,t qq `F i ´X i s,t , νN´1 X´i t ¯ds ˇˇF t  .
Note that estimate (2.12b) provides a uniforme bound on u i r.s. Thus, by taking the limit in the last expression one gets

λ i " νN´1 X´i t ı " lim inf τ Ñ`8 1 τ E "ż τ t L i p X i s,t , ᾱi t p X i s,t qq `F i ´X i s,t , νN´1 X´i t ¯ds ˇˇF t  .
On the other hand, one easily checks that pᾱ 1 t , ..., ᾱN t q tě0 is a Nash equilibrium for any initial data V " pV 1 , ..., V N q owing to (3.6).

Remark 3.6. Note that the problem structure decouples the "fictitious" dynamics (3.4), and allows to compute the controls.

3.1.2. The case of a discounted cost functional. Set ρ 1 , ..., ρ N ą 0. We consider now the case where the i-th player seeks to minimize the following discounted cost functional: where all the functions are defined in the same way as in the previous case, with analogous notations and assumptions. One checks that a similar result to Proposition 3.5 holds, i.e. that the following problem:

(3.8) J i ρ i `t, V,
(3.9)

d Zi t " ᾱi ´Z i t , Dv i ρ " νN´1 Z´i t ı `Z i t ˘¯dt `?2σ i dW i t , Zi 0 " V i , i " 1, ..., N,
characterizes a Nash equilibrium on r0, T s associated to the cost functional (3.8).

3.2. The mean field limit N Ñ `8. We address now the convergence problem when the number of players goes to infinity, assuming that all the players are indistinguishable. Assume that: D i " D; ρ i " ρ; σ i " σ; σ 1 i " σ 1 ; F i " F ; H i " H; and ᾱi " ᾱ so that L i " H ˚for all 1 ď i ď N , where H ˚is the Legendre transform of H with respect to the p variable. We suppose also that LpV i q " m 0 P C 2`γ pQq for any i " 1, ..., N.

For simplicity we shall use the notations X t :" `X1 t , ..., X N t ˘and Z t :" `Z1 t , ..., Z N t ȋnstead of Xt :" `X Zi 0 " V i , i " 1, ..., N. Our main result in this section says that at the mean field limit N Ñ 8, one recovers the quasi-stationary systems (1.6) and (1.5), which respectively correspond to (3.10) and (3.11). Note that systems (1.6) and (1.5) can be rewritten on the form of Mckean Vlasov equations:

(1.6) $ ' & ' % B t m ´σ∆m ´div pmH p px, Durmptqsqq " 0 in Q T , mp0q " m 0 in Q, and (1.5) $ ' & ' % B t µ ´σ∆µ ´div pµH p px, Dv ρ rµptqsqq " 0 in Q T , µp0q " m 0 in Q.
Thus, one can use the usual coupling arguments (see e.g. [START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF][START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF]) to deduce the convergence. The main theorem of this section is the following: The analysis of the limit transition N Ñ `8 is essentially based on continuous dependence estimates, and therefore the mean field analysis is identical for both systems. Thus, we shall give the details only for system (3.10).

Let us introduce the following artificial systems:

(3.12) On the other hand, the following holds

$ ' ' & ' ' % dY i " ´Hp `Y i t , Du rmptqs pY i t q ˘dt `?2σ dW i t , 0 ď t ď T, Ȳ i 0 " V i , i
|T 3 ptq `T4 ptq| ď C| Xi t ´Y i t |.
The key step is the estimation the non-local term E d 1 pm, μN Y q; we use the following estimate due to Horowitz and Karandikar (see [START_REF] Rachev | Mass Transportation Problems. Vol I: Theory; Vol II: Applications, Probability and Its Applications[END_REF]Theorem 10.2.7]):

E d 1 pmptq, μN
Yt q ď κ d N ´1{pd`8q @t P r0, T s, where the constant κ d ą 0 depends only on d. Using the symmetry of the joint probability law (3.14) and the last estimate, we infer that

E ˇˇX i t ´Y i t ˇˇď C ż t 0 ˆ1 N 1{pd`8q `E ˇˇX i s ´Y i s ˇˇ˙ds,
which concludes the proof.

Recall the following definition and characterizations of chaotic measures [START_REF] Sznitman | Topics in propagation of chaos[END_REF].

Definition 3.9. Let π N be a symmetric joint probability measure on Q N and π P PpQq.

We say that π N is π-chaotic if for any k ě 1 and any continuous functions φ 1 , ..., φ k on Q one has

lim N ż k ź l"1 φ l dπ N " k ź l"1 ż φ l dπ.
Lemma 3.10. Let X N be a sequence of random variables on Q N whose the joint probability law π N is symmetric, and π P PpQq. Then the following assertions are equivalent:

(i) π N is π-chaotic;
(ii) the empirical measure νN X N converges in law toward the deterministic measure π; (iii) for any continuous function φ on Q, it holds that

lim N E ˇˇˇż φ d `ν N X N ´π˘ˇˇˇ" 0.
Combining Lemma 3.10 and Proposition 3.8, we deduce the propagation of chaos for system (3.13). Proposition 3.11. For any t P r0, T s, if m N ptq is the joint probability law of Xt :" p Xj t q 1ďjďN , then m N ptq is mptq-chaotic.

Proof. Let φ be a Lipschitz continuous function on Q. From Proposition 3.8, we have that

E ˇˇˇż φ dpμ N Xt ´μ N Yt q ˇˇˇď }φ} Lip N N ÿ k"1 E ˇˇX i t ´Y i t ˇˇď }φ} Lip C T N 1{pd`8q .
Invoking the fact that LpY t q " b N i"1 mptq and Lemma 3.10, it holds that lim N E ˇˇˇż φ dpmptq ´μ N Yt q ˇˇˇ" 0.

The claimed result follows from

E ˇˇˇż φ dpμ N Xt ´mptqq ˇˇˇď E ˇˇˇż φ dpμ N Xt ´μ N Yt q ˇˇˇ`E ˇˇˇż φ dpmptq ´μ N Yt q ˇˇˇ.
We are now in position to prove Theorem 3.7.

Proof of Theorem 3.7. Observe that for any two random variables X, Y, one has

d 1 pLpXq, LpYqq ď E |X ´Y| .
Under the monotonicity condition (1.2), uniqueness holds for system (4.2). In all this section p λ, ū, mq denotes the unique solution to (4.2). Observe that m " e ´ū { e ´ū , so that the following holds (4.3) 1{κ ď m ď κ for some constant κ ą 0.

The main result of this section is the following:

(4.7b) }wrπs} C 2 ď C 1 }π} 2 .
Proof. Existence and uniqueness of regular solutions to such problems are discussed in Section 2. Estimates (4.7a)-(4.7b) are a direct consequence of the uniqueness and the continuous dependence estimates (2.12a)-(2.12b).

Next we give the following technical Lemma.

Lemma 4.3.

There exists a constant κ ą 0 such that }π{ m} 2 ď κ}Dpπ{ mq} 2 , for any π P V 1,2 pQq :" tπ P W 1,2 pQq { ă π ą" 0u.

Proof. As usual, the result is obtained by contradiction. In fact, if our claim is not satisfied one can find a sequence pπ n q P V 1,2 pQq such that for any n ě 1, (4.8) }π n { m} 2 " 1 and

1 n ě }Dpπ n { mq} 2 .
By Sobolev embeddings, pπ n { mq n converges (up to a subsequence) to some π in L 2 pQq. Using (4.8) it follows that π is constant, i.e. π " C. Moreover,

C " ż Q mπ " lim n ż Q π n " 0;
this provides the desired contradiction owing to (4.8).

Combining these elements one can prove the main Theorem of this section.

Proof of Theorem 4.1. Let pς, w, πq be a smooth solution to (4.5) for any t ě 0.

Estimates of Lemma 4.2 conclude the proof.

Remark 4.4. One notices that the previous proof can be adapted to show that (1.5) converges exponentially fast to (1.3) when the discount rate ρ is small enough, under the same assumptions of Theorem 4.1. In fact, setting π :" µ ´μ, the same arguments leading to (4.13) also provide d dt }πptq 2 {μ} 1 ď ´1{ C0 }πptq 2 {μ} 1 `M }πptq 2 {μ} 2 1 `Cρ}πptq 2 {μ} 1 .

Therefore, the same conclusion holds when ρ is small enough.

Remark 4.5. In practice, this convergence results can help to understand the emergence of highly-rational equilibria in situations with myopic decision-making mechanisms. For instance in [12, Section 2.2.2] the authors consider a decision-making mechanism for pedestrian dynamics that is very similar to the mechanism described in Section 3. Theorem 4.1 can apply for this kind of models, in the case of a quadratic running cost, and a monotonous coupling function (pedestrians dislike congested areas) which satisfies (H1) and (H5). For instance F can take the following form:

F px, mq " ż R d φ ˚mpyqφpx ´yq dy,
where ˚is the usual convolution product (in R d ), and φ is a smooth, even function with compact support.

APPENDIX A. ELEMENTARY FACTS ON THE FOKKER-PLANCK EQUATION

Let V : r0, T s ˆQ Ñ R be a given bounded vector field, which is continuous in time and H ölder continuous in space, and we consider the following Fokker-Planck equation:

(A.1)

#

B t m ´σ∆m ´divpmV q " 0 in p0, T q ˆQ, mp0q " m 0 in Q;

and the following stochastic differential equation:

(A.2) dX t " V pt, X t q dt `?2σ dB t t P p0, T s, X 0 " Z 0 , where pB t q is a standard d-dimensional Brownian motion over some probability space pΩ, F, Pq and Z 0 P L 1 pΩq is random and independent of pB t q. Under these assumptions, there is a unique solution to (A.2) and the following hold:

Lemma A.1. If m 0 " LpZ 0 q then, mptq " LpX t q is a weak solution to (A.1) and there exists a constant C T ą 0 such that, for any t, s P r0, T s,

d

(2. 1 )

 1 sup mPPpQq }F p., mq} Lip ă 8; (H2) the Hamiltonian H : QˆR d ÝÑ R is locally Lipschitz continuous, and Z d -periodic with respect to the first variable; (H3) H p exists and is locally Lipschitz continuous; (H4) D x H p and H pp exist and are locally Lipschitz continuous; (H5) there exists a constant κ F ą 0 such that, for any m, m 1 P PpQq, }F p., mq ´F p., m 1 q} 8 ď κ F d 1 pm, m 1 q;

  assumption (H3) and estimate (2.7a), we have for any t ‰ s and x P Q |Gpx; νptqq ´Gpx; νpsqq| ď C ρ,Hp d 1 pνptq, νpsqq,

	so that
	sup	sup
	xPQ

t‰s |Gpx; νptqq ´Gpx; νpsqq| |t ´s| 1{2 ď C δ,ρ,Hp ă 8.

  Dν. `Hp px, Dvq ´Hp px, Dv 1 q ˘˙,

	By standard duality techniques, we deduce that
	1 2 µ 1 so that d dt }νptq} 2 2 `σ}Dνptq} 2 2 " ´ˆż Q νDν.H p px, Dvq `żQ
	(2.10)	1 2	d dt	}νptq} 2 2 `σ 2	}Dνptq} 2 2 ď C ˆ}G} 8 }νptq} 2 2	`żQ	|µ 1 Dw| 2

  uniformly as α varies in any bounded subset, and asymptotically super-linear, i.e. lim

	inf
	|α|Ñ`8

xPQ L i px, αq{|α| " `8,

Theorem 3 . 7 .

 37 For any t P r0, T s, it holds that:

	lim N	max 1ďiďN	d 1	`L `Xi t ˘, mptq ˘" 0;
	lim N	max 1ďiďN	d 1	`L `Zi t ˘, µptq ˘" 0;
	lim N	› › urmptqs ´Eu	"	νN Xt	‰› › 8 " 0;
	lim N	ˇˇλrmptqs ´Eλ	"	νN Xt	‰ˇˇ"	0; and
	lim N	› › v ρ rµptqs ´Ev ρ		"	νN Zt	‰› › 8 " 0.

  Next we compare the trajectories of (3.12) and (3.13), and show that they are increasingly close on r0, T s when N Ñ `8.

	Observe that systems (3.12)-(3.13) are well-posed, and that the uniqueness of the solution
	to (1.6) provides that
					L `Y 1 t , ..., Y N t ˘" b N i"1 mptq.
	On the other hand, note that
	(3.14)			L ´X t ξp1q	, ...,	XξpNq t	¯" L ´X 1 t , ..., XN t	īs
	fulfilled for any permutation ξ, and any t P r0, T s. In addition, one checks that
	(3.15)			max 1ďiďN	sup 0ďtďT	E ˇˇX i t ´X i t ˇˇď	C T N ´1
	holds thanks to the continuous dependence estimate (2.12b), since
				sup 0ďtďT	max 1ďiďN	d 1 ´ν N Xt , νN´1 X ´i t	¯ď C N ´1 .
	Proposition 3.8. Under assumptions of this section, it holds that
	max 1ďiďN t ˇˇď C d sup 0ďtďT E ˇˇX i t ´Y i dt " Xi t ´Y i t ı " H p `Y i t , DurmptqspY i t q ˘´H p ´X i t , Du	"	νN Xt	‰	t q p Xi "
					T 1 ptq `T2 ptq `T3 ptq `T4 ptq,
	where
		T 1 ptq :" H p pY i t , DurmptqspY i t qq ´Hp pY i t , Du	"	νN Yt	‰	pY i t qq,
		T 2 ptq :" H p pY i t , Du	"	νN Yt	‰	pY i t qq ´Hp pY i t , Du	"	νN Xt	‰	pY i t qq,
		T 3 ptq :" H p pY i t , Du	"	νN Xt	‰	pY i t qq ´Hp p Xi t , Du	"	νN Xt	‰	pY i t qq,
	and
		T 4 ptq :" H p p Xi t , Du	"	νN Xt	‰	pY i t qq ´Hp p Xi t , Du	"	νN Xt	‰	p Xi t qq.
	Using the continuous dependence estimate (2.12b) one gets
	|T 2 ptq| ď	C N	N j"1 ÿ	| Xj t	´Y j t |, and |T 1 ptq| ď C d 1 pmptq, μN Yt q.
				" 1, ..., N ;
	and
	$ ' '	d Xi " ´Hp ´X i t , Du	"	νN
	&
	(3.13)
	'
	'
	%

Xt

‰

p Xi t q ¯dt `?2σ

dW i t , 0 ď t ď T, Xi 0 " V i , i " 1, ..., N. T N ´1{pd`8q .

Proof. For any i P t1, ..., N u and t P r0, T s, one has

  divp mDwq " m∆w `xD m, Dwy " ´m p´∆w `xDū, Dwyq .Hence, using (4.9), (4.7b) and the monotonicity of F , one has { m} 1 ď ´1{C 0 }πptq 2 { m} 1 `M }πptq 2 { m}2 1 ,for some C 0 , M ą 0, thanks to (4.3) and Young's inequality. For any R 0 ă

	so that (4.9) and (4.10) We infer that divp mDwq " ´m ˆ´ςptq D ´π m ¯" Dπ `πDū m ´1 2 |Dw| 2 `F px, m `πptqq ´F px, mq , ˙, which provides in particular dt ż Q π 2 2 m " ż Q π m " ∆π `divpπDūq `divpπDwq ´m ˆ´ςptq ´1 2 |Dw| 2 `F px, m `πptqq ´F px, mq ˙ı " ż Q " ´|Dπ `πDū| 2 m `π|Dw| 2 2 ´πxDπ, Dwy m ´π2 xDū, Dwy m ´żQ π pF px, m `πptqq ´F px, mqq ď ż Q ´m ˇˇˇD ˆπptq m ˙ˇˇˇ2 `C}πptq} 3 2 `1{κ}Dπptq} 2 }πptq} 2 2 . ď ´1{κ ż Q ˇˇˇD ˆπptq m ˙ˇˇˇ2 `C p}πptq} 2 `}Dπptq} 2 q }πptq} 2 2 . (4.11) Using Lemma 4.3, one easily checks that (4.12) }πptq} 2 `}Dπptq} 2 ď C › › › › D ˆπptq m ˙› › › › 2 . Thus the following holds (4.13) d dt }πptq 2 1 ? κM C 0 , the last * differential inequality entails that B d 1{M C 0 }πptq 2 { m} 1 ď 1 `´1 M C 0 }πp0q 2 { m} 1 ´1¯e t{C 0

. Recall that D m " ´e´ū Dū{ ă e ´ū ą" ´mDū, t π ´∆π ´divpπDūq`m ˆ´ςptq ´1 2 |Dw| 2 `F px, m `πptqq ´F px, mq ˙´divpπDwq " 0.

  1 pmptq, mpsqq ď C T p1 `}V } 8 q|t ´s| 1{2 . Proof. The first assertion is a straightforward consequence of It ô's formula. On the other hand, for any 1-Lipschitz continuous function φ and any t ě s, one has ż T d φpxq dpmptq ´mpsqqpxq ď E|φpX t q ´φpX s q| ď E|X t ´Xs |

	"ż t
	ď E

s

|V pu, X u q| du `?2σ|B t ´Bs |  ď }V } 8 pt ´sq `a2σpt ´sq.

Hence, combining Lemma 3.8 and estimate (3.15) Remark 3.12. Note that the two main arguments in the proof of Theorem 3.7 are the continuous dependence estimate, and symmetry with respect to states of the other players.

EXPONENTIAL CONVERGENCE TO THE ERGODIC MFG EQUILIBRIUM

We prove in this section the exponential convergence of the quasi-stationnary system (1.6) to the ergodic equilibrium assuming that σ 1 " σ and Hpx, pq " |p| 2 {2. The proofs rely on algebraic properties of the equations, the continuous dependence estimates (Lemma 2.11), and the monotonicity condition (1.2). Throughout this section we suppose that assumptions (H1),(H5), and (H6) are fulfilled. In addition, we assume that the coupling F satisfies the monotonicity condition:

For the sake of simplicity we set σ " σ 1 " 1.

In this framework the quasi-stationary MFG system (1.6) takes the following form, (4.1)

System (4.1) has a unique global (in time) classical solution thanks to Theorem 2.8. Consider the following ergodic Mean Field Games problem:

Theorem 4.1. There exists R 0 ą 0 such that if

then the following holds for some constants A, δ ą 0:

|λptq ´λ| `}uptq ´ū} C 2 `}mptq ´m} 2 ď Ae ´δt for any t ě 0.

This convergence result reveals that our decision-making mechanism lead to the emergence of a Mean Field Games equilibrium, under the conditions mentioned above. This can also be interpreted as a phase transition from a non-equilibrium state to an equilibrium state (see also [START_REF] Mehta | Synchronization of Coupled Oscillators is a Game[END_REF][START_REF]Bifurcation Analysis of a Heterogeneous Mean-field Oscillator Game[END_REF]). Agents reach this equilibrium by adjusting and selfcorrecting. We believe that this convergence result holds true in more general cases. For instance, one can show that an analogous convergence result holds for system (1.5) when the discount rate ρ is small enough (c.f. Remark 4.4).

Let pλ, u, mq be the solution to (4.1), and set (4.4) ς :" λ ´λ, w :" u ´ū, and π :" m ´m.

The triplet pς, w, πq is a solution to the following system of equations: πp0q " m 0 ´m in Q, ă w ą" 0 in p0, 8q.

The following preliminary Lemma states the dependence of w and ς on π in the first equation of (4.5). Lemma 4.2. Let be a probability measure on Q which is absolutely continuous with respect to the Lebesgue measure, and such that " m `π, where π P L 2 pQq. There exists a unique periodic solution pςrπs, wrπsq in R ˆC2 pQq to the following problem:


S Schwartz

M Rebut

G Feugnet

J Colineau

J.-P Pocholle

Optimizing the electronic control loop of a solid-state ring laser gyroscope

We study in this Letter the dynamical effects of the limited bandwidth of the control electronics in a solid-state (Nd-YAG) ring laser gyroscope. We derive a stability condition for the rotation-sensing regime in the case of a first-order control loop, showing that the smallest measurable rotation speed depends directly on the cutoff frequency value. Our experimental measurements are in good agreement with this prediction.

Introduction:

The Ring Laser Gyroscope (RLG) is a high-precision rotation sensor used for inertial navigation [START_REF] Macek | Rotation Rate Sensing with Traveling-Wave Ring Lasers[END_REF][START_REF] Aronowitz | Lock-In and Intensity-Phase Interaction in the Ring Laser[END_REF][START_REF] Chow | The ring laser gyro[END_REF]. It typically consists of a ring laser cavity emitting two counterpropagating beams. When such a device is rotated, the eigenfrequency difference ) 2 /( π Ω between the counterpropagating waves is proportional to the speed of rotation θ & according to the following law :

θ π & S 2 = Ω
where S, which depends on the design of the cavity, is known as the scale factor. The frequency of the beat note signal between the counterpropagating waves thus provides a convenient and accurate measurement of angular motions. In order to ensure bidirectional operation, which is required to obtain the beat signal, a gaseous amplifying medium is typically used, circumventing mode competition [START_REF] Siegman | Lasers[END_REF] thanks to Doppler gain broadening (see [START_REF] Macek | Rotation Rate Sensing with Traveling-Wave Ring Lasers[END_REF][START_REF] Aronowitz | Lock-In and Intensity-Phase Interaction in the Ring Laser[END_REF][START_REF] Chow | The ring laser gyro[END_REF]). However, many industrial constraints arise from the use of a gas, which is usually the limiting factor in terms of lifetime, cost and reliability of the RLG. Another option is to use a solid-state gain medium, with an electronic feedback loop for intensity control and stabilization. This option has been previously described and achieved in [START_REF] Schwartz | Mode-Coupling Control in Resonant Devices: Application to Solid-State Ring Lasers[END_REF][START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF], assuming an infinite bandwidth for the theoretical modelization of the electronic feedback loop. In this Letter, we study analytically and experimentally the effects of a limited bandwidth, and we show, in the case of a first order loop, that the value of the smallest measurable rotation speed (i.e. the performance of the gyroscope) directly depends on the loop cutoff frequency. Our experimental results, obtained with a diode-pumped Nd-YAG ring laser gyroscope, are in good agreement with our theoretical predictions. This work could find applications in the design of a new generation of competitive diode-pumped solid-state ring laser gyroscopes.

Theoretical description: The total electric field inside the ring laser cavity is assumed to be the sum of two counterpropagating modes :

[ ]e E ) ( 2
) ( 1 ) ( ) (
( ) 2 ; 1 1 , 2 0 2 , 1 1 , 2 2 , 1 2 , 1 2 , 1 2 , 1 2 , 1 2 2 2 2 2 ) 1 ( d d N E N E T L E m i E E i t E + + + - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Ω + Δ - = σ γ (1)
where Δ is the difference between the loss coefficients of each mode, γ is the average value of these loss coefficients, N are defined by :

∫ ∫ - = = L kx i L x Ne L N x N L N 0 ) 1 ( 2 2 , 1 0 0 d 1 and d 1 1 , 2
, N being the population inversion density function, obeying the following equation :

2 2 1 1 1 1 2 ) 1 ( ikx ikx th e E e E T aN T N T W t N + - - + = ∂ ∂ - η
where W th is the pumping rate at laser threshold, T 1 is the population inversion lifetime (we assume T 1 =230 µs in what follows), η is the excess of pump power above threshold and a is the saturation parameter. We consider the regime of beat note operation, i.e. the regime where the intensities of both modes are approximately constant and equal and where the (angular) frequency difference between both modes is close to the ideal (angular) frequency difference Ω . In the high rotation rate limit, namely :

2 , 1 m >> Ω and 1 / T γη >> Ω (2)
the laser parameters for the beat regime have the following expressions (see reference [START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF]) :

B y t y t B E E M M << + = + with ) ( ) ( ~2 2 2 1 B x C t x t C E E M M << + = - , with ) ( ) ( ~2 2 2 1 ( ) ( ) 1 , with ) ( ) ( ãrg ãrg 0 0 1 2 << Φ Φ Φ + Φ = Ω - - M M t t t E E
where the functions x M , y M and M Φ oscillate with an angular frequency close to Ω while B, C and 0 Φ are slowly varying with respect to the time scale

Ω / 1
. When the loss coefficients of the counterpropagating modes are equal

(i.e. 0 = Δ
), this regime is not a solution of the laser equations (1) and hence does not occur in practice [START_REF] Siegman | Lasers[END_REF][START_REF] Schwartz | Mode-Coupling Control in Resonant Devices: Application to Solid-State Ring Lasers[END_REF][START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF]. To obtain the beat regime, one has to introduce an additional stabilizing coupling, which can be achieved for example by actively creating more losses for the strongest mode. This was modeled in [START_REF] Schwartz | Mode-Coupling Control in Resonant Devices: Application to Solid-State Ring Lasers[END_REF][START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF] by assuming differential losses of the following form :

0 with ~2 2 2 1 > ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - = Δ K E E aK . (3)
However, such a description does not take into account the way those differential losses are created experimentally, namely using an active electronic feedback loop acting on a Faraday magneto-optic element. An efficient yet simple way of designing the electronics configuration turns out to be the use of a first-order loop, resulting in differential losses Δ obeying the following equation :

) ( d d 0 aKC t - Δ - = Δ ω (4)
where ) 2 /( 0 π ω is the cutoff frequency and where the effects of the intensity fast modulations

) (t x M
on the electronics response have been neglected. In the limit of an infinite cutoff frequency, equation ( 4) is consistent with the previous model [START_REF] Chow | The ring laser gyro[END_REF]. Equations ( 1), together with equation ( 4), can be solved following the procedure described in [START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF]. A stationary solution is found for the slowly varying quantities B, C and Δ . Applying small perturbations like t e λ to this stationary solution leads to the following characteristic equation :

0 2 2 2 1 2 0 2 1 2 0 2 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Ω - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Ω - + T K T γη η ω γη ω λ λ . (5)
The beat regime is stable when all the roots of equation ( 5) have a negative real part, which finally yields (using the Routh-Hurwitz criterion) the following simple conditions :

2 1 2 2 1 2 0 2 and 2 T K T Ω > Ω > γη η γη ω . ( 6
)
In the infinite-bandwidth limit, conditions [START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF] reduce to the single condition on K, which is consistent with [START_REF] Schwartz | Mode-Coupling Control in Resonant Devices: Application to Solid-State Ring Lasers[END_REF][START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF]. In the more realistic case of a limited bandwidth, both conditions [START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF] have to be fulfilled to ensure the stability of the beat regime.

Experiment : in order to confirm experimentally the beat regime stability conditions (6), we used a diode-pumped Nd-YAG ring laser gyroscope configuration similar to the one described in [START_REF] Schwartz | Mode-Coupling Control in Resonant Devices: Application to Solid-State Ring Lasers[END_REF][START_REF] Schwartz | Oscillation regimes of a solid-state ring laser with active beat-note stabilization : From a chaotic device to a ring-laser gyroscope[END_REF]. Concerning the electronic control loop, we ensured a first order behaviour, in particular thanks to a current controlled output stage connected to the solenoid of the magneto-optic element, which allows to cancel the frequency response effects of the inductive load. Typical parameters for our control loop are , which implies in particular -see ( 6) -that the condition on 0 ω is more stringent than the condition on K, and will ultimately determine the value of the lowest measurable rotation speed cr θ & , according to the following formula : Conclusion : the influence of the bandwidth of the electronic control loop on the solid-state ring laser gyroscope dynamics has been studied in the case of a first-order configuration. It is found theoretically that the performance of the gyroscope directly depends on the cutoff frequency provided this latter is not too high. Experimental data show very good agreement with this prediction.

This result could find applications in the design of future solid-state highperformance inertial navigation devices.

  parameters describing backscattering on the cold cavity elements, σ is the emission cross section, L is the cavity length, T is the cavity round trip time and 0

.For

  The result is reported on figure 1. Our experimental laser parameters are γ =24.1 s -1 , S=1.37 kHz/(deg/s) and η =0.8. The precision in measuring cr θ & , typically ±5 deg/s, is mainly limited by the hysteretic behavior of the laser gyroscope at the boundary of the beat regime zone, due to the existence of metastable transient regimes with a frequency dependence very similar to the true stable beat regime. As can be seen on figure 1, the measured values for crθ & are in good agreement with our theoretical predictions (7) provided the cutoff frequency is not too high (typically below 500 Hz). For higher values of ) deriving (6) and (7) become more stringent than the second condition of (6) itself, which explains the fact that the agreement between equation (7) and the experimental data of figure1no longer stands. Instead, some other physical parameters such as the strength of the population inversion grating become determining in the laser gyroscope dynamics. We have also checked, considering equation (7), the dependence of cr θ & on the pumping parameter η . we have measured cr θ & =35 deg/s when η =0.8, and cr θ & =25 deg/s when η =0.4. The ratio between those two measured values is very close to 2 , which is consistent with equation (7).

Figure captions:Fig. 1

 1 Figure captions:

  Figure 1

Acknowledgements : the authors acknowledge constant support from Thales Aerospace Division. They also wish to thank Augustin Mignot and Mehdi Alouini for fruitful discussions.