Chapter 1 Introduction

Before discussing the state of the art and introducing the topics covered in this thesis, we proceed to motivate why we get interested in these problems.

The work is divided in two parts: the first is devoted to the systematic study of Mean Field Game (MFG) systems with control on the acceleration; in the second we address the problem of long time-average behavior of solutions to Hamilton-Jacobi equations associated with sub-Riemannian control systems and Aurby-Mathery Theory for these latter. We will explain below how these two topics are related.

Let us start describing the first. In the recent years there have been an increasing attention to the study of multi-agent systems with control on the acceleration, i.e., models of interacting individuals in which each player wants to control their acceleration instead of the classical control of the velocity. For this reason in Chapter 3 we study the well-posedness of MFG systems associated with such control problems. By using a relaxed notion of Nash equilibrium (MFG equilibrium) we provide existence, uniqueness and regularity results for the so-called mild solutions. We conclude by investigating the connections between these solutions and the PDEs system. Then, the aim of Chapter 4 is to study the long time-average behavior of solutions to the MFG system studied in Chapter 3 as the time horizon goes to infinity. The main issue for this is the lack of small time controllability that prevents to define the associated ergodic MFG system in the standard way. We conclude this first part addressing the problem of singular perturbation for "pure" control systems and for MFG with control of acceleration in Chapter 5. In particular, solving this problem we found a relation between MFG of acceleration and the classical system.

At this point, the difficulties in Chapter 4 lead us to the following question: are these issues common to more general control systems than the control of acceleration? To address this problem, we start with a general drift-less control system and in Chapter 6 we address the problem of the long time behavior of solutions to Hamilton-Jacobi equations. Note that, 4 CHAPTER 1. INTRODUCTION the case of control of acceleration does not fit into this class of systems since it has a linear non-zero drift. However, what we immediately realize is that sub-Riemannian control systems are locally small time controllable. So, by using new ideas which relies on the different geometry on the state space we prove the existence of a critical constant and of a critical viscosity solution to the ergodic Hamilton-Jacobi equation. Moreover, we study the well-posedness of the Lax-Oleinik semigroup and we prove the existence of a fixed-point. Finally, in Chapter 7 we extend the well-known Aubry-Mather theory for Tonelli Hamiltonian systems to the sub-Riemannian ones.

More details on the results and on the difficulties to achieve them are given in the following sections.

Mean field games

Since MFG is the common subject of the first part of this thesis, we introduce here the argument and describe the state of the art.

Game theory is a branch of mathematics which aims to describe the behavior of a group of interacting agents. Fix, for instance, this number to N ∈ N. Each player satisfy a certain dynamics that depends on the interaction with the other agents and they choose their strategy in order to minimize/maximize a certain cost functional. A fundamental tool in the analysis of these models is the notion of Nash equilibria, introduce by Nash in [START_REF] Nash | Non-coopertaive games[END_REF]. Roughly speaking, a strategy is called a Nash equilibrium if each agent is not interested to be the unique who changes strategy.

However, the study of the N-players games lead to several issues as N becomes large and, in this case, we are interested in describing the behavior of Nash equilibria as N → ∞. In order to overcame these difficulties, MFG system has been introduced by J.M. Lasry and P.L. Lions in [START_REF] Lasry | Jeux à champ moyen. I: Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II: Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games[END_REF] and a similar analysis was also developed, in the same years but independently, by P. Caines, M. Huang and R. Malhamé in [START_REF] Huang | Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF][START_REF] Huang | Large-population costcoupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ϵ-Nash equilibria[END_REF]. At the macroscopic level the model turns out to be described by a systems of PDEs: an Hamilton-Jacobi equation which describes the single agent's strategy and a Kolmogorov Fokker-Planck equation (continuity equation) which explains how the distribution of players evolves in time according to the optimal strategy provided by the first equation. Classically, the mean-field interaction term that coupled the two equation is given by a function of space and measure and, moreover, the drift appearing in the continuity equation depends on the value function satisfying the Hamilton-Jacobi equation. Let H : R d ×R d → R be an Hamiltonian function, let F : R d ×P(R d ) → R be the coupling function describing the interaction of the agents, let µ 0 ∈ P(R d ) be the initial distribution of players in space and let G : R d × P(R d ) → R be the terminal costs. Then, the simplest for of this system is the following

     -∂ t u(t, x) + H(x, D x u(t, x)) = F (x, m t ), (t, x) ∈ [0, T ] × R d ∂ t m t -div m t D p H(x, D x u(t, x)) = 0, (t, x) ∈ [0, T ] × R d m 0 = µ 0 , u(T, x) = G(x, m T ), x ∈ R d .
Let us describe heuristically the meaning of such a system. To do so, let L : R d × R d → R be the Lagrangian associated with the Hamiltonian H by taking the Legendre Transform. Then, each player choose is own strategy in order to minimize the cost functional of the form

T t L(γ(s), γ(s)) + F (γ(s), m s ) ds + G(γ(T ), m T )
where the control system is of the following simple form γ(s) = u(s), (s ∈ [0, T ]).

There is by now an extensive literature concerning MFG system of the above form concerning problems as existence, uniqueness and regularity of solutions depending on the assumptions on F . For an overview on the subject we refer the reader to [START_REF] Bensoussan | Mean field games and mean field type control theory[END_REF][START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF][START_REF] Carmona | Probabilistic theory of mean field games with applications II. Mean field games with common noise and master equations[END_REF][START_REF] Gomes | Regularity theory for mean-field game systems[END_REF][START_REF] Cardaliaguet | Notes on mean field games from p. -l. lions lectures at collége de france[END_REF], which is however far from being complete. So far, most of the literature concerns the analysis of the above system describing models in which the agents has control only of their velocity. However, in many applications, see for instance [START_REF] Cristiani | Multiscale modeling of pedestrian dynamics[END_REF], one might be interested in studying systems in which players needs to have control on their acceleration. In this case, proceeding heuristically as before, we have that each agent choose is strategy in order to minimize a cost functional of the form T t L(γ(s), γ(s), γ(s)) + F (γ(s), γ(s), m s ) ds + G(γ(T ), m T ) where the control system now has the form

γ(t) = v(t), v(t) = u(t).
Hence, the PDEs system is given by

     -∂ t u(t, x, v) + H(x, v, D x u(t, x, v), D v u(t, x, v)) = F (x, v, m t ), (t, x, v) ∈ [0, T ] × R 2d ∂ t m t -div x,v m t D p H(x, v, D x u(t, x, v), D v u(t, x, v)) = 0, (t, x, v) ∈ [0, T ] × R 2d m 0 = µ 0 , u(T, x, v) = G(x, v, m T ), (x, v) ∈ R 2d .
Note that, now the state space is not R d but R d × R d which takes into account not only the position but also the dependence of the strategy on the velocity v ∈ R d . Consequently, we also have that for any t ∈ [0, T ] the distribution m t is a probability measure on

R d × R d .
This is what motivated us at beginning of this project to analyze this problem and it is what Chapter 3 concerns, i.e., the study of the existence, uniqueness and regularity of solutions to the MFG system with control of acceleration. In particular, we consider the acceleration model as embedded into a more general setting which is the case of linear state equation linear, that is, a dynamics of the form

γ(t) = Aγ(t) + Bu(t)
for some constant matrices A and B.

MFG for linear control systems

Fixed a time horizon T > 0, we consider players having the following dynamics the whole space R d

γ(t) = Aγ(t) + Bu(t), ∀ t ∈ [0, T ] (1.1)
where A and B are real matrices and u is a measurable control function.

Each player aims to minimize a cost functional of the form T 0 L(γ(s), u(s), m s ) ds + G(γ(T ), m T ),

where, for each time t ∈ [0, T ], the probability measure m t on R d represents their distribution. In this framework the MFG system reads as

       -∂ t V (t, x) + H(x, D x V (t, x), m t ) = 0, (t, x) ∈ [0, T ] × R d ∂ t m t + div m t D p H(x, D x V (t, x), m t ) = 0, (t, x) ∈ [0, T ] × R d m 0 = m 0 , V (T, x) = G(x, m T ), ∀ x ∈ R d (1.3)
where the Hamiltonian H : R d × R d → R defined by

H(x, p, m) = sup u∈R k -Ax + Bu, p -L(x, u, m) .
One can immediately observe that if the Lagrangian L is of Tonelli type (strictly convex and coercive w.r.t. control variable) then H fails to be Tonelli. Hence, using the standard approach in MFG via fixed-point methods would lead us to several issues.

Therefore, in order to overcome this issue we solve the problem via the Lagrangian approach (see, for instance, [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF] and [START_REF] Mazanti | Minimal-time mean field games[END_REF]). That is, we define the metric space Γ T = γ ∈ AC([0, T ]) : γ(t) satisfy (1.1), γ(0) ∈ R d , endowed with uniform metric • ∞ and we consider Borel probability measures η supported on Γ T . Then, we restrict the attention to probability distributions on R d of the form m t = e t ♯η where e t : Γ T → R d denotes the evaluation map and ♯ stands for the push-forward operator. This correspond to consider only flow of measures concentrated on trajectories satisfying (1.1).

Let us describe the results of this work. The first problem we deal with is the definition of MFG equilibria for this class of problems. So, given an initial distribution of players m 0 ∈ P(R d ) we say that η ∈ P(Γ T ) is a MFG equilibrium if it is supported on minimizing curves of (1.2), with starting point in spt(m 0 ). Then, we prove that such equilibria exist (Theorem 3.13) and having this at our disposal we give the definition of mild solutions, (V, m) ∈ C([0, T ] × R d ) × C([0, T ]; P(R d )), of our MFG problem. For these, we study the existence, the uniqueness and the regularity. In particular, we show that {e t ♯η} t∈[0,T ] is 1 2 -Hölder continuous in time (Theorem 3.17) and, consequently, the value function V is locally semiconcave on [0, T ] × R d linearly in space and with fractional semiconcave modulus in time (Theorem 3.18). Moreover, by standard tools of optimal control theory we get that V is locally Lipschitz continuous (Theorem 3.20).

Under an extra growth assumption on the Lagrangian, we also show that there exists a MFG equilibrium such that the flow of measures {e t ♯η} t∈[0,T ] is Lipschitz continuous in time. This yields to linear semiconcavity estimates for the value function V both in space and in time. In conclusion, we show that the notion of mild solution is strictly related with the classical definition of weak solutions for the MFG system. Indeed, we prove that they coincide, in the sense that: a mild solution is a weak solution and vice versa (Theorem 3.30).

After this work was submitted, similar results were obtained in [1] for the special case of mean field games with control on acceleration.

Ergodic behavior of MFG of acceleration

In this Chapter we focus the attention on a special case of system (3.21). Indeed, we consider the case of control of acceleration which can be written as γ(t)

γ(t) = 0 1 0 0 γ(t) γ(t) + 0 1 0 u(t)
, t ≥ 0.

CHAPTER 1. INTRODUCTION

In this case, we have that the MFG system is given by

           -∂ t u T (t, x, v) + 1 2 |D v u T (t, x, v)| 2 -D x u T (t, x, v), v = F (x, v, m T t ), in [0, T ] × T d × R d ∂ t m T t -v, D x m T t -div m T t D v u T (t, x, v) = 0, in [0, T ] × T d × R d u T (T, x, v) = g(x, v, m T T ), m T 0 (x, v) = m 0 (x, v) in T d × R d .
(1.4)

During the last years, the question of the long time behavior of solutions of (standard) MFG systems has attracted a lot of attention. Results describing the long-time average of solutions were obtained in several context: see [START_REF] Cardaliaguet | Long time average of mean field games[END_REF][START_REF] Cardaliaguet | Long time average of mean field games with a nonlocal coupling[END_REF], for second order systems on T d , and [START_REF] Cardaliaguet | Long time average of first order mean field games and weak KAM theory[END_REF][START_REF] Cannarsa | Long-time behavior of first-order mean field games on Euclidean space[END_REF][START_REF] Cannarsa | Weak kam approach to first-order mean field games with state constraints[END_REF], for first order systems on T d , R d and for state constraint case respectively. Recently, Cardaliaguet and Porretta studied the long time behavior of solutions for the so-called Master equation associated with a second order MFG system, see [START_REF] Cardaliaguet | Long time behavior of the master equation in mean field game theory[END_REF]. In view of the results obtained in these works one would expect the limit of u T /T to be described by the following ergodic system

       1 2 |D v u(x, v)| 2 -D x u(x, v), v = F (x, v, m), (x, v) ∈ T d × R d -v, D x m -div mD v u(x, v) = 0, (x, v) ∈ T d × R d T d ×R d m(dx, dv) = 1.
(

The main issue of this work is that this ergodic system makes no sense. Indeed, as we explain below, even for problems without mean field interaction, we cannot expect to have a solution to the corresponding ergodic Hamilton-Jacobi equation (the first equation in (1.5)). As the drift of the continuity equation (the second equation in (1.5)) is given in terms of solution to the ergodic Hamilton-Jacobi equation, there is no hope to formulate the problem in this way. As far as we know, this is the first time this kind of problem is faced in the literature.

To overcome the issue just described, we first study the ergodic Hamilton-Jacobi equation without mean field interaction. More precisely, in the first part we investigate the existence of the limit, as T tends to infinity, of u T (0, •, •)/T , where now u T solves the Hamilton-Jacobi equation (without mean field interaction)

-∂ t u T (t, x, v) + 1 2 |D v u T (t, x, v)| 2 -D x u T (t, x, v), v = F (x, v), in [0, T ] × T d × R d u T (T, x, v) = 0 in T d × R d .
Here F : R d × R d → R is periodic in space (the first variable) and coercive in velocity (the second one). Following the seminal paper [START_REF] Lions | Homogenization of hamilton-jacobi equations[END_REF], it is known that the existence of the limit of u T /T is related with the existence of a corrector, namely to a solution of the ergodic Hamilton-Jacobi equation:

-D x u(x, v), v + 1 2 |D v u(x, v)| 2 = F (x, v) + c, (x, v) ∈ ×T d × R d ,
for some constant c. However, we stress again the fact that due to the lack of coercivity and due to the lack of small time controllability of our model, we do not expect the existence of a continuous viscosity solutions of the ergodic equation (see, however, this reference [START_REF] Esteve | The turnpike property and the long-time behavior of the hamilton-jacobi equation[END_REF] on this point). This problem has been overcome in several other frameworks: we can quote for instance [START_REF] Oliu-Barton | A uniform tauberian theorem in optimal control. advances in dynamic games[END_REF][START_REF] Cardaliaguet | Homogenization and enhancement for the G-equation[END_REF][START_REF] Xin | Periodic homogenization of the inviscid G-equation for incompressible flows[END_REF][START_REF] Cardaliaguet | Ergodicity of hamilton-jacobi equations with a non coercive non convex hamiltonian in R 2 /Z 2[END_REF][START_REF] Barles | Some homogenization results for non-coercive Hamilton-Jacobi equations[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF][START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for bellman-isaacs equations[END_REF][START_REF] Artstein | The value function of singularly perturbed control systems[END_REF][START_REF] Buckdahn | On representation formulas for long run averaging optimal control problem[END_REF][START_REF] Giga | Large-time asymptotics for onedimensional Dirichlet problems for Hamilton-Jacobi equations with noncoercive Hamiltonians[END_REF][START_REF] Gaitsgory | On a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems[END_REF], for related problems see also [5,[START_REF] Khlopin | Uniform Tauberian theorem in differential games[END_REF] and the references therein. Following techniques developed in [START_REF] Artstein | The value function of singularly perturbed control systems[END_REF] we prove in the first part of Theorem 4.2 that the limit of u T /T exists and is equal to a constant. However, this convergence result does not suffice to handle our MFG system of acceleration: indeed, we also need to understand, when the map F also depends on the extra time dependent parameter {m t } t≥0 , how this ergodic constant depends on this. For doing so, we follow ideas from weak-KAM theory (see for instance [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF]) and characterize the ergodic constant in terms of closed probability measures: namely, we prove in the second part of Theorem 4.2 that, for any

(x, v) ∈ T d × R d , lim T →+∞ u T (0, x, v) T = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw)
where C is the set of Borel probability measures µ on T d × R d with suitable finite moments and which are closed in the sense that, for any test function

φ ∈ C ∞ c (T d × R d ), T d ×R d ×R d D x φ(x, v), v + D v φ(x, v
), w η(dx, dv, dw) = 0, (see also Definition 4.1).

We now come back to our MFG of acceleration (1.4). In view of the characterization of the ergodic constant for the Hamilton-Jacobi equation without mean field interaction, it is natural to describe an equilibrium for the ergodic MFG problem with acceleration as a fixed-point problem on the Wasserstein space. We say that ( λ, μ) ∈ R × C is a solution of the ergodic MFG problem of acceleration if

λ = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ) µ(dx, dv, dw) = T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ) μ(dx, dv, dw),
where π :

T d × R d × R d → T d × R d
is the canonical projection onto the first two variables. We show that such an ergodic MFG problem with acceleration has a solution and that the associated ergodic constant λ is unique under the following monotonicity condition (first introduced in [START_REF] Lasry | Jeux à champ moyen. I: Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen. II: Horizon fini et contrôle optimal[END_REF]): there exists a constant M F > 0 such that for any m 1 , m 2 ∈ P(

T d × R d ) 10 CHAPTER 1. INTRODUCTION
see (1) in Theorem 4.5. The main result of this Chapter is the fact that, if (u T , m T ) solves the MFG system of acceleration (1.4), then u T (0, x, v)/T converges, as T tends to infinity, to the unique ergodic constant λ of the ergodic MFG problem, see (2) in Theorem 4.5. The main technical step for this is to rewrite the MFG system in terms of time-dependent closed measure (a kind of occupation measure in this set-up), see Theorem 4.25, and to understand the long-time average of these measures.

Singular perturbation problem

Here we address the singular perturbation problem for control systems of acceleration and of MFG systems with control on the acceleration. The main goal of this analysis is the behavior of such MFG system when the acceleration costs goes to zero. So, the study of the singular problem without mean-field interaction is used to understand the expected behavior of the system. Hence, we first study the limit behavior of the solutions to the Hamilton-Jacobi equation

-∂ t u ε + 1 2 ε |D v u ε | 2 -D x u ε , v -L 0 (x, v) = 0, (t, x, v) ∈ [0, T ] × R 2d u ε (T, x, v) = g(x), (x, v) ∈ R 2d .
as ε → 0. As already pointed out in the previous Chapters, the Hamiltonian

H(x, v, p x , p v ) = 1 2 ε |p v | 2 -p x , v -L 0 (x, v)
fails to be strictly convex and coercive w.r.t. momentum variables. So, also in this case, we solve the problem by using variational technics observing that the value function u ε can be represented as

u ε (t, x, v) = inf γ(t)=x γ(t)=v T t ε 2 |γ(s)| 2 + L 0 (γ(s), γ(s)) ds + g(γ(T )) .
However, since this represents the test bench for the study of the singular perturbation problem for MFG with control of acceleration we immediately focus the attention on the latter describing it in details. The system we consider here is given by

     -∂ t u ε + 1 2 ε |D v u ε | 2 -D x u ε , v -L 0 (x, v, m ε t ) = 0, (t, x, v) ∈ [0, T ] × R 2d ∂ t µ ε t -D x µ ε t , v -1 ε div v (µ ε t D v u ε ) = 0, (t, x, v) ∈ [0, T ] × R 2d µ ε 0 = µ 0 , u ε (T, x, v) = g(x, m ε T ), (x, v) ∈ R 2d (1.6)
where u ε : [0, T ] × R 2d → R is the value function, µ ∈ C([0, T ]; P 1 (R 2d )) is the joint distribution of position and velocity of a typical agent and m ε t = π 1 ♯µ ε t with π 1 : R d × R d → R d the projection map onto the first variable. MFG systems with control on the acceleration describes in general models in which the interacting agents controls their acceleration. Here, the systems we are interested in are those in which the acceleration cost vanishes, as it can be easily observed from (1.6).

The Lagrangian L 0 appearing in the system is assumed to be smooth in space and in velocity variables and to have Tonelli type dependence on v. We refer to (M3) below for what concern the dependence of L 0 on the measure variable. One can immediately recognize that the underlying minimization problem associated with the above PDEs systems has the following form

inf γ(t)=x γ(t)=v T t ε 2 |γ(s)| 2 + L 0 (γ(s), γ(s), m ε s ) ds + g(γ(T ), m ε T )
for any initial position and velocity (x, v) ∈ R 2d . The singular perturbation problem has been widely studied for control problems and, more recently, for differential games. For an overview on the subject, which is far from being complete, we refer the reader to [START_REF] Artstein | The value function of singularly perturbed control systems[END_REF][START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for bellman-isaacs equations[END_REF][START_REF] Artstein | Singularly perturbed ordinary differential equations with dynamic limits[END_REF], and references therein. For these kind of problems, the general structure is to consider a classical controlled dynamic coupled with one that depends on a small parameter ε > 0. Then, as ε → 0 the limit system turns out to be defined only on R d where the unperturbed system is defined. Some type of perturbation problems in MFG have been studied, recently, in [START_REF] Cardaliaguet | Long time average of first order mean field games and weak KAM theory[END_REF][START_REF] Cannarsa | Long-time behavior of first-order mean field games on Euclidean space[END_REF][START_REF] Cannarsa | Weak kam approach to first-order mean field games with state constraints[END_REF][START_REF] Cardaliaguet | Ergodic behavior of control and mean field games problems depending on acceleration[END_REF] where the authors study the long time-average behaviour of solutions to first order MFG system and in [START_REF] Cesaroni | Homogenization of a mean field game system in the small noise limit[END_REF][START_REF] Lions | Homogenization of the backwardforward mean field games systems in periodic enviromets[END_REF] where the authors study the homogenisation problem for second order MFG system. Note that, in homogenisation the structure of the MFG system might be lost in the limit (as proved in [START_REF] Cesaroni | Homogenization of a mean field game system in the small noise limit[END_REF]) which is not the case here, as we will show in Theorem 5. 3.

Indeed, going back to the MFG system (1.6) we prove that (u ε , m ε ), where we recall that m ε t is the space marginal of the solution µ ε t for any t ∈ [0, T ], converges (up to subsequence) to a solution (u 0 , m 0 ) to the classical MFG system

       (i) -∂ t u 0 (t, x) + H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d (ii) ∂ t m 0 t -div m 0 t D p H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d m 0 0 = m 0 , u 0 (T, x) = g(x, m 0 T ), x ∈ R d (1.7)
where H 0 : R d × R d → R is the Hamiltonian associated with the Lagrangian L 0 . As observed so far, we can see from (1.7) that the limit ε → 0 leads to the elimination of the velocity as state variable, whose dynamics was controlled via the perturbation ε. At this point, we again want to stress the fact that the Lagrangian L 0 in (1.6) depends only on the space marginal of the measure µ ε . First, this comes from the elimination of the velocity as state variable and so, also for in the analysis of the measure µ ε , the limit does no see the behavior of the second marginal. Moreover, we are interested in connecting the MFG system of acceleration with the classical one which we know depends only on a flow of probability measures in space which describes the motion of the agents.

Let us briefly explain the method of proof. We first show that u ε is equibounded and m ε is tight (see Lemma 5.11 and Theorem 5.14). Thus, as a first consequence we get that, up to a subsequence, there exists m 0 ∈ C([0, T ]; P 1 (R d ) such that m ε → m 0 in C([0, T ]; P 1 (R d )). Then, we proceed with the analysis of the value function u ε : we show that u ε (t, •, v) is equi-Lipschitz continuous, u ε (•, x, v) is equicontinuous and u ε (t, x, •) has decreasing oscillation w.r.t. ε (see Lemma 5.16 and Proposition 5.17). We finally address the locally uniform convergence of u ε , showing that there exists a subsequence ε k ↓ 0 such that (u ε k , m ε k ) converges to a solution (u 0 , m 0 ) of (1.7) (see Theorem 5.19, Proposition 5.20 and Corollary 5.22). The main issues in proving the above results are due to the lack of strict convexity and the lack of superlinearity of the Hamiltonian in system (1.6). In particular, these and the fact that Lagrangian L 0 is non-autonomous motivated us to use a variational approach instead of a PDEs approach since the latter creates series difficulties in estimating uniformly the gradient of u ε w.r.t. velocity variable. We recall that such gradient plays a key role in understanding the limit state space since it capture the behaviour of the velocity as state variable in R d × R d .

Ergodic behavior of sub-Riemannian control systems

In recent years, increasing attention has been devoted to control systems of the form

γ(t) = m i=1 u i f i (γ(t)) (1.8)
where f i are m ∈ {1, • • • , d} vector fields defined on R d , with sublinear growth, and controls u i are measurable functions on R m . The main assumption on the model is the so-called Chow condition (also known as Hördmander condition in PDE), i.e., the fact that iterated Lie brackets of f 1 , . . . , f m generate the whole tangent space at any point. Indeed, this condition implies that the system is controllable, that is, given any two points in the state space one can find a control that generates a path which joins the two points. Such systems are naturally associated with a new metric on the state space-the sub-Riemannian metric-which in general fails to be equivalent to the classical Euclidean metric, see for instance [2,[START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Rifford | Sub-Riemannian geometry and optimal transport[END_REF][START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF].

Given a Lagrangian L : R d × R m → R, an initial position x ∈ R d , and a time horizon T > 0 we consider the problem of minimizing the functional

u → T 0 L(γ x u (s), u(s)) ds
over the space of all measurable controls u : [0, T ] → R m , where γ x u denotes the solution of (1.11) such that γ(0) = x. The first part of this work is devoted to the analysis of the long-time average behavior of the value function of the above problem as T → ∞, that is, the existence of the limit of V T (x)/T as T → ∞ where

V T (x) = inf u T 0 L(γ x u (t), u(t)) dt.
In particular, we prove that such a limit exists locally uniformly and is independent of the initial position

x ∈ R d , that is, lim T →∞ 1 T V T (x) = α(L) .
(1.9)

Following [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF], it is known that the existence of the limit in (1.9) is related to the existence of a critical constant c ∈ R and of a viscosity solution χ to the ergodic Hamilton-Jacobi equation

H(x, Dχ(x)) = c (x ∈ R d ) (1.10)
where

H(x, p) = sup u∈R m m i=1 u i p, f i (x) -L(x, u) .
The existence of the critical constant for equation (1.12), in a certain sub-Riemannian setting, was obtained in [3] by a technique based on optimal transport. The analysis in [3] covers compact manifolds and families of 3-generating vector fields (i.e., a step-2 Lie algebra).

Our analysis, unlike [3], is performed on a noncompact state space equipped with a general bracket-generating distribution. The lack of compactness is a major difficulty that we overcome by condition (L3) below, which ensures the existence of a compact attractor for all minimizing trajectories. We observe that an assumption of the same type was used, in [START_REF] Cannarsa | Long-time behavior of first-order mean field games on Euclidean space[END_REF], to study the long-time behavior of first order Mean Field Games systems on Euclidean space and, in [START_REF] Ishii | The vanishing discount problem for Hamilton-Jacobi equations in the Euclidean space[END_REF], to investigate the limit behavior of discounted Hamilton-Jacobi equations on the whole space.

By analyzing the limit behavior of the discounted Hamilton-Jacobi equation associated with (1.12), we deduce that the ergodic equation admits solutions for c = -α(L) (Theorem 6.13). Then we construct a specific solution of such an equation which coincides with its Lax-Oleinik evolution.

CHAPTER 1. INTRODUCTION

Our interest in such a solution is motivated by the fact that we need it to derive a further characterization of the ergodic constant as the minimum of the Lagrangian action on closed measures. As we will show in the following Chapter, this is a crucial step to investigate the related Mather and Aubry sets, on which ergodic solutions have important regularity properties.

We recall that for Tonelli, or even more general, Hamiltonians on a compact or a non-compact manifold, the existence of solutions to the ergodic equation (1.12) has a long history going back to the seminal paper [START_REF] Lions | Homogenization of hamilton-jacobi equations[END_REF]. Among the many papers that have been published on the subject, when the state space is compact and the Hamiltonian is Tonelli we refer, for instance, to [START_REF] Fathi | Existence of c 1 critical subsolutions of the hamilton-jacobi equation[END_REF][START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF] and references therein. If the state space fails to be compact and the Hamiltonian is Tonelli or quasi-Tonelli we refer, for instance, to [START_REF] Barles | Ergodic type problems and large time behaviour of unbounded solutions of Hamilton-Jacobi equations[END_REF], [START_REF] Barles | Large time behavior of unbounded solutions of first-order Hamilton-Jacobi equations in R N[END_REF], [START_REF] Fathi | Weak KAM theorem on non compact manifolds[END_REF], [START_REF] Ishii | Asymptotic solutions for large time of Hamilton-Jacobi equations in Euclidean n space[END_REF], [START_REF] Ichihara | Long-time behavior of solutions of Hamilton-Jacobi equations with convex and coercive Hamiltonians[END_REF], [START_REF] Ishii | The vanishing discount problem for Hamilton-Jacobi equations in the Euclidean space[END_REF].

However, when the Hamiltonian is not coercive the problem of finding solutions to (1.12) is open. This issue has been addressed in specific frameworks: we quote for instance [START_REF] Cardaliaguet | Homogenization and enhancement for the G-equation[END_REF], [START_REF] Xin | Periodic homogenization of the inviscid G-equation for incompressible flows[END_REF], [START_REF] Cardaliaguet | Ergodicity of hamilton-jacobi equations with a non coercive non convex hamiltonian in R 2 /Z 2[END_REF], [START_REF] Barles | Some homogenization results for non-coercive Hamilton-Jacobi equations[END_REF], for the ergodic problem associated with the so-called G-equation or other noncoercive Hamiltonians. Moreover, we refer to [5,[START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for bellman-isaacs equations[END_REF] [53], [START_REF] Oliu-Barton | A uniform tauberian theorem in optimal control. advances in dynamic games[END_REF] for more on second order differential games.

We want to point out that some of the results of this work are specific to affine-control systems without drift. Indeed, in the presence of a drift, the existence of a continuous viscosity solution to (1.12) with a noncoercive Hamiltonian remains a challenging problem. We also mention systems with control on the acceleration (see, for instance, [START_REF] Cardaliaguet | Ergodic behavior of control and mean field games problems depending on acceleration[END_REF]) for which it has been proved that, due to the lack of small time local controllability, there are no continuous viscosity solutions to the associated ergodic Hamilton-Jacobi equation.

Aubry-Mather theory for sub-Riemannian control systems

In Chapter 6, we have studied the asymptotic behaviour as T → +∞ of the value function

V T (x) = inf u(•) T 0 L(γ x u (t), u(t)) dt (x ∈ R d )
where (1.11) and {f 1 , . . . , f m } are linearly independent smooth vector fields satisfying the so-called Lie algebra rank condition. Observe that the above assumptions ensure that system (1.11) is small time locally controllable. By using such a property and assuming the existence of a compact attractor for the optimal trajectories of (1.11), we proved that V T (x)/T converges to a constantα(L), the critical constant of L-as T → +∞, uniformly on all bounded subsets of R d .

L is a Tonelli Lagrangian, controls u : [0, T ] → R m (1 ≤ m ≤ d) are square integrable functions, γ x u is the solution of the sub-Riemannian state equation γ(t) = m i=1 u i (t)f i (γ(t)) a.e. t ∈ [0, T ] γ(0) = x ,
As is well known, the convergence of the above time averages entails the (locally uniform) convergence as λ ↓ 0 of the Abel means {λv λ } λ>0 , where

v λ (x) = inf u(•) ∞ 0 e -λt L(γ x u (s), u(s)) ds (x ∈ R d ) .
This fact in turn allows to construct a corrector χ, that is, a continuous viscosity solution of the so-called ergodic Hamilton-Jacobi equation

α(L) +H(x, Dχ(x)) = 0 (x ∈ R d ) , (1.12)
where H is defined by

H(x, p) = sup u∈R m m i=1 u i p i , f i (x) -L(x, u) (x, p) ∈ R d × R d . (1.13)
The above analysis is by now classical in the Tonelli case, that is, when both L(x, v) and H(x, p) are smooth functions, strictly convex and superlinear in v and p, respectively. Moreover, in such settings, the critical constant has a powerful variational interpretation in terms of probability measures minimizing the Lagrangian action on the tangent bundle. This connection is well explained by the celebrated Aubry-Mather and weak KAM theories (see, for instance, [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF][START_REF] Sorrentino | Action-minimizing methods in Hamiltonian dynamics[END_REF][START_REF] Mather | Action minimizing invariant measures for positive definite Lagrangian systems[END_REF]] and the references therein).

However, it is easy to see that the Hamiltonian in (1.13) fails to be Tonelli, in general. So, the classical weak KAM theory does not apply to minimization problems for sub-Riemannian control systems which are, on the other hand, quite relevant for both theory ( [2]) and applications ( [START_REF] Jean | Complexity of nonholonomic motion planning[END_REF]). Introducing new ideas and techniques to make this extension possible is the purpose of this work.

To be more precise, we point out that the underlying geometry on the state space, namely the sub-Riemannian structure induced by the family of vector fields {f i } i=1,...,m on R d (see for instance [2,[START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Rifford | Sub-Riemannian geometry and optimal transport[END_REF] and references therein), plays a crucial role in our approach. Moreover, in order to improve the natural regularity of correctors-which would just be Hölder continuous, see [START_REF] Cannarsa | Asymptotic analysis for hamiltonjacobi equations associated with sub-riemannian control systems[END_REF]-we restrict the analysis to the class of sub-Riemannian systems that admit no singular minimizing controls different from zero. Then, owing to [START_REF] Cannarsa | Semiconcavity results for optimal control problems admitting no singular minimizing controls[END_REF], we know that correctors are locally semiconcave, hence locally Lipschitz, on R d . Finally, in order to deal with unbounded state and control spaces, we assume the existence of a compact attractor for all optimal trajectories as is customary in this kind of situations.

CHAPTER 1. INTRODUCTION

We now proceed to describe the main results of this Chapter. First, extending the classical notion of closed measures on the tangent bundle (see, e.g., [START_REF] Fathi | Existence of c 1 critical subsolutions of the hamilton-jacobi equation[END_REF]), we introduce the class C F of closed probability measures adapted to the sub-Riemannian structure and we show that the critical constant α(L) is the minimum of the Lagrangian action on C F (Theorem 7.8). In this context, it is worth noting that closed measures are naturally supported on the distribution associated with {f i } i=1,...,m , which in our case reduces to

R d × R m .
Then, we introduce and study the Aubry set A from a dynamical and topological point of view, proving that A is a nonempty compact subset of R d (Theorem 7.22), invariant for the class of calibrated curves for Peierl's barrier (7.26) (Proposition 7.28). Moreover, we show that any critical solution to (1.12) is differentiable along the range of the vector fields {f i } i=1,••• ,m at any point x ∈ A (see Theorem 7.27 establishing horizontal differentiability).

Papers extracted:

We conclude this introduction quoting the papers which has been extracted from the work in this thesis.

Chapter 2

Preliminaries

In this chapter we collect some preliminary definitions and results that we are going to use throughout this thesis. In particular, they concern:

1. Wasserstein spaces and Wasserstein distance, for which we refer to [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF] for more details.

2. Sub-Riemannian geometry and sub-Riemannian control systems on R d , see [START_REF] Montgomery | A tour of subriemannian geometries, their geodesics and applications[END_REF][START_REF] Rifford | Sub-Riemannian geometry and optimal transport[END_REF]2].

3. Weak KAM theory for Tonelli Hamiltonian systems for which we refer to [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF][START_REF] Fathi | Weak KAM theorem on non compact manifolds[END_REF][START_REF] Sorrentino | Action-minimizing methods in Hamiltonian dynamics[END_REF][START_REF] Mather | Action minimizing invariant measures for positive definite Lagrangian systems[END_REF]].

Measure Theory

Let (X, d) be a metric space (in the work, we use

X = R d or X = R d × R m ).
Denote by B(X) the Borel σ-algebra on X and by P(X) the space of Borel probability measures on X. The support of a measure µ ∈ P(X), denoted by spt(µ), is the closed set defined by

spt(µ) := x ∈ X : µ(V x ) > 0 for each open neighborhood V x of x .
We say that a sequence

{µ k } k∈N ⊂ P(X) is weakly- * convergent to µ ∈ P(X), denoted by µ k w * -→ µ, if lim n→∞ X f (x) dµ n (x) = X f (x) dµ(x), ∀f ∈ C b (X).
There exists an interesting link between the weak- * convergence and the convergence of the support of the measures, see [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF]Proposition 5.1.8]. Indeed, if {µ j } j∈N ⊂ P(X) weakly- * converges to µ ∈ P(X) then

∀ x ∈ spt(µ) ∃x j ∈ spt(µ j ) : lim j→∞ x j = x. (2.1)

CHAPTER 2. PRELIMINARIES

For p ∈ [1, +∞), the Wasserstein space of order p is defined as

P p (X) := m ∈ P(X) : X d(x 0 , x) p dm(x) < +∞ ,
for some (and thus all) x 0 ∈ X. Given any two measures m and m ′ in P p (X), define

Π(m, m ′ ) := λ ∈ P(X×X) : λ(A×X) = m(A), λ(X×A) = m ′ (A), ∀A ∈ B(X) . (2.
2) The Wasserstein distance of order p between m and m ′ is defined by

d p (m, m ′ ) = inf λ∈Π(m,m ′ ) X×X d(x, y) p dλ(x, y) 1/p .
The distance d 1 is also commonly called the Kantorovich-Rubinstein distance and can be characterized by a useful duality formula (see, for instance, [START_REF] Villani | Topics in optimal transportation[END_REF]) as follows

d 1 (m, m ′ ) = sup X f (x) dm(x) - X f (x) dm ′ (x) | f : X → R is 1-Lipschitz , (2.3) for all m, m ′ ∈ P 1 (X).
Let K be a subset of P(X). We say that the set K has uniformly integrable p-moment with respect some (and thus any) x ∈ X if and only if

lim i→∞ X\B i (x)
d(x, x) p µ(dx) = 0, uniformly with respect to µ ∈ K.

⊂ P p (X) be such that sup µ∈K X |x| r µ(dx) < ∞.
Then the set K is tight. If, moreover, r > p then K is relatively compact for the d p distance.

Let X 1 , X 2 be metric spaces, let µ ∈ P(X 1 ) and let f : X 1 → X 2 be a µ measurable map. Then, we denote by f ♯µ ∈ P(X 2 ) the push-forward of µ through f defined by

f ♯µ(B) := µ(f -1 (B)), ∀ B ∈ B(X 2 ).
More generally, in integral form, it reads as

X 1 φ(f (x)) µ(dx) = X 2 φ(y) f ♯µ(dy).
We conclude this introductory section recalling the so-called disintegration theorem.

Theorem 2.4 (Disintegration Theorem). Let X and Y be Radon separable metric spaces, let µ be a Borel probability measure on X and let π : X → Y be Borel map. Define ν = π♯µ ∈ P(Y ). Then there exists a µ-a.e. uniquely determined Borel measurable family of probability measures {ν y } y∈Y ⊂ P(X) such that

ν y (X\π -1 (y)) = 0, for µ -a.e. y ∈ Y, and X f (x)µ(dx) = Y π -1 (y) f (x)ν y (dx) ν(dy)
for every Borel map f : X → [0, +∞].

Sub-Riemannian control

A class of nonholonomic drift-less systems on R d is a control system of the form

γ(t) = m i=1 f i (γ(t))u i (t), t ∈ [0, +∞) (2.4)
Such a system induces a distance on R d in the following way. First, we define the sub-Riemannian metric to be the function g :

R d × R m → R ∪ {∞} given by g(x, v) = inf m i=1 u 2 i : v = m i=1 f i (x)u i . If v ∈ span{f 1 (x), . . . , f m (x)} then the infimum is attained at a unique value u x ∈ R m and g(x, v) = |u x | 2 .
Then, since g(γ(t), γ(t)) is measurable, being the composition of the lower semicontinuous function g with a measurable function, we can define the length of an absolutely continuous curve γ :

[0, 1] → R d as length(γ) = 1 0 g(γ(t), γ(t)) dt.

CHAPTER 2. PRELIMINARIES

In conclusion, one defines the sub-Riemannian distance as

d SR (x, y) = inf (γ,u)∈Γ x→y 0,1 length(γ)
where

Γ x→y 0,1
denotes the set of all trajectory-control pairs such that u ∈ L 2 (0, 1; R m ), γ solves (2.4) for such a control u, γ(0) = x and γ(1) = y. Following [START_REF] Cannarsa | Semiconcavity results for optimal control problems admitting no singular minimizing controls[END_REF] it is possible to represent the sub-Riemannian distance as follows

d SR (x, y) = inf T > 0 : ∃ (γ, u) ∈ Γ x→y 0,T , |u(t)| ≤ 1 a.e. t ∈ [0, T ] (2.5)
for any x, y ∈ R d . Moreover, again from [START_REF] Cannarsa | Semiconcavity results for optimal control problems admitting no singular minimizing controls[END_REF] the sub-Riemannian distance can be characterised in terms of the sub-Riemannian energy: setting

e SR (x, y) = inf (γ,u)∈Γ x→y 0,1 1 0 g(γ(t), γ(t)) dt, one can prove that d SR (x, y) = e SR (x, y) (2.6)
(see, for instance, [START_REF] Cannarsa | Semiconcavity results for optimal control problems admitting no singular minimizing controls[END_REF]Lemma 11]). Among the many properties of these systems we are interested in the controllability. For such a system, controllability can be obtained by using the Lie algebra generated by f 1 , . . . , f m , which is defined as follows. Set

∆ 1 = span{f 1 , . . . , f m }
and, for any integer s ≥ 1,

∆ s+1 = ∆ s + [∆ 1 , ∆ s ] where [∆ 1 , ∆ s ] := span{[X, Y ] : X ∈ ∆ 1 , Y ∈ ∆ s }. The Lie algebra gener- ated by f 1 , . . . , f m is defined as Lie(f 1 , . . . , f m ) = s≥1 ∆ s . We say that system (2.4) satisfies Chow's condition if Lie(f 1 , . . . , f m )(x) = R d for any x ∈ R d , where Lie(f 1 , . . . , f m )(x) = {X(x) : X ∈ Lie(f 1 , . . . , f m )}.
Equivalently, for any x ∈ R d there exists an integer r ≥ 1 such that ∆ r (x) = R d . The minimum integer with such a property is called the degree of nonholonomy at x and will be denoted by r(x). Chow's condition is also known as the Lie algebra rank condition (LARC) in control theory and as the Hörmander condition in the context of PDEs. (i) Heisenberg group: We consider the system in R

3      ẋ(t) = u(t), ẏ(t) = v(t), ż(t) = u(t)y(t) -v(t)x(t)
In this case, the matrix of the system is given by

A(x, y, z) =   1 0 0 1 y -x  
and the columns of such matrix satisfy the Hördmander condition:

X 1 = (1, 0, y), X 2 = (0, 1, -x) and [X 1 , X 2 ] = (0, 0, 2) generate R 3 .
(ii) Grushin type systems: Consider a control system of the form

ẋ(t) = u(t), ẏ(t) = φ(x(t))v(t)
for a nonzero continuous function φ(x) with sub-linear growth. The classical Grushin system in R 2 is obtained taking φ(x) = x. Then, the dynamics is given by the matrix A(x, y) = 1 0 0 x whose columns satisfy the Hördmander condition: Besides controllability, another important consequence of Chow's condition is the well-known Ball-Box Theorem, see for instance [2,Theorem 10.67]. Of particular interest to us is a corollary of such a theorem which gives Hölder equivalence between the Euclidean distance and the sub-Riemannian one. First, we observe that for any x ∈ R d a continuity argument ensures the existence of a neighborhood U x of x such that

X 1 = (1, 0) and [X 1 , X 2 ] = (0,
∆ r(x) (y) = R d , ∀ y ∈ U x .
(2.7)

Thus, given a compact set K there exists a finite cover given by {U x i } i=1,...,N and a set of integers {r(x i )} i=1,...,N such that (2.7) holds on

U x i with r(x) = r(x i ). Taking r = max i=1,...,N r(x i ) we obtain ∆ r (y) = R d , ∀ y ∈ K. (2.8)
We call degree of nonholonomy of K the minimum integer such that (2.8) holds true and we denote it by r(K). Moreover, we recall that a family of vector fields {f i } i=1,...,m is an equi-regular distribution on R d if there exists

r 0 ≥ 1 such that ∆ r 0 (x) = R d for any x ∈ R d .
Corollary 2.7. For any compact set K ⊂ R d there exist two constants c1 , c2 > 0 such that

c 1 |x -y| ≤ d SR (x, y) ≤ c 2 |x -y| 1 r(K) , ∀ x, y ∈ K. (2.9)
Furthermore, we recall that the topology induced by (R d , d SR ) coincides with the topology induced by the Euclidean distance on R d ([2, Theorem 3.31]). In particular, from this result, we obtain that a set is compact in

(R d , d SR ) if and only if it is compact in R d w.r.t. Euclidean distance.
We conclude this preliminary part with a brief introduction to singular controls. Let x 0 ∈ R d and fix t > 0. The end-point mapping associated with system (2.4) is the function

E x 0 ,t : L 2 (0, t; R m ) → R d defined as E x 0 ,t (u) = γ(t)
where γ is a solution of (2.4) associated with u such that γ(0) = x 0 . Under the assumption that the vector field f i has sub-linear growth for any i = 1, . . . m it is known that E x 0 ,t is of class C 1 on L 2 (0, t; R m ). Then, we say that a control ū ∈ L 2 (0, t; R m ) is singular for E x 0 ,t if dE x 0 ,t (ū) is not surjective. Moreover, defining the function

H 0 : R d × R d × R m → R as H 0 (x, p, u) = m i=1 u i p, f i (x)
we have the following well-known characterization of singular controls.

Theorem 2.8. A control u ∈ L 2 (0, t; R m ) is singular for E x 0 ,t if

and only if there exists an absolutely continuous arc

p : [0, t] → R d \{0} such that γ(s) = D p H 0 (γ(s), p(s), u(s)) -ṗ(s) = D x H 0 (γ(s), p(s), u(s)) with γ(0) = x 0 and D u H 0 (γ(s), p(s), u(s)) = 0, for a.e. s ∈ [0, t], that is, f i (γ(s)), p(s) = 0 for any s ∈ [0, t].
(i) For each (x, v) ∈ R d × R d , the Hessian D 2 vv L(x, v) is positive definite. (ii) For each A > 0 there exists B(A) ∈ R such that L(x, v) > A|v| + B(A), ∀(x, v) ∈ R d × R d . (iii) For each R > 0 A(R) := sup L(x, v) : |v| ≤ R < +∞.

Define the Hamiltonian

H : R d × R d → R associated with L by H(x, p) = sup v∈R n p, v -L(x, v) , ∀(x, p) ∈ R d × R d .
It is straightforward to check that if L is a Tonelli Lagrangian, then H defined above also satisfies (i), (ii), and (iii) in Definition 2.9. Such a function H is called a Tonelli Hamiltonian. Moreover, if L is a reversible

Lagrangian, i.e., L(x, v) = L(x, -v) for all (x, v) ∈ R d × R d , then H(x, p) = H(x, -p) for all (x, p) ∈ R d × R d .
Let us recall definitions of weak KAM solutions and viscosity solutions of the Hamilton-Jacobi equation

H(x, Du) = c, x ∈ R d , (2.10)
where c is a real constant. (i) For each continuous piecewise

C 1 curve γ : [t 1 , t 2 ] → R d , we have that u(γ(t 2 )) -u(γ(t 1 )) ≤ t 2 t 1 L(γ(s), γ(s))ds + c(t 2 -t 1 ); (ii) For each x ∈ R d , there exists a C 1 curve γ : (-∞, 0] → R d (resp. γ : [0, +∞) → R d ) with γ(0) = x such that u(x) -u(γ(t)) = 0 t L(γ(s), γ(s))ds -ct, ∀t < 0 (resp. u(γ(t)) -u(x) = t 0 L(γ(s), γ(s))ds + ct, ∀t > 0).
H(y 0 , Dψ(y 0 )) ≥ c; (iii) A function u : V → R is called a viscosity solution of equation (2.10)
if it is both a viscosity subsolution and a viscosity supersolution.

Definition 2.13 (Mañé critical value). The Mañé critical value of a Tonelli

Hamiltonian H is defined by

c(H) := inf c ∈ R : ∃ u ∈ C(R d ) viscosity sol. of H(x, Du) = c .
See [START_REF] Fathi | Weak KAM theorem on non compact manifolds[END_REF]Theorem 1.1] for the following weak KAM theorem for noncompact state spaces.

Theorem 2.14 (Weak KAM theorem). Let H be a Tonelli Hamiltonian. Then, there exists a global viscosity solution of equation

H(x, Du) = c(H), x ∈ R d .
In [START_REF] Fathi | Weak KAM theorem on non compact manifolds[END_REF], viscosity solutions are shown to coincide with backward weak KAM solutions. Observe that, as R d can be seen as a covering of the torus T d , Mañé's critical value can be characterized as follows:

c(H) = inf u∈C ∞ (R d ) sup x∈R d H(x, Du(x)).
We conclude this section by recalling the notion of Mather set and the role such a set plays for the regularity of viscosity solutions. Let L be a Tonelli Lagrangian. As is well known, the associated Euler-Lagrange equation, i.e.,

d dt D v L(x, ẋ) = D x L(x, ẋ), (2.11) generates a flow of diffeomorphisms ϕ L t : R d × R d → R d × R d , with t ∈ R, defined by ϕ L t (x 0 , v 0 ) = (x(t), ẋ(t))
, where x : R → R d is the maximal solution of (2.11) with initial conditions x(0) = x 0 , ẋ(0) = v 0 . It should be noted that, for any Tonelli Lagrangian, the flow ϕ L t is complete, see for instance [START_REF] Fathi | Weak KAM theorem on non compact manifolds[END_REF]. We recall that a Borel probability measure

µ on R d × R d is called ϕ L t - invariant, if µ(B) = µ(ϕ L t (B)), ∀t ∈ R, ∀B ∈ B(R d × R d ),
or, equivalently,

R d ×R d f (ϕ L t (x, v)) µ(dx, dv) = R d ×R d f (x, v) µ(dx, dv), ∀f ∈ C ∞ c (R d ×R d ).
We denote by M L the class of all ϕ L t -invariant probability measures. Definition 2.15 (Mather measures [59]). A probability measure µ ∈ M L is called a Mather measure for L, if it satisfies

R d ×R d L(x, v) µ(dx, dv) = inf ν∈M L R d ×R d L(x, v) ν(dx, dv).
In [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF], it was proved that

c(H) = -inf ν∈M L R d ×R d L(x, v) ν(dx, dv).

Denote by M *

L the set of all Mather measures. Observe that, if L (resp. H) is a reversible Lagrangian (resp. reversible Hamiltonian), then

-c(H) = inf x∈R d L(x, 0). The Mather set is the subset M 0 ⊂ R d × R d defined by M 0 = µ∈M * L spt(µ). We call M 0 = π 1 (M 0 ) ⊂ R d the
0 . Moreover, if (x, v) ∈ M 0 , then Du(x) = D v L(x, v)
and the map M 0 → R d × R d , defined by x → (x, Du(x)), is locally Lipschitz with a Lipschitz constant which is independent of u.

Chapter 3

Mild and weak solutions of Mean Field Games problems for linear control systems

Setting of the Mean Field Games problem

Assumptions

Let us consider a Lagrangian

L : R d × R k × P 1 (R d ) → R and a function G : R d × P 1 (R d ) → R satisfying the following. (L1) For any m ∈ P 1 (R d ), the map (x, u) → L(x, u, m) is of class C 2 (R d × R k
) and the map m → L(x, u, m), from P 1 (R d ) to R, is Lipschitz continuous with respect to the d 1 distance, i.e.

Q L := sup (x,u)∈R d ×R k m 1 , m 2 ∈P 1 (R d ) m 1 ̸ =m 2 |L(x, u, m 1 ) -L(x, u, m 2 )| d 1 (m 1 , m 2 ) < +∞. (L2) The map (x, m) → G(x, m) is of class C b (R d × P 1 (R d )) and for every m ∈ P 1 (R d ) the map x → G(x, m) belongs to C 1 b (R d ). (L3) (i) There exist a constant C 0 such that Id C 0 ≤ D uu L(x, u, m) ≤ C 0 Id, ∀ (x, u, m) ∈ R d × R k × P 1 (R d ).
(ii) There exists a constant C 1 ≥ 0 such that for any

(x, u, m) ∈ R d × R k × P 1 (R d ) D 2 xu L(x, u, m) ≤ C 1 (1 + |u|).

CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG

(iii) There exists a constant C 2 ≥ 0 such that for any

(x, u, m) ∈ R d × R k × P 1 (R d ) |L(x, 0, m)| + |D x L(x, 0, m)| + |D u (x, 0, m)| ≤ C 2 .
Remark 3.1. Note that, from (L3), it is not difficult to check that there exist c 0 ≥ 0 and c 1 ≥ 0 such that

c 0 |u| 2 -c 1 ≤ L(x, u, m) ≤ c 1 + 1 c 0 |u| 2 ∀ (x, u, m) ∈ R d × R k × P 1 (R d ).
Fix a time horizon T > 0. Let A and B be real matrices, d × d and d × k, respectively, and consider the control system defined by

γ(t) = Aγ(t) + Bu(t), t ∈ [0, T ] (3.1)
where u : [0, T ] → R k is a summable function. For all x ∈ R d we denote by γ(• ; x, u) the solution of the differential equation (3.1) such that γ(0) = x and define the metric space

Γ T = γ(•; x, u) : x ∈ R d , u ∈ L 1 (0, T ; R k ) ⊂ AC([0, T ]; R d )
endowed with the uniform norm, denoted by • ∞ . Moreover, set

Γ T (x) = γ ∈ Γ T : γ(0) = x .
For any x ∈ R d , any u ∈ L 1 (0, T ) and any flow of probability measures m ∈ C([0, T ]; P 1 (R d )) define the functional

J(x, u, {m t } t ) = T 0 L(γ(t, x, u), u(t), m t ) dt + G(γ(T, x, u), m T ),
and the associated optimal control problem inf

u∈L 2 (0,T ; R k ) J(x, u, {m t } t ). (3.2)
Notice that the restriction to controls u ∈ L 2 (0, T ; R k ) is due to the structure assumptions we imposed on L.

We proceed now to prove some estimates on the optimal controls and the associated optimal trajectories. Proposition 3.2. Assume (L1) -(L3). Then, there exists a real positive constant K such that for any x ∈ R d , any m ∈ C([0, T ]; P 1 (R d )) and any optimal control u * of (3.2), we have that

u * 2 ≤ K.
Proof. By Remark 3.1 and the optimality of u * we deduce that

c 1 T + G ∞ ≥ J(x, 0, {m t } t ) ≥ J(x, u * , {m t } t ) ≥ c 0 T 0 |u * (t)| 2 dt-c 1 T -G ∞ .
Therefore, from the above inequalities we deduce that

u * 2 2 = T 0 |u * (t)| 2 dt ≤ 2 c 0 (c 1 T + G ∞ ) =: K 2 .
Thus, the proof is complete.

= Aγ * (t) + Bu * (t) 2 ≤ A 1 2 γ * 2 + B 1 2 u * 2 ≤ A 1 2 T 0 |γ * (t)| 2 dt 1 2 + B 1 2 K ≤ A 1 2 T 1 2 C 1 + |x| + B 1 2 K.

CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG

Thus, for any t, s ∈ [0, T ] such that s ≤ t we get

|γ * (t) -γ * (s)| ≤ t s | γ * (τ )| dτ ≤ γ * 2 |t -s| 1 2 ≤ A 1 2 T 1 2 C 1 + |x| + B 1 2 K |t -s| 1 2
Which completes the proof.

In order to express our MFG problem in terms of the Lagrangian formulation we are going to give a special structure to the continuous flow of probability measures {m t } t∈[0,T ] . Let α > 1, let m 0 be a Borel probability measure in P α (R d ), and denote by [m 0 ] α the α-moment of m 0 , i.e.,

[m 0 ] α = R d |x| α m 0 (dx). (3.3)
Let R be a real constant such that R ≥ [m 0 ] α and define the following space of probability measures on

Γ T P m 0 (Γ T , R) = η ∈ P(Γ T ) : Γ T γ α 2 η(dγ) ≤ R, e 0 ♯η = m 0
where e t (γ) = γ(t) is the evaluation map. Note that the sets P m 0 (Γ T , R) are compact subsets of P(Γ T ) with respect to d 1 distance. Indeed, for any r > 0 define the following sets

C r = {γ ∈ Γ T : |γ(0)| ≤ r, γ 2 ≤ r},
which are compact by Ascoli-Arzela Theorem. Observe, also, that by definition

C c r ⊂ {γ ∈ Γ T : γ 2 > r} ∪ {γ ∈ Γ T : |γ(0)| > r}. Thus, given η ∈ P m 0 (Γ T , R) we have that η({γ ∈ Γ T : |γ(0)| > r}) = m 0 (B c r )
which goes to zero as r → +∞. Moreover, by Bienaymé-Tchebychev inequality we obtain

η({γ ∈ Γ T : γ 2 > r}) ≤ R r α .
Therefore, we get

η(C c r ) ≤ R r α + m 0 (B c r )
which in turn yields the compactness of P m 0 (Γ T , R).

Remark 3.5. There exist at least one constant R ≥ [m 0 ] α such that the set P m 0 (Γ T , R) is non-empty. Indeed, fixed a Borel probability measure m 0 ∈ P α (R d ), consider the map p : R d → Γ T such that

x → p[x](t) := e tA x, ∀t ∈ [0, T ]
and define the measure η = p[•]♯m 0 ∈ P(Γ T ). Note that, for any x ∈ R d the curve e tA x is an admissible curve associated with the control u ≡ 0.

Then, the following holds:

1. for any bounded continuous function f on R d , we have that e 0 ♯η = m 0 . Indeed,

R d f (x) e 0 ♯η(dx) = Γ T f (γ(0)) η(dγ) = Γ T f (γ(0)) p♯m 0 (dγ) = R d f (p[x](0)) m 0 (dx) = R d f (x) m 0 (dx);
2. the α-moment of η is bounded:

Γ T γ α 2 η(dγ) = R d ṗ[x] α 2 m 0 (dx) ≤ A e T ∥A∥ α R d |x| α m 0 (dx) ≤ A e T ∥A∥ α [m 0 ] α .
Therefore, taking R ≥ A e T ∥A∥ α [m 0 ] α we have that η ∈ P m 0 (Γ T , R).

Definitions and first properties

For any x ∈ R d , any u ∈ L 1 (0, T ) and any η ∈ P m 0 (Γ T , R), define the functional

J η (x, u) = T 0 L(γ(t, x, u), u(t), e t ♯η) dt + G(γ(T, x, u), e T ♯η)
and the associated optimal control problem inf

u∈L 2 (0,T ; R k ) J η (x, u). (3.4)
We denote by Γ * η (x) the set of curves associated with an optimal control u * (3.4), i.e. Proof. We are going to prove only the point (1), see [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF]Lemma 3.2] for a proof of (2) and (3). Given η ∈ P m 0 (Γ T , R) we have that

Γ * η (x) = γ(• ; x, u * ) : J η (x, u * ) = inf u∈L 2 (0,T ;R k ) J η (x, u) .
R d |x| α e t ♯η(dx) = Γ T |γ(t)| α η(dγ) ≤ Γ T γ α ∞ η(dγ) ≤ C 0 ,
where the last inequality holds by definition of P m 0 (Γ T , R). So, by Theorem 2.3 the family of measures {e t ♯η} t∈[0,T ] is tight in P α (R d ) with respect to the d 1 distance since by assumption α > 1.

Remark 3.8. Note that, in (3.5) the constant R in independent of t ∈ [0, T ] and η. Indeed, as explained so far it is fixed a priori such that R ≥ [m 0 ] α .

Mean Field Games equilibria: Existence and Uniqueness

At this point, it is not difficult to prove that for any given α > 0 and any given initial measure m 0 ∈ P α (R d ) there exists R 0 ≥ 0 such that for any R ≥ R 0 there exists at least one Mean Field Games equilibrium η ∈ P m 0 (Γ T , R) and that, under a classical monotonicity assumption, such an equilibrium is unique.

For the sake of completeness, we give below the key ideas and steps to prove the existence of a Mean Field Games equilibrium, following the appoach in [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF].

Given m 0 ∈ P α (R d ) and given η ∈ P m 0 (Γ T , R) we recall that by Theorem 2.4 there exists a unique Borel measurable family of probability measures

{η x } x∈R d on Γ T such that η(dγ) = R d η x (dγ) m 0 (dx) spt(η x ) ⊂ Γ T (x), m 0 -a.e., x ∈ R d .
Define the set-valued map

E : P m 0 (Γ T , R), d 1 ⇒ P m 0 (Γ T , R), d 1
that associates with any η ∈ P m 0 (Γ T , R) the set

E(η) = ν ∈ P m 0 (Γ T , R) : spt(ν x ) ⊂ Γ * η (x), m 0 -a.e. .
It is easy to realize that a given η ∈ P m 0 (Γ T , R) is a Mean Field Games equilibrium if and only if η is a fixed point of the above set-valued map, that is, η ∈ E(η). Therefore, in order to prove the existence of Mean Field Games equilibria, we appeal to Kakutani-Fan-Glicksberg's fixed point theorem, see for instance [4,Corollary 17.55], which provides conditions under which the set-valued map E has a fixed point.

We check the validity of such conditions in the following Lemmas.

Lemma 3.9. Assume (L1) -(L3). Let R ≥ [m 0 ] α . For any

x i → x in R d , any η i ⇀ * η in P m 0 (Γ T , R) and any γ i ∈ Γ * η i (x i ) such that γ i → γ in Γ T we have that γ ∈ Γ * η (x).
Proof. Since γ i ∈ Γ * η i (x i ) we know that there exists a sequence of optimal controls u i ∈ L 2 (0, T ) such that γ i (•) = γ i (•, x i , u i ) for every t ∈ [0, T ]. Moreover, from Proposition 3.2 we get that u i 2 ≤ K. Therefore, up to a subsequence, we obtain that there exists ū ∈ L 2 (0, T ) such that u i ⇀ ū in L 2 . Hence, we are reduced to prove that

1. γ(•) = γ(•, x, ū); 2. J η (x, ū) ≤ J η (x, u) for every u ∈ L 2 ([0, T ]),

Point 1:

By definition of γ i , we know that γ i (t) = e At x + t 0 e A(t-s) Bu i (s) ds.

CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG

Let v be a vector on R d , then

v, γ i (t) = v, e At x + t 0 v, e A(t-s) Bu i (s) ds = v, e At x + t 0 (e A(t-s) B) * v, u i (s) ds.
Thus, letting i → ∞ by the weak L 2 convergence of u i we obtain that

v, γ(t) = v, e At x + t 0 v, e A(t-s) B ū(s) ds.
This concludes the proof of point 1.

Point 2:

We now prove that

J η (x, ū) ≤ lim inf i→∞ J η i (x i , u i ).
By (L2) and the convergence of γ i in Γ T and that of η i , it follows that

G(γ i (T ), e T ♯η i ) → G(γ(T ), e T ♯η).
Therefore, it suffices to prove that

T 0 L(γ(t), ū(t), e t ♯η) dt ≤ lim inf i→∞ T 0 L(γ i (t), u i (t), e t ♯η i ) dt. Now, T 0 L(γ(t), ū(t), e t ♯η) -L(γ i (t), u i (t), e t ♯η) dt = T 0 L(γ(t), ū(t), e t ♯η) -L(γ(t), u i (t), e t ♯η) dt A + T 0 L(γ(t), u i (t), e t ♯η) -L(γ i (t), u i (t), e t ♯η i ) dt B .
By assumption (L3) (iii) and Lipschitz condition (L1) we have that B → 0 as i → 0. Thus, we have to prove now that the functional

Λ(u) = T 0 L(γ(t), u(t), e t ♯η) dt
is weakly lower semicontinuous with respect to the L 2 topology. Define, for every λ ∈ R,

X λ = {u ∈ L 2 (0, T ) : Λ(u) ≤ λ}.
By assumption (L3) on convexity of the Lagrangian L with respect to controls, we get that the sets X λ are convex. Furthermore, such sets are closed in the strong L 2 topology. Indeed, ) for some u * ∈ L 2 (0, T ) solution of (3.4) associated with η. Define, now, the measure η as follows

if {u i } i∈N ⊂ X λ is such that u i → u ∞ in L 2 then u i → u ∞ a.e
η(A) = R d δ γx (A) m 0 (dx) for any A ∈ B(Γ T ).
Thus, we need to prove that η ∈ P m 0 (Γ T , R). Indeed, e 0 ♯η = m 0 by definition and

Γ T γ α 2 η(dγ) = R d γx α 2 m 0 (dx) ≤ R d C2 α (1 + |x|) α m 0 (dx),
where the last inequality holds by Lemma 3.4. Therefore, we deduce that

Γ T γ α 2 η(dγ) ≤ Cα 2 R d |x| α m 0 (dx) + 1 ≤ Cα 2 ([m 0 ] α + 1). Hence, taking R ≥ R(α, [m 0 ] α ), where R(α, [m 0 ] α ) := Cα 2 ([m 0 ] α + 1)
we obtain that η ∈ P m 0 (Γ T , R). Consequently, that E(η) is non-empty. The proof of convexity is a straightforward application of [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF]Lemma 3.5].

In conclusion, for any η ∈ P m 0 (Γ T , R) the sets E(η) are compact, with respect to the d 1 distance, since E(η) ⊂ P m 0 (Γ T , R) which is compact.

∈ P α (R d ) and R ≥ R(α, [m 0 ] α )
, where R(α, [m 0 ] α ) is defined as in Lemma 3.11. Thus, by Theorem 3.13 we have that there exists at least one Mean Field Games equilibrium η ∈ P m 0 (Γ T , R).

From now on, we denote by γ(s; t, x, u) the solution to the following control system

γ(s) = Aγ(s) + Bu(s), s ∈ [t, T ] γ(t) = x, ( 3.6)
where u : [t, T ] → R k belongs to L 2 (t, T ; R k ). Moreover, we introduce the following notation

m η t = e t ♯η, (3.7)
for any η ∈ P m 0 (Γ T , R).

Definition 3.14 (Mild solutions of Mean Field Games problem).

We

say that (V, m) ∈ C([0, T ] × R d ) × C([0, T ], P α (R d )
) is a mild solution for the Mean Field Games problem if there exists a Mean Field Games equilibrium η ∈ P m 0 (Γ T ) such that (i) m t = m η t for all t ∈ [0, T ];

(ii) V can be represented as the value function of the optimal control problem (3.4), that is

V (t, x) = inf u∈L 2 (0,T ; R k ) T t L(γ(s; t, x, u), u(s), m η s ) ds + G(γ(T ; t, x, u), m η T ) (3.8) for all (t, x) ∈ [0, T ] × R d .
Note that the above definition is well-posed since we have proved so far that there exists at least one Mean Field Games equilibrium and the map

[0, T ] → P α (R d ) t → e t ♯η
is continuous with respect to d 1 . Moreover, for the same reasons we know that there exists at least one mild solution of the Mean Field Games problem.

In order to study the uniqueness of mild solutions, we focus the attention on a particular Lagrangian function, that is

L(x, u, m) := ℓ(x, u) + F (x, m), ( 3.9)
where ℓ and F satisfy the assumptions (L1)-(L3). Definition 3.15 (Monotonicity). We say that Ψ :

R d × P 1 (R d ) → R is monotone if R d Ψ(x, m 1 ) -Ψ(x, m 2 ) (m 1 -m 2 )(dx) ≥ 0, (3.10
)

for all m 1 , m 2 ∈ P 1 (R d ).
We say that Ψ is strictly monotone if (3.10) holds true and

R d Ψ(x, m 1 ) -Ψ(x, m 2 ) (m 1 -m 2 )(dx) = 0 ⇐⇒ F (x, m 1 ) = F (x, m 2 )
for any x ∈ R d .

Theorem 3.16 (Uniqueness of mild solutions). Assume (L1) -(L3).

Let F and G be strictly monotone. Then, for any Mean Field Games equilibria η 1 and η 2 in P m 0 (Γ T , R) we have that the associated functionals J η 1 and J η 2 are equal. Consequently, if (V 1 , m 1 ) and (V 2 , m 2 ) are two mild solutions associated with the Mean Field Games equilibria η 1 and η

2 , then V 1 = V 2 .
We omit the proof of the Theorem 3.16 which is similar to the one of [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF]Theorem 4.1].

Further regularity of mild solutions

Throughout this section, given α > 1 fix m 0 ∈ P α (R d ) and R ≥ R(α, [m 0 ] α ), where R(α, [m 0 ] α ) in defined as in Lemma 3.11. At this point, we know that under assumptions (L1)-(L3) by Theorem 3.13 there exists at least one Mean Field Games equilibrium η ∈ P m 0 (Γ T , R). Furthermore, if the Lagrangian L is of the form (3.9), the coupling function F and the terminal costs G satisfy the strict monotonicity assumption, see Definition 3.15, then the mild solution is unique. For this reasons, from now on we fix R ≥ R(α, [m 0 ] α ). Now, we are going to prove that any Mean Field Games equilibrium generates a family of probability measures {m η t } t∈[0,T ] which is 1 2 -Hölder continuous in time. Consequently, any mild solution (V, m η ) is such that the value function V is locally Lipschitz continuous and locally fractionally semiconcave on [0, T ] × R d . Moreover, we will prove that there exists at least one Mean Field Games equilibrium η ∈ P α (Γ T , R) such that t → m η t is Lipschitz continuous.

Given the control system (3.1), the Hamiltonian associated with the Lagrangian function L is defined as

H(x, p, m) = sup u∈R k -p, Ax + Bu -L(x, u, m) .
The Hamiltonian H can be explicitly written as follows

H(x, p, m) = -p, Ax -L * (x, -B ⋆ p, m), ∀ (x, p, m) ∈ R d × R k × P α (R d )
(3.11) where L * denotes the Legendre Transform of L, i.e.,

L * (x, p, m) = sup u∈R k -p, u -L(x, u, m) .
Moreover, it is easy to check that there exists a constant c 2 ≥ 0 such that for any

(x, p, m) ∈ R d × R k × P α (R d ) |D p H(x, p, m)| ≤ c 2 (1 + |x| + |p|) (3.12)
and, from (i) in (L1) one can also deduce that there exists a constant c H ≥ 0 such that

|D p H(x, p, m) -D p H(y, q, m)| ≤ c H (|p -q| + |x -y|) (3.13)
for any (x, y) ∈ R 2d , (p, q) ∈ R 2k and m ∈ P α (R d ).

Local Lipschitz continuity and local fractional semiconcavity of the Value function

Let (V, m η ) a mild solution of the Mean Field Games problem associated with an equilibrium η ∈ P α (Γ T , R). In this section, we prove that the flow

FURTHER REGULARITY OF MILD SOLUTIONS

39

of measures {m η t } t∈[0,T ] are Hölder continuous and consequently, that the associated value function is locally semiconcave on [0, T ] × R d , linearly in space and with a fractional modulus of semiconcavity in time. Moreover, we show that the value function V is locally Lipschitz continuous on [0, T ] × R d . We conclude this section proving that, under some extra assumptions on the data, there exists at least one equilibrium η ∈ P α (Γ T , R) such that {m η t } t∈[0,T ] is Lipschitz continuous in time.

We recall that V is defined as the value function

V (t, x) = inf u∈L 2 (0,T ; R k ) T t L(γ(s; t, x, u), u(s), m η s ) ds + G(γ(T ; t, x, u), m η T ) .
Theorem 3.17 (Hölder continuity of equilibria). Assume (L1) -(L3).

Then, given any Mean Field Games equilibrium η, the map t → m η t is 1 2 -Hölder continuous in time.

Proof. By definition of d 1 , we have that

d 1 (m η t , m η s ) = inf φ∈Lip 1 (R d ) R d φ(x)(m η t -m η s )(dx) = inf φ∈Lip 1 (R d ) Γ T (φ(γ(t)) -φ(γ(s))η(dγ) ≤ Γ T |γ(t) -γ(s)|η(dγ),
where Lip 1 (R d ) is the set of Lipschitz continuous functions such that the Lipschitz constant is equal to 1.

We recall that, since η is a Mean Field Games equilibrium then it is supported on the set of all minimizing curves of problem (3.4). Therefore, by Lemma 3.4 and recalling that x = γ(0) we obtain

d 1 (m η t , m η s ) ≤ Γ T |γ(t) -γ(s)|η(dγ) ≤|t -s| 1 2 Γ T A 1 2 T 1 2 C 1 + |x| + B 1 2 K η(dγ) = κ([m 0 ] α ])|t -s| 1 2 ,
where the constant κ depends on the moment of m 0 which we know is bounded by construction. Thus, the proof is complete.

In order to prove the semiconcavity of the value function V , we need to add the following assumption on the Lagrangian L and terminal cost G: (L4) There exists two constants w L ≥ 0 and w G ≥ 0 such that for any λ ∈ [0, 1], any radius R > 0, any u ∈ R k , any x 0 , x 1 ∈ B R , and any

m ∈ P 1 (R d ) such that λL(x 0 , u, m) + (1 -λ)L(x 1 , u, m) -L(λx 0 + (1 -λ)x 1 , u, m) ≤ w L λ(1 -λ)|x 0 -x 1 | 2 ,
and

λG(x 0 , m) + (1 -λ)G(x 1 , m) -G(λx 0 + (1 -λ)x 1 , m) ≤ w G λ(1 -λ)|x 0 -x 1 | 2 .
Theorem 3.18 (Local fractional semiconcavity of V ). Assume (L1) -(L3). Let R be a positive radius. Then, there exists a constant Λ ≥ 0 such that for any

(t, x) ∈ [0, T ] × B R , any (h, δ) ∈ R × R such that (x + h, t + δ) ∈ [0, T ] × B R and (x -h, t -δ) ∈ [0, T ] × B R we have that V (t + δ, x + h) + V (t -δ, x -h) -2V (t, x) ≤ Λ |h| 2 + |δ| 3 2
.

Proof. We first prove that the value function V is locally semiconcave in space uniformly in time and then, that it is locally semiconcave in space and time.

Let R > 0 be a positive radius and fix

(t, x) ∈ [0, T ] × B R . Let h ∈ R d be such that x + h, x -h ∈ B R and let u * ∈ L 2 be an optimal control for (t, x) ∈ [0, T ] × B R . Then, define the following curves γ(s) =γ(s; t, x, u * ), s ∈ [t, T ] γ + (s) =γ(s; t, x + h, u * ), s ∈ [t, T ] γ -(s) =γ(s; t, x -h, u * ), s ∈ [t, T ].
Thus, we have that

V (t, x + h) + V (t, x -h) -2V (t, x) ≤ T t L(γ + (s), u * (s), m η s ) + L(γ -(s), u * (s), m η s ) -2L(γ(s), u * (s), m η s ) ds +G(γ + (T ), m η T ) + G(γ -(T ), m η T ) -2G(γ(T ), m η T ). (3.14)
Consider, first, the expression involving only the terminal costs:

G(γ + (T ), m η T ) + G(γ -(T ), m η T ) -2G(γ(T ), m η T ) = G(γ + (T ), m η T ) + G(γ -(T ), m η T ) -2G γ + (T ) + γ -(T ) 2 , m η T + 2G γ + (T ) + γ -(T ) 2 , m η T -2G(γ(T ), m η T ).
By (L1) and (L4) we deduce that

G(γ + (T ), m η T ) + G γ -(T ), m η T -2G γ + (T ) + γ -(T ) 2 , m η T ≤ w G |γ + (T ) -γ -(T )| 2 , 2G γ + (T ) + γ -(T ) 2 , m η T -2G γ(T ), m η T ≤ G ∞ |γ + (T ) + γ -(T ) -2γ(T )|.
From the definition of γ, γ + and γ -we have that these curves are solutions of (3.1). Therefore, we get that there exists a real positive constant W such that

|γ + (T ) -γ -(T )| 2 ≤ W |h| 2 , |γ + (T ) + γ -(T ) -2γ(T )| ≤ W |h| 2 .
Hence, we get 2 . By almost similar arguments, one can prove that also the integral term in (3.14) is bounded by a constant times |h| 2 . This proves that V is locally semiconcave in space uniformly in time.

G(γ + (T ), m η T ) + G(γ -(T ), m η T ) -2G(γ(T ), m η T ) ≤ W (w G + G ∞ ) |h|
We proceed to show that V is locally semiconcave on

[0, T ] × R d . Fix (t, x) ∈ [0, T ] × B R and let h ∈ R d , δ ∈ R be such that x + h, x -h ∈ B R
and 0 < t -δ < t + δ < T . Let u * be an optimal control for (t, x) and define the control

ū(s) = u * t + δ + s 2 , s ∈ [t -δ, t + δ].
By the Dynamic Programming Principle we get

V (t + δ, x + h) + V (t -δ, x -h) -2V (t, x) ≤ V (t + δ, x + h) + V (t + δ, γ(t + δ; t -δ, x -h, ū)) -2V (t + δ, γ(t + δ; t, x, u * )) I + t+δ t-δ L(γ(s; t -δ, x -h, u * ), ū(s), m η s ) ds -2 t+δ t L(γ(s; t, x, u * ), u * (s), m η s ) ds II .
Thus, by the first parte of the proof term I is bounded by a constant times |h| 2 + |δ| 2 . Now, we have to estimate term II. Let us denote, for simplicity, by γ -the curve γ(• ; t -δ, x -h, u * ). Then, by assumption (L1) we have that there exists a constant D ≥ 0 such that

II = 2 t+δ t L(γ -(2s -t -δ), u * (s), m η 2s-t-δ ) -L(γ(s), u * (s), m η s ) ds ≤ D t+δ t |γ -(2s -t -δ) -γ(s)| + d 1 (m η 2s-t-δ , m η s ) ds (3.15)
Since η is a Mean Field Games equilibrium we know by Theorem 3.17 that

{m η t } t∈[0,T ] is 1 2 -Hölder continuous in time with respect to the d 1 distance. Therefore, t+δ t d 1 (m η 2s-t-δ , m η s ) ds ≤ κ([m 0 ] α ) t+δ t |s -t -δ| 1 2 ds ≤ 2 3 κ([m 0 ] 1 )|δ| 3 2 .
(3.16)

Now, we have to estimate the distance between the curves γ -and γ. For that, we recall that since γ -and γ are solutions of (3.1) we know that

γ -(2s -t -δ) =e (s-t+δ)A (x -h) + 2s-t-δ t-δ e (τ -t+δ)A B ū(τ ) dτ , γ(s) =e (s-t)A x + s t e (τ -t)A Bu * (τ ) dτ .
From [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.6], without loss of generality, we can assume that u * belongs to L ∞ and consequently, ū ∈ L ∞ . Thus, we obtain that for any

s ∈ [t, t + δ] |γ -(2s -t -δ) -γ -(s)| ≤ e T ∥A∥ |h| + 2se T ∥A∥ B ū ∞ + (s -t)e T ∥A∥ B u * ∞ .
Therefore, we deduce that

t+δ t e T ∥A∥ |h| + 2se T ∥A∥ B ū ∞ + (s -t)e T ∥A∥ B u * ∞ ds ≤ δe T ∥A∥ |h| + 2e T ∥A∥ B ū ∞ + e T ∥A∥ B u * ∞ δ 2 . ( 3.17)
Hence, combining (3.16) and (3.17) with (3.15) yields the conclusion.

Remark 3.19. We note that Theorem 3.18 guarantees that the function x → V (t, x) is linearly semiconcave, locally uniformly in time.

The proof of the following theorem is given in Section 3.5.1 since the techniques we have used to prove it are classical in optimal control theory.

Theorem 3.20. Assume (L1) -(L3). V is locally Lipschitz continuous on

[0, T ] × R d .

Lipschitz regularity of Mean Field Games equilibrium

Define the following class of curves on P α (R d )

Lip(P α ) =      m ∈ C([0, T ]; P α (R d )) : sup t̸ =s t,s∈[0,T ] d 1 (m t , m s ) |t -s| < ∞      ,
and set

P Lip(Pα) m 0 (Γ T ) = η ∈ P m 0 (Γ T , R) : m η ∈ Lip(P 1 ) . Remark 3.21. The set P Lip(Pα) m 0 (Γ T ) is non-empty.
Following the construction we have done in Remark 3.5, let p : R d → Γ T be defined as

x → p[x](t) := e tA x, ∀ t ∈ [0, T ]
and define η = p♯m 0 . Therefore, by Remark 3.5, we only need to prove that m η ∈ Lip(P α ).

Indeed,

d 1 (m η t 1 , m η t 2 ) = sup ϕ∈1-Lip R d ϕ(x) m η t 1 (dx) -m η t 2 (dx) = sup ϕ∈1-Lip Γ T ϕ(γ(t 1 )) -ϕ(γ(t 2 )) η(dγ) = sup ϕ∈1-Lip Γ T ϕ(γ(t 1 )) -ϕ(γ(t 2 )) p♯m 0 (dγ) = sup ϕ∈1-Lip R d ϕ(p[x](t 1 )) -ϕ(p[x](t 2 )) m 0 (dx) = sup ϕ∈1-Lip R d ϕ(e At 1 x) -ϕ(e At 2 x) m 0 (dx) ≤ R d e At 1 x -e At 2 x m 0 (dx).
Since the function t → e At x is Lipschitz continuous in any compact subintervals of R we get the conclusion. (Γ T ). Let u * be an optimal control for (3.4) and let γ * be the minimizing curve generated by u * . Then, there exists a real positive constant

Q 1 such that γ * ∞ ≤ Q 1 (1 + |x|).
Proof. From the same reasoning in [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.6] one can prove that

u * ∞ ≤ Q 0 (1 + |x|). (3.18)
Hence, from the state equation we obtain

γ * (t) ∞ ≤ A γ * ∞ + B u * ∞ .
Thus, by Corollary 3.3 and by (3.18) we get

γ * (t) ∞ ≤ max{ A C 1 , B Q 0 }(1 + |x|). □ Remark 3.23.
We observe that the above result is a generalization of [START_REF] Cannarsa | Mild and weak solutions of mean field game problems for linear control systems[END_REF]Proposition 5.6] where we assumed that there exist two constants c 3 , c 4 such that for any

(x, p, m) ∈ R d × R k × P α (R d ) D x H(x, p, m), p ≥ c 3 |p| 2 -c 4 .

CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG

We recall the definition of the set-valued map E given in the Section 3, that is,

E : P m 0 (Γ T , R), d 1 ⇒ P m 0 (Γ T , R), d 1 with E(η) = ν ∈ P m 0 (Γ T , R) : spt(ν x ) ⊂ Γ * η (x), m 0 -a.e. .

Lemma 3.24. Assume (L1) -(L3). Then, E(P

Lip(Pα) m 0 ) ⊂ P Lip(Pα) m 0 . Proof. Fix η ∈ P Lip(Pα) m 0
and let µ be a Borel probability measure in E(η). We need to prove that for any

t 1 , t 2 ∈ [0, T ], with t 1 < t 2 sup t̸ =s t,s∈[0,T ] d 1 (m t , m s ) |t -s| < ∞.
Hence

d 1 (m µ t 1 , m µ t 2 ) = sup ϕ∈Lip 1 (R d ) R d ϕ(x) m µ t 1 (dx) -m µ t 1 (dx) = sup ϕ∈Lip 1 (R d ) Γ T ϕ(γ(t 1 )) -ϕ(γ(t 2 ) µ(dγ) ≤ Γ T γ(t 1 ) -γ(t 2 ) µ(dγ) ≤ |t 1 -t 2 | Γ T γ ∞ | µ(dγ) ≤|t 1 -t 2 | Γ T Q 1 (1 + |x|) µ(dγ),
where the last inequality follows by Proposition 3.22. Therefore, recalling that x = γ(0) and µ belongs to P m 0 (Γ T , R), we obtain the conclusion.

Theorem 3.25 (Existence of Lipschitz Mean Field Games equilibria). Assume (L1) -(L3). Then, there exist at least one Mean Field Games equilibrium such that the associated family of measures {m η t } t∈[0,T ] belongs to Lip(P α ).

Proof. It is sufficient to prove that the set-valued map E : P

Lip(Pα) m 0 (Γ T ) ⇒ P Lip(Pα) m 0
has a fix point and in order to prove it we want to use Kakutani's fixed point theorem.

We recall that by Lemma 3.12 we have that the map E has closed graph and so also the restriction of E on P

Lip(Pα) m 0 . Moreover, since P Lip(Pα) m 0 ⊂ P m 0 (Γ T , R) we have that P Lip(Pα) m 0
is compact.Therefore, all the assumptions of Kakutani's fixed point theorem are satisfied and this concludes the proof.

Corollary 3.26. Assume (L1) -(L3). Let η ∈ P α (Γ T , R) be a Lipschitz

Mean Field Games equilibrium and let (V, m η ) be a mild solution associated with η. Then, the value function V is locally semiconcave on [0, T ]×R d with a linear modulus of semiconcavity.

Mean Field Games: PDEs system

Optimal syntesis

In order to deduce the PDE system for our Mean Field Games problem, we have to derive first some optimality conditions for the following problem:

J(x, u) = inf γ∈Γ T (x) g(γ(T )) + T 0 L(t, γ(t), u(t)) dt . (OC)
As usual, let V be the value function of the above (OC) problem.

Proposition 3.27. Assume (L1) -(L3). Let (t 0 , x 0 ) ∈ [0, T ] × B R and let p 0 be a point in D * x V (t 0 , x 0 ).
Then, there exists a pair of curves (γ, p) solving the Hamiltonian system

γ(t) = -D p H(t, γ(t), p(t)), γ(t 0 ) = x 0 ṗ(t) = D x H(t, γ(t), p(t)), p(t 0 ) = p 0 such that γ is a minimizer of V (t 0 , x 0 ). In particular, if V (t 0 , •) is differen- tiable at x 0 then γ is the unique minimizer of V (t 0 , x 0 ). Proof. Let p 0 be a point in D * x V (t 0 , x 0 ) such that (t 0 , x 0 ) ∈ [0, T ] × B R
. By definition of reachable gradient, there exists a sequence {x k } k∈N such that

x k → x 0 -p 0 = lim k→∞ D x V (t 0 , x k ).
Let ūk and γk be, respectively, an optimal control and an optimal trajectory with starting point (t 0 , x k ). By the maximum principle, we have that there exists an absolutely continuous arc pk such that

-ṗk (t) = A * pk (t) + D x L(t, γk (t), ūk (t)) pk (T ) = Dg(γ k (T )). (3.19)
By the maximum principle we know that

γk (t) = -D p H(t, γk (t), pk (t)) ṗk (t) = D x H(t, γk (t), pk (t)). (3.20)
Since the sequence {x k } k∈N is convergent, by Corollary 3.3 and Proposition 3.22 we obtain that {γ k } k∈N is equibounded and equicontinuous.

Moreover, by (3.19) we have that for any t ≥ t 0

pk (t) = e (T -t)A * Dg(γ k (T )) + T t e (s-t)A * D x L(s, γk (s), ūk (s)) ds.
Thus, it easily follows that also the sequence of dual arcs {p k } k∈N is equibounded and equicontinuous. Therefore, there exist an absolutely continuous arc p and a curve γ such that pk → p and γk → γ, uniformly as k → ∞.

From (L3), we have that there exists a constant κ ≥ 0 such that

|D x L(t, x, u)| ≤ κ(1 + |u| 2 ).
Moreover, since x ∈ B R we deduce by [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.6] that there exists a constant κ ≥ 0 such that u k ∞ ≤ κ. Consequently, we obtain that

D x L(t, γk (t), ūk (t)) weakly converges in L 2 (0, T ; R d ) to D x L(t, γ(t), ū) as k → ∞.
Therefore, passing to the limit in (3.19) we get that p is a solution of the limit equation and by the maximum principle the pair (γ, p) solves system (3.20). In conclusion, as k → ∞ in the value function we obtain that the curve γ is a minimizer for (t 0 , x 0 ).

Weak solutions

In this section, we consider the case of splitted Langrangian, that is L is of the form (3.9).

We recall that, given the control system (3.1), the Hamiltonian associated with the Lagrangian function L is defined as

H(x, p) = sup u∈R k -p, Ax + Bu -ℓ(x, u) .
For α > 1, let m 0 ∈ P α (R d ) be a Borel probability measure and introduce the following Mean Field Games PDEs system

       -∂ t V (t, x) + H(x, D x V (t, x)) = F (x, m t ), (t, x) ∈ [0, T ] × R d ∂ t m t + div m t D p H(x, D x V (t, x)) = 0, (t, x) ∈ [0, T ] × R d m 0 = m 0 , V (T, x) = G(x, m T ), ∀ x ∈ R d . (3.21) Definition 3.28 (Weak solutions). We say that (V, m) ∈ W 1,∞ ([0, T ] × R d ) × C([0, T ], P α (R d ))
is a weak solution of the Mean Field Games PDEs system if:

(i) m is a solution in the sense of distribution of the continuity equation, i.e. for any test function

φ ∈ C 1 c ([0, T ) × R d ) we have that - R d φ(0, x) m 0 (dx) = T 0 R d ∂ t φ(t, x) -D x φ(t, x), D p H(x, D x V (t, x)) m t (dx). (ii) V (t, •) is differentiable on spt(m t )
and V is a continuous viscosity solution of Hamilton-Jacobi equation.

Remark 3.29. We recall that by classical optimal control theory, see for instance [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF], the following holds:

1. from the maximum principle one can deduce that any minimizer γ of problem (3.4) has the same regularity of the data, thus in this case we obtain that γ ∈ C 2 ;

2. given a Mean Field Games equilibrium η we have that for any x ∈ spt(m η t ) the value function V is differentiable since the value function of an optimal control problem with a strictly convex Hamiltonian (with respect to p) is known to be differentiable in the interior of any optimal trajectory, see for instance [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 6.4.7] and [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF]Proposition 4.4].

Theorem 3.30 (Equivalence between mild and weak solutions). Assume (L1)-(L4).

Fix α > 1 and let m 0 ∈ P α (R d ) be an absolutely continuous with respect the Lebesgue measure and with compact support. Then,

(V, m) ∈ C([0, T ] × R d ) × C([0, T ], P α (R d ))

is a mild solution of the Mean Field Games problem if and only if it is a weak solution of system (3.21).

Proof. First, we show that any mild solutions (V, m η ) is a weak solution.

Let V be the value function defined as in Definition 3.14, in expression (3.8). Then, it is well-known that it is a continuous viscosity solution of the Hamilton-Jacobi equation in system (3.21) and satisfies the terminal condition. Hence, we are left to prove that m η is a solution of the continuity equation in system (3.21) in the sense of distributions.

Indeed, for any

φ ∈ C 1 c ([0, T ) × R d ), we have that d dt R d φ(t, x) m η t (dx) = d dt Γ T φ(t, γ(t)) η(dγ) = Γ T ∂ t φ(t, γ(t)) + D x φ(t, γ(t)), γ(t) η(dγ),
where the last integral is well-posed by point (1) in Remark 3.29. Since η ∈ P m 0 (Γ T , R) is a Lipschitz Mean Field Games equilibrium we know that η is supported on the minimizers of problem (3.4). So, from the Maximum Principle we know that

γ(t) = -D p H γ(t), D x V (t, γ(t)) .
Therefore,

d dt R d φ(t, x) m η t (dx) = Γ T ∂ t φ(t, γ(t)) + D x φ(t, γ(t)), γ(t) η(dγ) = Γ T ∂ t φ(t, γ(t)) -D x φ(t, γ(t)), D p H(γ(t), D x V (t, γ(t))) η(dγ) = R d ∂ t φ(t, x) -D x φ(t, x), D p H(x, D x V (t, x)) m η t (dx),
where the last integral in above series of equality is well-posed by point (2) in Remark 3.29. The conclusion follows by integrating the above equalities over [0, T ]. Now, let (V, m) be a weak solution of Mean Field Games system. Since V is a viscosity solution of the Hamilton-Jacobi equation we know that it can be represented by the formula (3.8) in Definition 3.14. Hence, we only have to prove that there exists a Mean Field Games equilibrium η such that m t = e t ♯η.

Since m is a solution of the continuity equation in the sense of distributions, by the superposition principle [7, Theorem 8.2.1] we know that there exists a probability measure µ ∈ P(Γ T ) such that m t = e t ♯µ and µ-a.e. is a solution of the following equation

γ(t) = -D p H(γ(t), D x V (t, γ(t))), t ∈ [0, T ]. (3.22)
As m 0 = e 0 ♯µ, by Theorem 2.4 there exists a family of Borel probability measures µ x , for any x ∈ spt(m 0 ), such that

µ(dγ) = R d µ x (dγ)m 0 (x) dx.
Since m 0 is absolutely continuous with compact support and the value function V is locally Lipschitz continuous, it follows that m 0 -a.e. and µ x -a.e. γ is a solution of (3.22) such that γ(0) = x. Therefore, by the optimal synthesis explained above, such a curve γ is a minimizer of the underlying optimal control problem and from (3.13) it is also the unique solution of (3.22). Hence, the measures µ x are supported on minimizing curves of the optimal control problem. Consequently, µ is a Mean Field Games equilibrium for m 0 .

APPENDIX 49

The following result is an immediate consequence of Theorem 3.30 and Theorem 3.16.

Corollary 3.31. Assume (L1)-(L4).

Assume that F is strictly monotone, in the sense of definition Definition 3.15. Let η 1 , η 2 ∈ P m 0 (Γ T , R) be two Lipschitz Mean Field Games equilibria and let (V 1 , m η 1 ), (V 2 , m η 2 ) be, respectively, the weak solutions of system (3.21).

Then, V 1 ≡ V 2 .

Appendix

Proof of Theorem 3.20

We divide the proof in two steps: first, we prove that V is locally Lipschitz in space and then, we prove that it is locally Lipschitz in both the variables.

Let R be a positive radius and denote by B R the ball of radius R centered in the origin on

R d . Fix x ∈ B R and h ∈ R d such that x + h ∈ B R . Then, given an optimal control u * associated with (t, x) ∈ [0, T ] × B R we get that V (t, x + h) -V (t, x) ≤ T t L(γ(s; t, x + h, u * ), u * (s), m η s ) -L(γ(s; t, x, u * ), u * (s), m η s ) ds +G(γ(T ; t, x + h, u * ), m η T ) -G(γ(T ; t, x, u * ), m η T ). (3.23)
Thus, we have to estimate the distance between two admissible paths: the one starting in (t, x) and the other one starting in (t, x + h). Recall that

γ(s; t, x, u) = e (s-t)A x + s t e (τ -t)A Bu * (τ ) dτ , ∀ s ∈ [t, T ] to obtain |γ(s; t, x + h, u * ) -γ(s; t, x, u * )| ≤ e T ∥A∥ |h|, ∀ s ∈ [t, T ].
Therefore, by assumption (L2) we get

G(γ(T ; t, x + h, u * ), m η T ) -G(γ(T ; t, x, u * ), m η T ) ≤ G ∞ e T ∥A∥ |h|.
So, we just have to bound the integral term in (3.23). By assumption (L3),

CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG

we have that

T t L(γ(s; t, x + h, u * ), u * (s), m η s ) -L(γ(s; t, x, u * ), u * (s), m η s ) ds = T t 1 0 D x L(λγ(s; t, x + h, u * ) + (1 -λ)γ(s; t, x, u * ), u * (s), m η s , γ(s; t, x + h, u * ) -γ(s; t, x, u * ) ds ≤ T t 1 0 D x L(λγ(s; t, x + h, u * ) + (1 -λ)γ(s; t, x, u * ) u * (s), m η s , γ(s; t, x + h, u * ) -γ(s; t, x, u * ) ds ≤ T t 1 0 c 2 1 + |u * (s)| γ(s; t, x + h, u * ) -γ(s; t, x, u * ) ds ≤ T c 2 e T ∥A∥ |h| + c 2 √ T u * 2 |h| = c 2 T e T ∥A∥ + c 2 √ T K |h|,
where u * 2 ≤ K by Proposition 3.2. Then, we conclude that

V (t, x + h) -V (t, x) ≤ c 2 T e T ∥A∥ + c 2 √ T K + G ∞ e T ∥A∥ |h|.
By similar considerations, one can easily prove that the reverse inequality also holds true. Therefore, we have that V is locally Lipschitz in space.

We now prove that V is locally Lipschitz in space and time on

[0, T ]×B R for any R > 0. Fix t ∈ [0, T ], x ∈ B R and let δ ∈ R be such that t+δ ∈ [0, T ].
We recall that, by the Dynamic Programming Principle we know that

V (t, x) = inf u∈L 2 V (t + δ, γ(t + δ; t, x, u)) + t+δ t L(γ(s; t, x, u), u(s), m η s ) ds .
(3.24) Moreover, by [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.6] we know that, under the assumptions (L1)-(L4), for any η ∈ P m 0 (Γ T ) and any

x ∈ R d , problem (3.4) is equiva- lent to the following one inf u∈L ∞ (0,T ;R k ) J η (x, u).
Thus, we can minimize over the set of bounded controls. Let the control u * ∈ L ∞ be optimal for V (t, x). By (3.24) we deduce that for any ϵ ≥ 0

V (t, x) + ϵ ≥ t+δ t L(γ(s; t, x, u * ), u * (s), m η s ) ds + V (t + δ, γ(t + δ; t, x, u * )).

APPENDIX 51

Hence, we have that

V (t + δ, x) -V (t, x) ≤ V (t + δ, x) -V (t + δ, γ(t + δ; t, x, u * )) - t+δ t L(γ(s; t, x, u * ), u * (s), m η s ) ds + ϵ ≤ c 2 T e T ∥A∥ + c 2 √ T K + G ∞ e T ∥A∥ |x -γ(t + δ; t, x, u * )| +δ c 1 + 1 c 0 u * ∞ , (3.25)
where the last inequality holds true by the first step of the proof and assumption (L3). Moreover, since the curve γ(•; t, x, u * ) is Lipschitz continuous in time, we know that the first term of the right-hand side is bounded by a constant times δ. Thus, the proof of first estimate is complete.

On the other hand, again by (3.24) we know that taking u ≡ 0 we have that

V (t, x) ≤ V (t + δ, γ(t + δ; t, x, 0)) + t+δ t L(γ(s; t, x, 0), 0, m η s ) ds.
Therefore, adding and subtracting the term V (t + δ, x) we get that

V (t, x) -V (t + δ, x) ≤ V (t + δ, γ(t + δ; t, x, 0)) -V (t + δ, x) + t+δ t L(γ(s; t, x, 0), 0, m η s ) ds.
Hence, by the same considerations as in (3.25) we get the result.

Chapter 4

Ergodic behavior of control and mean field games problems depending on acceleration 4.1 Setting and assumptions of the problems

Calculus of variation with acceleration

In our first main result we study the large time average of an optimal control problem of acceleration. Let L :

T d × R d × R d → R be the Lagrangian function defined as L(x, v, w) = 1 2 |w| 2 + F (x, v)
where F : T d × R d → R satisfies the following assumptions:

(F1) F is globally continuous with respect to both variables;

(F2) there exists α > 1 and there exists a constant c F ≥ 1 such that for any

(x, v) ∈ R d × R d 1 c F |v| α -c F ≤ F (x, v) ≤ c F (1 + |v| α ) (4.1)
and, without loss of generality, we assume

F (x, v) ≥ 0 for an (x, v) ∈ T d × R d ; (F3) there exists a constant C F ≥ 0 such that |D x F (x, v)| + |D v F (x, v)| ≤ C F (1 + |v| α ).

54CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

Let Γ be the set C 1 curves γ : [0, +∞) → T d (endowed with the local uniform convergence of the curve and its derivative) and for

(t, x, v) ∈ [0, T ]× T d × R d let Γ t (x, v) be the subset of Γ such that γ(t) = x and γ(t) = v. Define the functional J t,T : Γ → R as J t,T (γ) = T t 1 2 |γ(s)| 2 + F (γ(s), γ(s)) ds, if γ ∈ H 2 (0, T ; T d ), (4.2)
and

J t,T (γ) = +∞ if γ ∈ H 2 (0, T ; T d ),
and let V T (t, x, v) denote the value function associated with the functional J t,T , i.e.

V T (t, x, v) = inf γ∈Γt(x,v) J t,T (γ). (4.3)
Let H be the Hamiltonian associated with the Lagrangian L, that is for any

(x, v, p v ) ∈ T d × R d × R d , H(x, v, p v ) = 1 2 |p v | 2 -F (x, v),
where

p v ∈ R d denotes the momentum variable associated with v ∈ R d .
Then, it is not difficult to see that the value function V T is a continuous viscosity solution of the following Hamilton-Jacobi equation on

[0, T ] × T d × R d : -∂ t V T (t, x, v) -D x V T (t, x, v), v + 1 2 |D v V T (t, x, v)| 2 = F (x, v), V T (T, x, v) = 0 in T d × R d .
Our aim is to characterize the behavior of V T (0, •, •) as T → +∞. To state the result, we need the notion of closed measure, which requires another notation: we set

P α,2 (T d × R d × R d ) = µ ∈ P(T d × R d × R d ) : T d ×R d ×R d |w| 2 + |v| α µ(dx, dv, dw) < +∞ endowed with the weak- * convergence. Definition 4.1 (Closed measure). Let η ∈ P α,2 (T d × R d × R d ). We say that η is a closed measure if for any test function φ ∈ C ∞ c (T d × R d ) the following holds T d ×R d ×R d D x φ(x, v), v + D v φ(x, v), w η(dx, dv, dw) = 0.
We denote by C the set of closed measures.

SETTING AND ASSUMPTIONS OF THE PROBLEMS

55

Theorem 4.2 (Main result 1). Assume that F satisfies assumptions (F1) and (F2). Then, the following limits exist:

lim T →+∞ 1 T V T (0, x, v) = lim T →+∞ inf γ∈Γ 0 (x,v) 1 T J T (γ)
and are independent of

(x, v) ∈ T d × R d . Moreover, if F satisfies also (F3) then lim T →∞ 1 T V T (0, x, v) = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw). Remark 4.3.
1. If we denote by λ the above limits, the convergence of

V T (0, x, v) -λT
is a completely open problem in this context. This is related to the lack of solution of the ergodic HJ equation.

2. The (strong) structure condition on L and the fact that the problem is periodic in the x variable can probably be relaxed: this would require however more refined and technical estimates and we have chosen to work in this simpler framework.

Mean Field Games of acceleration

In our second main result, we consider a mean field game problem of acceleration. The Lagrangian function L :

T d × R d × R d × P 1 (T d × R d ) → R now takes the form L(x, v, w, m) = 1 2 |w| 2 + F (x, v, m)
where (F2') there exists α > 1 and a constant c F ≥ 1 such that for any

F : T d × R d × P 1 (T d × R d ) → R
(x, v, m) ∈ R d × R d × P 1 (T d × R d × R d ) 1 c F |v| α -c F ≤ F (x, v, m) ≤ c F (1 + |v| α )
and, without loss of generality, we assume F (x, v, m) ≥ 0 for any

(x, v, m) ∈ T d × R d × P 1 (T d × R d × R d ); (F3') there exists a constant C F ≥ 0 such that, for any (x, v, m) ∈ R d × R d × P 1 (T d × R d × R d ), |D x F (x, v, m)| + |D v F (x, v, m)| ≤ C F (1 + |v| α ).

56CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

We consider the time-dependent MFG system on [0

, T ] × T d × R d        -∂ t u T (t, x, v) -D x u T (t, x, v), v + 1 2 |D v u T (t, x, v)| 2 = F (x, v, m T t ), ∂ t m T t -v, D x m T t -div m T t D v u T (t, x, v) = 0, u T (T, x, v) = g(x, v, m T T ), in T d × R d , m T 0 = m 0 ∈ P(T d × R d ). (4.4)
where the terminal condition of the Hamilton-Jacobi equation satisfies the following:

(G1) (x, v) → g(x, v, m) belongs to C 1 b (T d × R d ) for any m ∈ P(T d × R d ) (without loss of generality we assume g(x, v, m) ≥ 0) and m → g(x, v, m) is Lipschitz continuous with respect to the d 1 distance, uni- formly in (x, v) ∈ T d × R d .
We recall that (u T , m T ) is a solution of (4.4) if u T is a viscosity solution of the first equation and m T is a solution in the sense of distributions of the second equation.

Our aim is to understand the averaged limit of u T as T → +∞. For this we define the ergodic MFG problem, inspired by the characterization of the limit in Theorem 4.2. Let us recall that the notion of closed measure was introduced in Definition 4.1 and that C denotes the set of closed measures.

Definition 4.4 (Solution of the ergodic MFG problem). We say that

( λ, μ) ∈ R × C is a solution of the ergodic MFG problem if λ = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ) µ(dx, dv, dw) = T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ) μ(dx, dv, dw). (4.5)

Theorem 4.5 (Main result 2). Assume that F and G satisfy (F1'), (F2') and (G1).

1. There exists at least one solution ( λ, μ) ∈ R × C of the ergodic MFG problem (4.5). Moreover, if F satisfies the following monotonicity assumption: there exists

M F > 0 such that for m 1 , m 2 ∈ P(T d × R d ) T d ×R d F (x, v, m 1 ) -F (x, v, m 2 ) (m 1 (dx, dv) -m 2 (dx, dv)) ≥ M F T d ×R d F (x, v, m 1 ) -F (x, v, m 2 ) 2 dxdv, ( 4
T →+∞ 1 T T d ×R d u T (0, x, v) m 0 (dx, dv) = λ.

Ergodic behavior of control of acceleration 4.2.1 Existence of the limit

Before proving the main result of this section, Proposition 4.13, we need a few preliminary lemmas.

Lemma 4.6. Assume that F satisfies (F1) and (F2). Then, for any (x, v) ∈ T d × B R , with R ≥ 0, and for any T > 0, we have

1 T V T (0, x, v) ≤ c F (1 + R α ).
Remark 4.7. The result also holds when F = F (t, x, v) depends also on time, provided that F is continuous and satisfies (F2) with a constant c F independent of t.

Proof. Define the curve ξ(t) = x + tv, for t ∈ [0, T ]. Then, by definition of the value function V T , we have

V T (0, x, v) ≤ J T (ξ) = T 0 F (x + tv, v) dt ≤ T c F (1 + R α ).
Lemma 4.8. Assume that F satisfies (F1) and (F2). Let θ ≥ 1, (x 0 , v 0 ) and (x, v) be in

T d × B R for some R ≥ 1.
Then, there exists a constant C 2 ≥ 0 (depending only the constants α and c F in (F2)) and a curve σ :

[0, θ] → R d such that σ(0) = x 0 , σ(0) = v 0 and σ(θ) = x, σ(θ) = v and J θ (σ) ≤ C 2 (R 2 θ -1 + R α θ). (4.7)
Remark 4.9. The result also holds when F = F (t, x, v) depends also on time, provided that F is continuous and satisfies (F2) with a constant c F independent of t.

Proof. Define the following parametric curve

σ(t) = x 0 + v 0 t + Bt 2 + Ct 3 , t ∈ [0, θ]. 58CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL Choosing B = 3(x -x 0 ) -θv -2θv 0 )θ -2 C = (-2(x -x 0 ) + θ(v + v 0 ))θ -3 , we have that σ(0) = x 0 , σ(0) = v 0 and σ(1) = x, σ(1) = v.
By definition of the functional J θ we get

J θ (σ) = θ 0 1 2 |σ(t)| 2 + F (σ(t), σ(t)) dt ≤ θ 0 1 2 |2B + 6Ct| 2 + c F (1 + |v 0 + 2tB + 3t 2 C| α ) dt ≤ C 2 (R 2 θ -1 + R α θ),
for some constant C 2 depending on the constants α and c F in (F2) only.

Lemma 4.10. Let T ≥ 2 and (x, v) ∈ T d × B R 0 for some R 0 ≥ c 2 α F . Let γ ∈ Γ(x, v) be optimal for V T (0, x, v). Then for any λ ≥ 2 there exists γ ∈ Γ(x, v) with γ(T ) = x, γ(T ) = v and J T (γ) ≤ J T (γ) + C 3 (λ 2 R 2 0 + R α 0 λ -α T ),
where the constant C 3 depends on α and c F only.

Remark 4.11. The result also holds when F = F (t, x, v) depends also on time, provided that F is continuous and satisfies (F2) with a constant c F independent of t. In addition, by the construction in the proof, there exists τ > 0 such that γ = γ on [0, τ ] and

T τ ( 1 2 | γ(t)| 2 + c F (1 + | γ(t)| α ))dt ≤ C 3 (λ 2 R 2 0 + R α 0 λ -α T ).
Finally, the map which associates γ and τ to γ is measurable.

Proof. Let τ := sup{t ≥ 0, | γ(t)| ≤ λR 0 } if |γ(T -1)| > λR 0 , T -1 otherwise. If τ ≥ T -2, we set γ(t) = γ(t) for t ∈ [0, τ ], σ(t -τ ) for t ∈ [τ, T ],
where σ is the map built in Lemma 4.8 with

θ = T -τ , σ(0) = γ(τ ), σ(0) = γ(τ ), σ(T -τ ) = x, σ(T -τ ) = v. If τ < T -2, then we set γ(t) =    γ(t) for t ∈ [0, τ ], σ 1 (t -τ ) for t ∈ [τ, τ + 1], σ 2 (t -τ -1) for t ∈ [τ + 1, T ],
where σ 1 and σ 2 are the map built in Lemma 4.8 with θ

= 1, σ 1 (0) = γ(τ ), σ1 (0) = γ(τ ), σ 1 (1) = x, σ(1) = v and θ = T -τ -1 and σ 2 (0) = σ 2 (T - τ -1) = x and σ2 (0) = σ2 (T -τ -1) = v respectively. Note that γ(T ) = x and γ(T ) = v.
In order to estimate J T (γ), we first show that τ cannot be too small: namely we claim that

τ ≥ T 1 - c F (1 + R α 0 ) 1 c F (λR 0 ) α -c F -1. (4.8)
Indeed, let us first recall that by Lemma 4.6 we have

J T (γ) ≤ c F (1 + R α 0 )T.
On the other hand, by assumption (F2) and the fact that

| γ(t)| > λR 0 on [τ, T -1]
and that F ≥ 0, we also have that

J T (γ) = T 0 1 2 |γ(t)| 2 + F (γ(t), γ(t)) dt ≥ T -1 τ 1 c F | γ(t)| α -c F dt ≥ (T -τ -1) 1 c F (λR 0 ) α -c F . So (4.8) holds for R 0 ≥ c 2/α
F . We estimate J T (γ) in the case τ < T -2, the other case being similar and easier. Note that | γ(τ )| ≤ λR 0 . By Lemma 4.8 and the fact that F ≥ 0, we have

J T (γ) = τ 0 ( 1 2 |γ(t)| 2 + F (γ(t), γ(t)))dt + 1 0 ( 1 2 |σ 1 (t)| 2 + F (σ 1 (t), σ1 (t)))dt + T -τ -1 0 ( 1 2 |σ 2 (t)| 2 + F (σ 2 (t), σ2 (t)))dt ≤ J T (γ) + C 2 ((λR 0 ) 2 + (λR 0 ) α + R 2 0 (T -τ -1) -1 + R α 0 (T -τ -1)).
In view of (4.8) this implies that

J T (γ) ≤ J T (γ) + C 3 (λ 2 R 2 0 + R α 0 λ -α T ),
for a constant C 3 depending on α and c F only.

Next we prove that the (x, v) → V T (0, x, v) have locally uniformly bounded oscillations. Lemma 4.12. There exists a constant M 1 (R) ≥ 0 such that for any (x, v) and

(x 0 , v 0 ) in T d × B R we have that V T (0, x, v) -V T (0, x 0 , v 0 ) ≤ M 1 (R).

60CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

Proof. Let γ * be a minimizer for V T (0, x 0 , v 0 ) and let σ : [0, 1] → T d be such that σ(0) = x, σ(0) = v and σ(1) = x 0 , σ(1) = v 0 as in Lemma 4.8 for θ = 1. Define

γ(t) = σ(t), t ∈ [0, 1] γ * (t -1), t ∈ [1, T ].
Then γ ∈ Γ 0 (x, v) and, by Lemma 4.8 and the assumption that F ≥ 0, we have that

V T (0, x, v) -V T (0, x 0 , v 0 ) ≤ 1 0 1 2 |σ(t)| 2 + F (σ(t), σ(t)) dt + T 1 1 2 |γ * (t -1)| 2 + F (γ * (t -1), γ * (t -1)) dt -V T (0, x 0 , v 0 ) ≤ 2C 2 R 2 + T -1 0 1 2 |γ * (t)| 2 + F (γ * (t), γ * (t)) dt -V T (0, x 0 , v 0 ) ≤ 2C 2 R 2 - T T -1 1 2 |γ * (t)| 2 + F (γ * (t), γ * (t)) dt ≤ 2C 2 R 2 ,
which is the claim.

Proposition 4.13 (Existence of the limit). Assume that F satisfies (F1) and (F2). Then, for any (x, v) ∈ T d × R d , the following limits exist:

lim T →+∞ 1 T V T (0, x, v) = lim T →+∞ 1 T inf γ∈Γ 0 (x,v) J T (γ).
In addition the convergence is locally uniform in (x, v) and the limit is independent of (x, v).

Proof. Fix R 0 ≥ c 2/α F such that |v| ≤ R 0 . Let {T n } n∈N and let {γ n } n∈N be a sequence of minimizers for V Tn (0, x, v) such that T n → ∞ as n → ∞ and lim inf T →∞ 1 T V T (0, x, v) = lim n→∞ 1 T n J Tn (γ n ).
For λ ≥ 2, let us define γn is in Lemma 4.10. Then we know that γn (T ) = x, γn (T ) = v and

J Tn (γ n ) ≤ J Tn (γ n ) + C 3 (λ 2 R 2 0 + R α 0 λ -α T n ). (4.9)
Let us define γn as the periodic extension of the curve γn , i.e. γn is T nperiodic and it is equal to γn on [0, T n ]. Then, taking γn as competitors for J T we obtain that lim sup

T →∞ inf γ∈Γ 0 (x,v) 1 T J T (γ) ≤ lim sup T →∞ 1 T J T (γ n ) = 1 T n J Tn (γ n ) ≤ 1 T n J Tn (γ n ) + C 3 (λ 2 R 2 0 T -1 n + R α 0 λ -α ) ,
where the equality holds true since we are taking the limit of a periodic function and the last inequality holds by (4.9). We get the conclusion letting n → ∞ and then λ → ∞, indeed: as n → ∞ we deduce that lim sup

T →∞ inf γ∈Γ 0 (x,v) 1 T J T (γ) ≤ lim n 1 T n J Tn (γ n ) + C 3 R α 0 λ -α = lim inf T →+∞ inf γ∈Γ 0 (x,v) 1 T J T (γ) + C 3 R α 0 λ -α
and then, taking the limit as λ → ∞ we get lim sup

T →∞ inf γ∈Γ 0 (x,v) 1 T J T (γ) ≤ lim inf inf γ∈Γ 0 (x,v) 1 T J T (γ).
As the (V T (0, •, •)) have locally bounded oscillation (Lemma 4.12), the above convergence is locally uniform and the limit does not depend on (x, v).

Characterization of the ergodic limit

In this part we characterize the limit given in Proposition 4.13 in term of closed measures. The proof of the main result, Proposition 4.22, where this characterization is stated, is technical and requires several steps. Here are the main ideas of the proof. By using standard results on occupational measures, one can obtain in a relatively easy way that

λ := lim T →∞ inf γ∈Γ 0 (x 0 ,v 0 ) 1 T J T (γ) ≥ inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw),
where C denotes the set of closed probability measures (see Definition 4.1). The difficult part of the proof is the opposite inequality. The first step for this is a min-max formula (Theorem 4.15) which gives, by using the characterization of closed measures, that

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw) = sup φ∈C ∞ c (T d ×R d ) inf (x,v)∈T d ×R d - 1 2 |D v φ(x, v)| 2 -D x φ(x, v), v + F (x, v) .
In order to exploit this inequality, one just needs to find a map

φ ∈ C ∞ c (T d × R d ) for which - 1 2 |D v φ(x, v)| 2 -D x φ(x, v), v + F (x, v) 62CHAPTER 4.

ERGODIC BEHAVIOR OF ACCELERATION MODEL

is almost equal to λ. This is not easy because the corrector of our ergodic problem does not seem to exist (at least in the usual sense) because of the lack of controllability and, if it existed, it certainly would not be smooth with a compact support. The standard idea in this set-up is to use instead the approximate corrector, i.e., the solution V δ to

δV δ (x, v) + 1 2 |D v V δ (x, v)| 2 + D x V δ (x, v), v = F (x, v) in T d × R d .
However, this approximate corrector has not a compact support either (it is even coercive, see Proposition 4.16) and δV δ does not converge uniformly to -λ, but only locally uniformly. We overcome these issues by an extra approximation argument (Lemma 4.18).

Let us first explain why closed measures pop up naturally in our problem. To see this, let (x 0 , v 0 ) ∈ T d × R d be an initial position and let γ T (x 0 ,v 0 ) be an optimal trajectory for V T (0, x 0 , v 0 ). We define the family of Borel probability measures {µ T } T >0 as follows: for any function

φ ∈ C ∞ c (T d × R d × R d ) T d ×R d ×R d φ(x, v, w) µ T (dx, dv, dw) = 1 T T 0 φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)) dt. (4.10)
Lemma 4.14. Assume that F satisfies (F1) and (F2). Let the family of probability measures {µ T } T >0 be defined by eq. (4.10). Then, {µ T } T >0 is tight and there exists a closed measure µ * such that, up to a subsequence, µ T ⇀ * µ * as T → +∞.

Proof. We first prove that {µ T } T >0 its a tight family of probability measures. Indeed, by assumption (F2) for

(x 0 , v 0 ) ∈ T d × R d we know that 1 T V T (0, x 0 , v 0 ) = 1 T T 0 1 2 |γ T (x 0 ,v 0 ) (t)| 2 + F (γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)) dt = T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ T (dx, dv, dw) ≥ T d ×R d ×R d 1 2 |w| 2 + 1 c F |v| α -c F µ T (dx, dv, dw).
On the other hand, by Lemma 4.6 we have that

1 T V T (0, x 0 , v 0 ) ≤ C 1
where C 1 only depends on the initial point (x 0 , v 0 ). Therefore, we obtain that

T d ×R d ×R d 1 2 |w| 2 + 1 c F |v| α µ T (dx, dv, dw) ≤ C 1
which implies that {µ T } T >0 is tight. By Prokhorov theorem there exists a measure µ

* ∈ P(T d × R d × R d ) such that up to a subsequence µ T ⇀ * µ * as T → +∞.
We now show that the measure µ * is closed in the sense of Definition 4.

1. Let φ ∈ C ∞ c (T d × R d ) be a test function and let R ≥ 0 be such that φ(x, v) = 0 for any (x, v) ∈ T d × B c R . Moreover, define τ * = sup{t ∈ [0, T ] : | γT (x 0 ,v 0 ) (t)| ≤ R}, if | γ(x 0 ,v 0 ) (T )| > R T, if | γ(x 0 ,v 0 ) (T )| ≤ R and let σ * : [τ * , τ * + 1] → T d be as in Lemma 4.8 such that σ * (τ * ) = γ T (x 0 ,v 0 ) (τ * ), σ * (τ * ) = γT (x 0 ,v 0 ) (τ * ) and σ * (τ * + 1) = x 0 , σ * (τ * + 1) = v 0 . Moreover, define γ(t) = γ T (x 0 ,v 0 ) (t), t ∈ [0, τ * ] σ * (t), t ∈ (τ * , τ * + 1].
Then we get

T d ×R d ×R d D x φ(x, v), v + D v φ(x, v), w dµ T (x, v, w) = 1 T T 0 D x φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)), γT (x 0 ,v 0 ) (t) + D v φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)), γT (x 0 ,v 0 ) (t) dt = 1 T τ * +1 0 D x φ(γ T (t), γT (t)), γT (t) + D v φ(γ T (t), γT (t)), γT (t) dt - 1 T τ * +1 τ * D x φ(σ * (t), σ * (t)), σ * (t) + D v φ(σ * (t), σ * (t)), σ * (t) dt + T τ * D x φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)), γT (x 0 ,v 0 ) (t) + D v φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)), γT (x 0 ,v 0 ) (t) dt
One can immediately observe that by construction the last integral is 0 (since φ has a support in T d × B R ) and by the definition of γ one also has that the first one is 0. The behavior of the second is also immediate because, as φ is bounded,

1 T τ * +1 τ * D x φ(σ * (t), σ * (t)), σ * (t) + D v φ(σ * (t), σ * (t)), σ * (t) dt = 1 T (φ(σ * (τ * + 1), σ * (τ * + 1)) -φ(σ * (τ * ), σ * (τ * )) → 0, as T → +∞.
The proof is thus complete.

64CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

The next step consists in formulating in two different ways the expected limit of Proposition 4.13. Theorem 4.15 (Minmax formula). Assume that F satisfies (F1) and (F2). Then, the following equality holds true:

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw) = sup φ∈C ∞ c (T d ×R d ) inf (x,v)∈T d ×R d - 1 2 |D v φ(x, v)| 2 -D x φ(x, v), v + F (x, v) . (4.11)
Proof. By definition of a closed measure we can write

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw) = inf µ∈P 2,α (T d ×R d ×R d ) sup φ∈C ∞ c (T d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw).
Our aim is to use the min-max Theorem (see Theorem 4.34 below). We use for this the notation introduced in Appendix A and set

A = C ∞ c (T d × R d ), B = P 2,α (T d × R d × R d ) and for any (φ, µ) ∈ A × B L(φ, µ) := T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw).
Let us choose φ * (x, v) = 0 and

c * = 1 + inf µ∈P 2,α (T d ×R d ×R d ) sup φ∈C ∞ c (T d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw).
Note that c * is finite (since it is bounded below by assumption eq. (4.1) and bounded above for µ = δ (x 0 ,0,0) for any x 0 ∈ T d ). In addition, the set

B * = {µ ∈ B : L(φ * , µ) ≤ c * }
is nonempty and tight, and thus compact, in

P 2,α (T d × R d × R d )
for the weak- * convergence. Finally, we have

c * ≥ 1 + sup φ∈C ∞ c (T d ×R d ) inf µ∈P 2,α (T d ×R d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw).

ERGODIC BEHAVIOR OF CONTROL OF ACCELERATION 65

Therefore, Theorem 4.34 states that

inf µ∈P 2,α (T d ×R d ×R d ) sup φ∈C ∞ c (T d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw) = sup φ∈C ∞ c (T d ×R d ) inf µ∈P 2 (T d ×R d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w µ(dx, dv, dw) = sup φ∈C ∞ c (T d ×R d ) inf (x,v,w)∈T d ×R d ×R d 1 2 |w| 2 + F (x, v) -D x φ(x, v), v -D v φ(x, v), w = sup φ∈C ∞ c (T d ×R d ) inf (x,v)∈T d ×R d - 1 2 |D v φ(x, v)| 2 -D x φ(x, v), v + F (x, v) .
This complete the proof.

Next we introduce and study the discounted problem associated with eq. (4.2). For any δ > 0 and any

(x, v) ∈ T d × R d we define J δ : Γ → R ∪ {+∞} as J δ (γ) = +∞ 0 e -δt 1 2 |γ(t)| 2 + F (γ(t), γ(t)) dt if γ is absolutely continuous with +∞ 0 e -δt 1 2 |γ(t)| 2 + | γ(t))
| α dt < +∞, and J δ (γ) = +∞ otherwise. We define the associated value function (the approximate corrector)

V δ (x, v) = inf γ∈Γ 0 (x,v) J δ (γ).
(4.12)

We recall that V δ is the unique continuous viscosity solution with a polynomial growth of the following Hamilton-Jacobi equation

δV δ (x, v) + 1 2 |D v V δ (x, v)| 2 + D x V δ (x, v), v = F (x, v). (4.13)
As the convergence of V T (0, •, •)/T is locally uniform (by Lemma 4.12), we can apply the Abelian-Tauberian Theorem of [START_REF] Oliu-Barton | A uniform tauberian theorem in optimal control. advances in dynamic games[END_REF] and we have that for any

(x, v) ∈ T d × R d lim δ→0 + δV δ (x, v) = lim T →∞ 1 T V T (0, x, v) =: λ. (4.14)
In the proof of the main result of this section (Proposition 4.22) we will have to smoothen the map V δ . This involves some local regularity properties of V δ , which is the aim of the next result. (ii) {V δ (x, v)} δ>0 has locally uniformly bounded oscillation, i.e. there exists a constant M (R) ≥ 0 such that for any

(x 0 , v 0 ), (x, v) ∈ T d × B R V δ (x, v) -V δ (x 0 , v 0 ) ≤ M (R).
(iii) there exists a constant C ≥ 0 such that for any

(x, v) ∈ T d × R d C-1 |v| α -Cδ -1 ≤ V δ (x, v) ≤ c F δ -1 (|v| α + 1); (4.15) (iv) the map x → V δ (x,
|D x V δ (x, v)| ≤ C δ (1 + |v| α ). (4.16) Proof. (i) Fix (x, v) ∈ T d × B R and define a competitor γ : [0, +∞] → T d such that γ(t) = x + tv.
By definition and eq. ( 4.1) we get

δV δ (x, v) ≤ δ ∞ 0 e -δt F (γ(t), γ(t)) ds ≤ c F (1 + |v| α ) ≤ c F (1 + R α ).
On the other hand, we have by (F2) that F ≥ 0 and thus V δ ≥ 0, which completes the proof of (i).

(ii) Let (x 0 , v 0 ), (x, v) ∈ T d × B R be fixed points, let γ * be a minimizer for V δ (x 0 , v 0 ) and let σ be defined as in Lemma 4.8 such that σ(0) = x, σ(0) = v and σ(1) = x 0 , σ(1) = v 0 . We define a new curve γ : [0, +∞) → T d as follows

γ(t) = σ(t), t ∈ [0, 1] γ * (t -1), t ∈ (1, +∞). Then V δ (x, v) -V δ (x 0 , v 0 ) ≤ 1 0 e -λt 1 2 |γ(t)| 2 + F (γ(t), γ(t)) dt + +∞ 1 e -λt 1 2 |γ(t)| 2 + F (γ(t), γ(t)) dt -V λ (x 0 , v 0 ).
(4.17)

ERGODIC BEHAVIOR OF CONTROL OF ACCELERATION

67 By a change of variable, we have that

+∞ 1 e -δt 1 2 |γ * (t)| 2 + F (γ * (t), γ * (t)) dt =e -δ ∞ 0 e -δs 1 2 |γ * (s)| 2 + F (γ * (s), γ * (s)) ds = e -δ V δ (x 0 , v 0 ).
Therefore, we obtain that

+∞ 1 e -δt 1 2 |γ(t)| 2 + F (γ(t), γ(t)) dt -V δ (x 0 , v 0 ) ≤ e -δ -1 V δ (x 0 , v 0 ) ≤ δ|V δ (x 0 , v 0 )| ≤ c F (1 + R α ), (4.18)
where the last inequality holds true by (i). Moreover, by construction of σ in Lemma 4.8 we have that

1 0 e -δt 1 2 |σ(t)| 2 + F (σ(t), σ(t)) dt ≤ J 1 (σ) ≤ C 2 (R 2 + R α ). (4.19)
Combining together inequality eq. ( 4.18) and eq. (4.19) in eq. ( 4.17) we get (ii):

V δ (x, v) -V δ (x 0 , v 0 ) ≤ c F (1 + R α ) + C 2 (R 2 + R α ) =: M (R).
(iii) For some constants M 1 and M 2 we have that the map Z :

T d ×R d → R such that Z(x, v) = M -1 1 |v| α -M 2 δ -1 is a subsolution of eq. (4.13), indeed δZ(x, v) + 1 2 |D v Z(x, v)| 2 + D x Z(x, v), v -F (x, v) ≤ δM -1 1 |v| α -M 2 + 1 2 M -2 1 α 2 |v| 2(α-1) -c -1 F |v| α + c F . As 2(α -1) ≤ α, since α ∈ (1, 2], we get, for M 1 and M 2 large enough, δZ(x, v) + 1 2 |D v Z(x, v)| 2 + D x Z(x, v), v -F (x, v) ≤ 0.
By comparison we obtain V δ ≥ Z, which proves the first inequality in eq. (4.15).

In the same way, considering the map

Z(x, v) = c F δ -1 (|v| α + 1), we have δZ(x, v) + 1 2 |D v Z(x, v)| 2 + D x Z(x, v), v -F (x, v) ≥ c F (|v| α + 1) + 1 2 δ -2 (c F α) 2 |v| 2(α-1) -c F |v| α -c F ≥ 0, 68CHAPTER 4

. ERGODIC BEHAVIOR OF ACCELERATION MODEL

so that Z is a supersolution. By comparison we conclude that the second inequality in (4.15) holds.

(iv) Let γ * be optimal for V δ (x, v) and let h ∈ R d . Then

V δ (x + h, v) ≤ +∞ 0 e -δt 1 2 |γ * (t)| 2 + F (γ * (t) + h, γ * (t)) dt ≤ V δ (x, v) + +∞ 0 e -δt (F (γ * (t) + h, γ * (t)) -F (γ * (t), γ * (t))) dt ≤ V δ (x, v) + +∞ 0 e -δt c F (1 + | γ * (t)| α )|h| dt, (4.20)
where the last inequality holds true by assumption (F3). Moreover, by eq. ( 4.15) we deduce that there exists a constant

C δ ≥ 0 such that +∞ 0 e -δt (c -1 F | γ * (t)| α -c F ) dt ≤ V δ (x, v) ≤ C δ (1 + |v| α ).
Therefore, by eq. ( 4.20) we deduce that

V δ (x + h, v) -V δ (x, v) ≤ C δ (1 + |v| α )|h|,
which implies that V δ is locally Lipschitz continuous in space and proves (iv).

We now strengthen a little the convergence in (4.14):

Proposition 4.17. Assume that F satisfies (F1)-(F3). Then

λ = lim δ→0 + inf (x,v)∈T d ×R d δV δ (x, v),
with λ defined in eq. (4.14).

Proof. First we note that, by (i) in Proposition 4.16, the convergence in (4.14) is locally uniform. Fix R ≥ 0 such that

c -1 F R α -c F > λ. (4.21)
Then, for any ε > 0, there exists δ ε > 0 such that for any δ ∈ (0, δ ε ) we have that inf

(x,v)∈T d ×B R δV δ (x, v) ≥ λ -ε . (4.22) Fix (x, v) ∈ T d × R d and let γ * δ be a minimizer for V δ (x, v). We define τ δ = inf{t ∈ [0, +∞] : | γ * δ (t)| ≤ R}, if {t ∈ [0, +∞] : | γ * δ (t)| ≤ R} = ∅ +∞, if {t ∈ [0, +∞] : | γ * δ (t)| ≤ R} = ∅.

ERGODIC BEHAVIOR OF CONTROL OF ACCELERATION

69

By Dynamic Programming Principle we get

V δ (x, v) = τ δ 0 e -δt 1 2 |γ * δ (t)| 2 + F (γ * δ (t), γ * δ (t)) dt+e -δτ δ V δ (γ * δ (τ δ ), γ * δ (τ δ ))
and by assumption eq. ( 4.1) and definition of τ δ we deduce that

δV δ (x, v) ≥ (c -1 F R α -c F )(1 -e -δτ δ ) + e -δτ δ δV δ (γ * δ (τ δ ), γ * δ (τ δ )). (4.23) If τ δ is finite, we have that | γ * δ (τ δ )
| is bounded by R and thus, by (4.21) and eq. ( 4. [START_REF] Cannarsa | Some characterizations of optimal trajectories in control theory[END_REF] we deduce that for any δ ∈ (0, δ ε )

δV δ (x, v) ≥ λ(1 -e -δτ δ ) + e -δτ δ (λ -ε) ≥ λ -ε .
By (4.21) and (4.23) the same inequality also holds if τ δ = +∞. Hence, we obtain that lim

δ→0 + inf (x,v)∈T d ×R d δV δ (x, v) ≥ λ -ε .
By eq. (4.14) we infer that

λ = lim δ→0 + δV δ (0, 0) ≥ lim δ→0 + inf (x,v)∈T d ×R d δV δ (x, v) ≥ λ -ε,
which implies the desired result since ε is arbitrary.

As V δ is coercive, we cannot use it directly as a test function to test the fact that a measure is closed. To overcome this issue we approximate V δ by family of Lipschitz maps (V R δ ).

Lemma 4.18 (Approximate problem 1). Assume that F satisfies assumption (F1)-(F3). Let R > 0 and define

F R (x, v) = min{F (x, v), R} for any (x, v) ∈ T d × R d . Let V R
δ be the unique continuous and bounded viscosity solution to

δV R δ (x, v)+ 1 2 |D v V R δ (x, v)| 2 + D x V R δ (x, v), v = F R (x, v), (x, v) ∈ T d ×R d .
(4.24) Then, the following holds:

(i) V R
δ is globally Lipschitz continuous;

(ii) there are two positive constants c1,δ and c2,δ such that

δV R δ (x, v) ≥ c1,δ 1 + min{|v| α , R} -c2,δ (4.25
)

for any (x, v) ∈ T d × R d ;
(iii) there is a constant Cδ ≥ 0 such that

|D x V R δ (x, v)| ≤ Cδ 1 + min{|v| α , R} (4.26) for a.e. (x, v) ∈ T d × R d ; 70CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL (iv) V R δ converge, as R → +∞, uniformly on compact subsets of T d × R d to the map V δ defined in (4.12).
The proofs of (i) and (iv) are direct consequences of optimal control theory while the proofs of eq. (4.25) and eq. (4.26) follow the same argument as for eq. (4.15) and eq. (4.16), respectively and we omit these proofs. Lemma 4.19. Assume that F satisfies (F1) -(F3). Let F R and V R δ be defined in Lemma 4. [START_REF] Cannarsa | Existence and uniqueness for mean field games with state constraints. PDE models for multi-agent phenomena[END_REF]. Then we have that

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F R (x, v) µ(dx, dv, dw) ≥ inf (x,v)∈T d ×R d δV R δ (x, v). ( 4
. φ ∈ W 1,∞ (T d × R d ) ∩ C ∞ (T d × R d ). Indeed, let φ ∈ W 1,∞ (T d × R d ) ∩ C ∞ (T d × R d ) and for R > 1 let ξ R ∈ C ∞ c (R d ) be such that ξ R (x, v) = 1 for (x, v) ∈ T d ×B R , ξ R (x, v) = 0 for (x, v) ∈ T d ×R d \B 2R , 0 ≤ ξ R (x, v) ≤ 1 for T d × B 2R \B R and there exists a constant M ≥ 0 such that |Dξ R (x, v)| ≤ M R -1 for any (x, v) ∈ T d × R d . Set φ R = φξ R . Then, we have that φ R ∈ C ∞ c (T d × R d ),
Dφ R is uniformly bounded and converges locally uniformly to Dφ. For µ ∈ C we have:

T d ×R d ×R d D x φ R (x, v), v + D v φ R (x, v), w µ(dx, dv, dw) = 0. (4.28) Since µ ∈ P 2,α (T d × R d × R d )
, we can pass to the limit in eq. (4.28) as R → +∞ by dominate convergence. This proves that

T d ×R d ×R d D x φ(x, v), v + D v φ(x, v), w µ(dx, dv, dw) = 0 for φ ∈ W 1,∞ (T d × R d ) ∩ C ∞ (T d × R d ). Proof. Let ξ 1,ε ∈ C ∞ c (R d ) be such that spt(ξ 1,ε ) ⊂ B ε , ξ 1,ε (x) ≥ 0 and Bε ξ 1,ε (x) dx = 1, and define V R,ε δ (x, v) = V R δ ⋆ x ξ 1,ε (x, v)
where the mollification only holds in x. Then V ε δ satisfies the following inequality in the viscosity sense

δV ε δ (x, v) + 1 2 |D v V ε δ (x, v)| 2 + D x V ε δ (x, v), v ≤ F R ⋆ ξ 1,ε (x, v) ≤ F R (x, v) + C F ε(1 + min{|v| α , R})
where the last inequality holds true by (F3) and the definition of

F R . Now, let ξ 2,ε ∈ C ∞ c (R d ) be such that spt(ξ 2,ε ) ⊂ B ε , ξ 2,ε (v) ≥ 0 and Bε ξ 2,ε (v) dv = 1 and define φ ε,δ R (x, v) = ξ 2,ε ⋆ v V R,ε δ (x, v) (
where the the mollification now only holds in v). Then, by eq. ( 4.26) we have that

|ξ 2,ε ⋆ v ( D x V R,ε δ (x, •), • )(v) -D x φ ε,δ R (x, v), v | ≤ ε D x V R,ε δ L ∞ (Bε(x,v)) ≤ C δ ε(1 + min{|v| α , R}), which implies that δφ ε,δ R (x, v) + 1 2 |D v φ ε,δ R (x, v)| 2 + D x φ ε,δ R (x, v), v ≤ δφ ε,δ R (x, v) + 1 2 |D v φ ε,δ R (x, v)| 2 + ξ 2,ε ⋆ v D x V R,ε δ (x, v), v + C δ ε(1 + min{|v| α , R}) ≤ F R ⋆ ξ 2,ε (x, v) + C δ ε(1 + min{|v| α , R}) ≤ F R (x, v) + C 1,δ ε(1 + min{|v| α , R})
where the last inequality holds true by assumption (F3). Thus, so far we have proved that for any

(x, v) ∈ T d × R d δφ ε,δ R (x, v) + 1 2 |D v φ ε,δ R (x, v)| 2 + D x φ ε,δ R (x, v), v ≤ F R (x, v) + C 1,δ ε(1 + min{|v| α , R}). (4.29)
Moreover, in view of eq. (4.25) we deduce that there exists a constant C 2,δ ≥ 0 such that for any

(x, v) ∈ T d × R d we have that δφ ε,δ R (x, v) ≥ C -1 2,δ min{|v| α , R} -C 2,δ . (4.30)
We claim that for ε > 0 small enough, the following holds:

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F R (x, v) dµ(x, v, w) ≥ inf (x,v)∈T d ×R d δφ ε,δ R (x, v) -C 1,δ ε 1 + min{|v| α , R} .
inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F R (x, v) µ(dx, dv, dw) = inf µ∈P α,2 (T d ×R d ×R d ) sup ψ∈E(T d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F R (x, v) -D x ψ(x, v), v -D v ψ(x, v), w µ(dx, dv, dw) ≥ sup ψ∈E(T d ×R d ) inf µ∈P α,2 (T d ×R d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F R (x, v) -D x ψ(x, v), v -D v ψ(x, v), w µ(dx, dv, dw) ≥ inf µ∈P α,2 (T d ×R d ×R d ) T d ×R d ×R d 1 2 |w| 2 + F R (x, v) -D x φ ε,δ R (x, v), v -D v φ ε,δ R (x, v), w µ(dx, dv, dw) = inf (x,v)∈T d ×R d - 1 2 |D v φ ε,δ R (x, v)| 2 + F R (x, v) -D x φ ε,δ R (x, v), v ,
which proves eq. (4.31) thanks to eq. (4.29). Recalling (4.30), the right hand side of eq. (4.31) is coercive in v uniformly in ε for ε small. As in addition φ ε,δ R converges locally uniformly to V R δ as ε → 0, we obtain

lim ε→0 inf (x,v)∈T d ×R d δφ ε,δ R (x, v) -C 2,δ ε 1 + min{|v| α , R} = inf (x,v)∈T d ×R d δV R δ (x, v).
So we can let ε → 0 in (4.31) to obtain the result.

In the next step, we let R → +∞ in (4.27):

Lemma 4.21. Assume that F satisfies (F1) -(F3). Let V δ be defined in (4.12). Then

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw) ≥ inf (x,v)∈T d ×R d δV δ (x, v). (4.32)
Proof. We first consider the left-hand side of (4.27), for which we obviously have, by the definition of F R in Lemma 4.18,

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F R (x, v) µ(dx, dv, dw) ≤ inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw). (4.33)
As for the right hand side of eq. (4.27), we note that, if

(x R , v R ) ∈ T d × R d satisfies V R δ (x R , v R ) ≤ inf (x,v)∈T d ×R d V R δ (x, v) + R -1 ,
then, as V R δ ≤ V δ and (4.25) holds, we have

c1,δ 1 + min{|v R | α , R} -c2,δ ≤ inf (x,v)∈T d ×R d V δ (x, v) + R -1 .
This proves that v R remains bounded in R and we can find a subsequence of (x R , v R ), denoted in the same way, which converges to some We are now ready to prove the main result of this section.

(x, v) ∈ T d × R d as R → +∞. Then by local uniform convergence of V R δ to V δ , we obtain that inf (x,v)∈T d ×R d V δ (x, v) ≤ V δ (x, v) = lim R→+∞ V R δ (x R , v R ) = lim R→+∞ inf (x,v)∈T d ×R d V R δ (x, v).

Proposition 4.22 (Characterization with closed measures). Assume that F satisfies (F1) -(F3). For any

(x 0 , v 0 ) ∈ T d × R d we have that lim T →∞ 1 T V T (0, x 0 , v 0 ) = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw).
Proof. Let γ T (x 0 ,v 0 ) be a minimum for the problem

inf γ∈Γ 0 (x 0 ,v 0 ) J T (γ).
Let us define the probability measures µ T by

T d ×R d ×R d φ(x, v, w) dµ T (x, v, w) = 1 T T 0 φ(γ T (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t), γT (x 0 ,v 0 ) (t)) dt for any φ ∈ C ∞ c (T d × R d × R d )
. By Lemma 4.14, the (µ T ) converge, up to a subsequence (T n ), weak- * to a closed measure µ * . Therefore

lim T →∞ inf γ∈Γ 0 (x 0 ,v 0 ) 1 T J T (γ) = lim n→∞ 1 T n Tn 0 1 2 |γ Tn (x 0 ,v 0 ) (t)| 2 + F (γ Tn (x 0 ,v 0 ) (t), γTn (x 0 ,v 0 ) (t)) dt = lim n→∞ T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ Tn (dx, dv, dw) ≥ T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ * (dx, dv, dw).

74CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

Thus, taking the infimum over the set of closed measures C we obtain that lim

T →∞ inf γ∈Γ 0 (x 0 ,v 0 ) 1 T J T (γ) ≥ inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) dµ(x, v, w).
To obtain the opposite inequality, we note that, by (4.32) (which holds for any δ > 0) and Proposition 4.17, we have

inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v) µ(dx, dv, dw) ≥ lim δ→0 + inf (x,v)∈T d ×R d δV δ (x, v) = λ,
where λ defined in eq. (4.14). Then we can conclude thanks to (4.14).

Proof of Theorem 4.2. The existence of the limit and the fact that it does not depend on (x, v) is the main statement of Proposition 4.13 while the characterization of this limit is given by Proposition 4.22.

Asymptotic behavior of MFG with acceleration

We now turn to MFG problems of acceleration. In order to study the asymptotic behavior of these problems, we first need to describe the expected limit: the ergodic MFG problems of acceleration. The difficulty here is that, as explained in the previous part, we do not expect the existence of a corrector and therefore the ergodic MFG problem cannot be phrased in these terms.

We overcome this issue by using the characterization of the ergodic limit given by Theorem 4.2 in terms of closed measures. This suggests the definition of equilibria for ergodic MFG of acceleration (Definition 4.4). We prove the existence and the uniqueness of a solution in Proposition 4. [START_REF] Cannarsa | Asymptotic analysis for hamiltonjacobi equations associated with sub-riemannian control systems[END_REF]. In order to pass to the limit in the time-dependent MFG system of acceleration, we first need to rephrase the solution of this system in terms of closed measures (more precisely in terms of the so-called T -closed measures, see Definition 4.24). This is the aim of the second part of the section (Theorem 4.25). Thanks to this characterization, we are then able to conclude on the long time average and complete the proof of Definition 4.4.

Ergodic MFG with acceleration

Following Definition 4.1 we recall that

C ⊂ P α,2 (T d × R d × R d ) denotes the set of closed measures, i.e. µ ∈ C if it satisfies for any test function φ ∈ C ∞ c (T d × R d
) the following condition:

T d ×R d ×R d D x φ(x, v), v + D v φ(x, v), w µ(dx, dv, dw) = 0.

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 75

The candidate limit problem that we are going to study is the following fixed point problem: we look for a measure µ ∈ C such that

µ ∈ argmin η∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯µ) η(dx, dv, dw) (4.35)
where π : Proof. Let K be the set of probability measures µ ∈ C such that

T d × R d × R d , defined as π(x, v, w) = (x, v),
T d ×R d ( 1 2 |w| 2 + c -1 F |v| α ) µ(dx, dv, dw) ≤ 2c F ,
where α and c F are given by assumption (F2'). We endow K with the d 1 distance and define, for any µ ∈ K, the set Ψ(µ) as the set of minimizers η ∈ C of the map defined on C η →

T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯µ) η(dx, dv, dw) (4.36)
We also denote by Λ(µ) the value of this minimum. First, we show that the set-valued map Ψ is well-defined from K into K. Indeed, if µ ∈ K and η ∈ C is any minimum of (4.36), we have by assumption (F2') (setting η = δ (x 0 ,0,0) ∈ C for an arbitrary point x 0 ∈ T d ):

T d ×R d ×R d ( 1 2 |w| 2 + c -1 F |v| α -c F ) η(dx, dv, dw) ≤ T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ)) η(dx, dv, dw) ≤ T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ)) η(dx, dv, dw) ≤ c F .
So η belongs to K. Moreover, we observe that a solution of the ergodic MFG problem exists if the set-valued map Ψ has a fixed-point and we prove that this is the case using the Kakutani fixed-point theorem. Since α > 1, by the above considerations, we know that the space K is compact with respect to the d 1 distance. Thus, for any µ ∈ K, the set Ψ(µ) is convex and compact. It remains to check that Ψ has closed graph. Fix a sequence {µ j } j∈N ⊂ K and a sequence {η j } j∈N ⊂ K such that

µ j ⇀ d 1 µ, η j ⇀ d 1 η
, and η j ∈ Ψ(µ j ) ∀j ∈ N.

76CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

Let us show that η ∈ Ψ(µ). Note that η ∈ C. It remains to check that η minimizes (4.36). By standard lower-semi continuity arguments and continuity of F , we have:

T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ)) η(dx, dv, dw) ≤ lim inf j T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ j )) η j (dx, dv, dw).
(4.37)

We now check that the right-hand side is not larger that Λ(µ). Indeed, let η belong to Ψ(µ) and fix η > 0. As η belongs to K we can find R > 0 such that

(T d ×R d ×R d )\B R ( 1 2 |w| 2 + c F |v| α + c F ) η(dx, dv, dw) ≤ ε .
As π♯µ j converges to π♯µ for the d 1 distance, we have by assumption (F1') that, for j large enough,

lim j→+∞ sup (x,v)∈B R |F (x, v, π♯µ j ) -F (x, v, π♯µ)| ≤ ε .
So, by optimality of η j and the estimates above,

T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ j )) η j (dx, dv, dw) = Λ(µ j ) ≤ T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ j )) η(dx, dv, dw) ≤ B R ( 1 2 |w| 2 + F (x, v, π♯µ j )) η(dx, dv, dw) + B c R ( 1 2 |w| 2 + c F |v| α + c F ) η(dx, dv, dw) ≤ B R ( 1 2 |w| 2 + F (x, v, π♯µ)) η(dx, dv, dw) + 2 ε ≤ Λ(µ) + 2 ε .
Coming back to (4.37), this shows that

T d ×R d ×R d ( 1 2 |w| 2 + F (x, v, π♯µ)) η(dx, dv, dw) ≤ Λ(µ),
and therefore that η belongs to Ψ(µ). Therefore, applying Kakutani fixedpoint theorem we have that there exists a fixed point η of Ψ and this is a solution of the ergodic MFG problem. Now, we prove that under the monotonicity assumption eq. (4.6) the critical value is unique. Let ( λ1 , μ1 ) and ( λ2 , μ2 ) be two solutions of the

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 77 ergodic MFG problem. Then, by definition we have that, for

i = 1 or i = 2, λi = inf µ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ i ) µ(dx, dv, dw) = T d ×R d ×R d 1 2 |w| 2 + F (x, v, π♯μ i ) μi (dx, dv, dw). (4.38)
Thus, exchanging the role of μ1 and μ2 as competitor for λ1 and λ2 , respectively, we get We first take the difference between eq. (4.39) and eq. (4.38) for i = 2 and we get

λ1 ≤ T d ×R d ×R d 1 2 |w| 2 + F (x, v,
λ1 -λ2 ≤ T d ×R d ×R d F (x, v, π♯μ 1 ) -F (x, v, π♯μ 2 ) dμ 2 (dx, dv, dw).
Taking the difference between eq. (4.39) for i = 1 and eq. ( 4.40) we get

λ1 -λ2 ≥ T d ×R d ×R d F (x, v, π♯μ 1 ) -F (x, v, π♯μ 2 ) dμ 1 (dx, dv, dw).
Thus, taking the difference of the above expressions we deduce that

0 ≥ T d ×R d ×R d F (x, v, π♯μ 1 ) -F (x, v, π♯μ 2 ) (μ 1 (dx, dv, dw)-μ 2 (dx, dv, dw))
which implies by monotonicity assumption eq. (4.6) that F (x, v, π♯μ 1 ) = F (x, v, π♯μ 2 ). Coming back to (4.39), it follows that λ1 = λ2 .

Representation of the solution of the time-dependent MFG system

We now consider the time-dependent MFG system (4.4). We have shown in Chapter 3 that such system has a solution (u T , m T ) and that the function u T can be represented as

u T (t, x, v) = inf γ∈Γt(x,v) T t 1 2 |γ(s)| 2 + F (γ(s), γ(s), m T s ) ds + g(γ(T ), γ(T ), m T T ) .
∈ P 1 (T d × R d ). If η ∈ C([0, T ]; P 1 (T d × R d × R d )), we say that η is a T -closed measure associated with m 0 if for any test function φ ∈ C ∞ c ([0, T ] × T d × R d ) the following holds T 0 T d ×R d ×R d ∂ t φ(t, x, v) + D x φ(t, x, v), v + D v φ(t, x, v), w η t (dx, dv, dw)dt = T d ×R d ×R d φ(T, x, v) η T (dx, dv, dw) - T d ×R d ×R d φ(0, x, v) m 0 (dx, dv). (4.42)
We denote by C T (m 0 ) the set of T -closed measures associated with

m 0 ∈ P 1 (T d × R d ).
The goal of the subsection is to prove the following equality:

Theorem 4.25. Assume that F satisfies (F1'), (F2') and g satisfies (G1).

Let M ≥ 0 and assume that

T d ×R d |v| α m 0 (dx, dv) ≤ M. (4.43)
Let (u T , m T ) be a solution to (4.4). Then

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) = T d ×R d u T (0, x, v) m 0 (dx, dv).
(4.44)

In addition, there exists a minimizer μT ∈ C T (m 0 ) of the problem in the left-hand side of (4.44) such that m T t = π♯μ T t , where π :

T d × R d × R d → T d × R d is the canonical projection on the two first coordinates, i.e. such that π(x, v, w) = (x, v).
The proof of Theorem 4.25 follows standard arguments but is slightly technical because the problem is stated in the whole space in velocity. The main problem is to regularize the map u T in order to have a smooth function with a compact support which satisfies a suitable (approximate) Hamilton-Jacobi inequality. The first step towards this aim is the following Lemma:

ξ R ≥ 0, ξ R (x, v) = 1 if (x, v) ∈ T d ×B R , 0 ≤ ξ R (x, v) ≤ 1 if (x, v) ∈ T d ×B 2R \B R and ξ R (x, v) = 0 if (x, v) ∈ T d ×B c R . Define f R : T d ×R d ×[0, T ] → R and g R : T d ×R d → R as f R = ξ R f and g R = ξ R g. Let u T
R be the viscosity solution of the following problem

     -∂ t u T R (t, x, v) + 1 2 |D v u T R (t, x, v)| 2 -D x u T R (t, x, v), v = f R (t, x, v), in [0, T ] × T d × R d u T (T, x, v) = g R (x, v), in T d × R d . (4.45)
Then, the following hold:

1. u T R has compact support;

2. u T R is Lipschitz continuous in space and velocity variable;

3. u T R converge, as R → +∞, locally uniformly to the solution u T of the following problem

     -∂ t u T (t, x, v) + 1 2 |D v u T (t, x, v)| 2 -D x u T (t, x, v), v = f (t, x, v), in [0, T ] × T d × R d u T (T, x, v) = g(x, v), in T d × R d .
The proof of the Lemma follows standard argument in optimal control and we omit it. Next we prove Theorem 4.25 in the simpler case where F and g are replaced by F R and g R : Proposition 4.27. Assume that F satisfies (F1') and (F2') and g satisfies (G1). Let (u T , m T ) be a solution of system eq. (4.4). For R > 0, let ξ R be a smooth cut-off function as in Lemma 4. [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]

and let us set

F R = ξ R F and g R = ξ R g. Let u T R be

the continuous viscosity solution of the following problem

     -∂ t u T R (t, x, v) + 1 2 |D v u T R (t, x, v)| 2 -D x u T R (t, x, v), v = F R (x, v, m T t ), in [0, T ] × T d × R d u R (T, x, v) = g R (x, v, m T T ), in T d × R d . (4.46) 80CHAPTER 4

. ERGODIC BEHAVIOR OF ACCELERATION MODEL

Then

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) = T d ×R d u T R (0, x, v) m 0 (dx, dv).
Proof. We first prove that

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) ≥ T d ×R d u T R (0, x, v) m 0 (dx, dv). (4.47)
We have that

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) = inf µ∈C([0,T ];P 1 (T d ×R d ×R d )) sup φ∈C ∞ c ([0,T ]×T d ×R d ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) + ∂ t φ(t, x, v) + D x φ(t, x, v), v + D v φ(t, x, v), w µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) -φ(T, x, v) µ T (dx, dv, dw) + T d ×R d φ(0, x, v) m 0 (dx, dv) ≥ sup φ∈C ∞ c ([0,T ]×T d ×R d ) inf µ∈C([0,T ];P 1 (T d ×R d ×R d )) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) + ∂ t φ(t, x, v) + D x φ(t, x, v), v + D v φ(t, x, v), w µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) -φ(T, x, v) µ T (dx, dv, dw) + T d ×R d φ(0, x, v) m 0 (dx, dv).
In the argument below, the constant c R depends on R and on the data and may change from line to line. Let ξ 1,ε = ξ 1,ε (x) be a smooth mollifier such that spt(ξ 1,ε ) ⊂ B ε , ξ 1,ε (x) ≥ 0 and Bε ξ 1,ε (x) dx = 1, and define

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 81

u ε,R 1 = u T R ⋆ x ξ 1,ε (t, x, v) (the convolution being in the x variable only). Let R ′ ≥ R be such that spt(u T R ), spt(F R )
and spt(g R ) are contained in B R ′ . Then, we have that u ε,R 1 satisfies the following inequality in the viscosity sense

-∂ t u ε,R 1 (t, x, v) + 1 2 |D v u ε,R 1 (t, x, v)| 2 -D x u ε,R 1 (t, x, v), v ≤ F R ⋆ ξ 1,ε (t, x, v) ≤ F R (x, v, m T t ) + C F ε(1 + |v| α )1 (x,v)∈T d ×B R ′ . Now, let ξ 2,ε = ξ 2,ε (v) be a smooth mollifier such that spt(ξ 2,ε ) ⊂ B ε , ξ 2,ε (v) ≥ 0 and Bε ξ 2,ε (v) dv = 1 and define u R,ε 2 = ξ 2,ε ⋆ v u R,ε 1 (t,
x, v) (the convolution being now in the v variable only). Then, by the Lipschitz regularity of u T R stated in Lemma 4.26 we have that

|ξ 2,ε ⋆ v D x u R,ε 1 (t, x, •), • (v)-D x u R,ε 2 (t, x, v), v | ≤ ε D x u R,ε 1 ∞ ≤ c R ε 1 (x,v)∈T d ×B R ′ .
Hence u ε,R 2 satisfies in the viscosity sense:

-∂ t u ε,R 2 (t, x, v) + 1 2 |D v u ε,R 2 (t, x, v)| 2 -D x u ε,R 2 (t, x, v), v ≤ F R (x, v, m T t ) + c R ε 1 (x,v)∈T d ×B R ′ .
We finally regularize u ε,R 2 in time. Let ξ 3,ε = ξ 3,ε (t) be a smooth mollifier such that spt(ξ 2,ε ) ⊂ B ε , ξ 2,ε (t) ≥ 0 and Bε ξ 2,ε (t) dt = 1 and define u R,ε

3 = ξ 3,ε ⋆ t u R,ε 2 (t, x, v) (convolution in time). Thus, u R,ε 3 , for any (t, x, v) ∈ (-∞, T -ε] × T d × R d , satisfies (in the classical sense) -∂ t u R,ε 3 (t, x, v) + 1 2 |D v u R,ε 3 (t, x, v)| 2 -D x u R,ε 3 (t, x, v), v ≤ ξ 3,ε ⋆ t F R (x, v, m T • )(t) + c R ε 1 (x,v)∈T d ×B R ′ .
By Theorem 3.25 in Chapter 3 we know that m T is Lipschitz continuous in time with respect to the

d 1 distance. Setting ûR ε (t, x, v) = u R,ε 3 (t -ε, x, v), ûR ε satisfies therefore -∂ t ûR ε (t, x, v) + 1 2 |D v ûR ε (t, x, v)| 2 -D x ûR ε (t, x, v), v ≤ F R (x, v, m T t ) + c R ε 1 (x,v)∈T d ×B R ′ . (4.48)
We note that ûR ε is smooth and has a compact support and converges uni-82CHAPTER

ERGODIC BEHAVIOR OF ACCELERATION MODEL

formly to u R as ε → 0. Using ûR ε as test function we get

sup φ∈C ∞ c ([0,T ]×T d ×R d ) inf µ∈C([0,T ];P 1 (T d ×R d ×R d )) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) + ∂ t φ(t, x, v) + D x φ(t, x, v), v + D v φ(t, x, v), w µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) -φ(T, x, v) µ T (dx, dv, dw) + T d ×R d φ(0, x, v) m 0 (dx, dv) ≥ inf µ∈C([0,T ];P 1 (T d ×R d ×R d )) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) + ∂ t ûR ε (t, x, v) + D x ûR ε (t, x, v), v + D v ûR ε (t, x, v), w µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) -ûR ε (T, x, v) µ T (dx, dv, dw) + T d ×R d ûR ε (0, x, v) m 0 (dx, dv) = inf (t,x,v)∈[0,T ]×T d ×R d 1 2 |D v ûR ε (t, x, v)| 2 + F R (x, v, m T t ) -∂ t ûR ε (t, x, v) + D x ûR ε (t, x, v), v + g R (x, v, m T T ) -ûR ε (T, x, v) + T d ×R d ûR ε (0, x, v) m 0 (dx, dv).
By eq. (4.48) we obtain that inf

(t,x,v)∈[0,T ]×T d ×R d 1 2 |D v ûR ε (t, x, v)| 2 + F R (x, v, m T t ) + ∂ t ûR ε (t, x, v) + D x ûR ε (t, x, v), v + g R (x, v, m T T ) -ûR ε (T, x, v) + T d ×R d ûR ε (0, x, v) m 0 (dx, dv) ≥ -c R ε + inf (x,v)∈T d ×R d g R (x, v, m T T ) -ûR ε (T, x, v) + T d ×R d ûR ε (0, x, v) m 0 (dx, dv).
As ε → 0 + we obtain eq. (4.47).

On the other hand, since u T R is a continuous viscosity solution we know that it can be represented as follows:

u T R (0, x, v) = inf γ∈Γ 0 (x,v) T 0 1 2 |γ(t)| 2 + F R (γ(t), γ(t), m T t ) dt + g R (γ(T ), γ(T ), m T T ) . (4.49) We define the measure ν ∈ C([0, T ]; P 1 (T d × R d × R d )) as T d ×R d ×R d φ(x, v, w) ν t (dx, dv, dw) = T d ×R d ×R d φ(γ (x,v) (t), γ(x,v) (t), γ(x,v) (t)) m 0 (dx, dv), for any φ ∈ C ∞ c (T d ×R d ×R d
) and any t ∈ [0, T ], where γ (x,v) is a measurable selection of minimizers of problem eq. (4.49), see Lemma 4.28. By the

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 83

regularity of the minimizers it is not difficult to prove that ν ∈ C T (m 0 ). Moreover, integrating the equality

u T R (0, x, v) = T 0 1 2 |γ (x,v) (t)| 2 + F R (γ (x,v) (t), γ(x,v) (t), m T t ) dt + g R (γ (x,v) (T ), γ(x,v) (T ), m T T )
against the measure m 0 we deduce that

T d ×R d u T R (0, x, v) m 0 (dx, dv) = T d ×R d T 0 1 2 |γ (x,v) (t)| 2 + F R (γ (x,v) (t), γ(x,v) (t), m T t ) dt m 0 (dx, dv) + T d ×R d g R (γ (x,v) (T ), γ(x,v) (T ), m T T ) m 0 (dx, dv) = T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) ν t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) ν T (dx, dv, dw) ≥ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw).
This completes the proof.

Proof of Theorem 4.25. Using the notation of Proposition 4.27 we know that for any R ≥ 0

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) = T d ×R d u T R (0, x, v) m 0 (dx, dv).
Then, on the one hand it is easy to see, by standard optimal control arguments, that for any

(x, v) ∈ T d ×R d we have that |u T R (0, x, v)| ≤ C1 (1+|v| α ). By Dominated Convergence Theorem we get lim R→+∞ T d ×R d u T R (0, x, v) m 0 (dx, dv) = T d ×R d u T (0, x, v) m 0 (dx, dv).

84CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL

On the other hand, without loss of generality we can define a cut-off function ξ R as in Proposition 4.27 such that F R and g R are non-decreasing in R. Thus

lim sup R→+∞ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) ≤ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw).
To prove the reverse inequality, let {R j } j∈N and {µ j t } j∈N ⊂ C T (m 0 ) be such that

lim inf R→+∞ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R (x, v, m T T ) µ T (dx, dv, dw) = lim j→+∞ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R j (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R j (x, v, m T T ) µ T (dx, dv, dw) = lim j→+∞ T 0 T d ×R d ×R d 1 2 |w| 2 + F R j (x, v, m T t ) µ j t (dx, dv, dw)dt + T d ×R d ×R d g R j (x, v, m T T ) µ j T (dx, dv, dw).
We claim that {µ j t } j∈N is tight. Indeed, the lower bound on F and g, there exists a constant C ≥ 0 such that

sup j T 0 T d ×R d ×R d |w| 2 µ j t (dx, dv, dw)dt ≤ C (4.50)
and thus it is enough to prove that the moment with respect to v is also bounded. In order to prove this bound, let

ψ ∈ C ∞ c (R d ) with ψ(0) = 0 and such that |Dψ(p)| ≤ 1. For φ(t, x, v) = (T -t)ψ(v),
T 0 T d ×R d ×R d -ψ(v) + (T -t) Dψ(v), w µ j t (dx, dv, dw)dt = -T T d ×R d ψ(v) m 0 (dx, dv) (4.51)

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 85

and by eq. (4.50) and Cauchy-Schwarz inequality we get

T 0 T d ×R d ×R d (T -t) Dψ(v), w µ j t (dx, dv, dw)dt ≤ T C 1/2 .
Thus, by eq. ( 4.51) we obtain that

T d ×R d ×R d ψ(v) µ j t (dx, dv, dw)dt ≤ C,
for some new constant C. If we choose ψ n such that ψ n (v) increases in n and converges to |v|, we get therefore

T 0 T d ×R d ×R d |v| µ j t (dx, dv, dw)dt ≤ C.
This implies that {µ j t } j∈N is tight and, up to a subsequence still denoted by µ j t , converges to some μ ∈ C T (m 0 ). Then, we have that

inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) ≤ T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) μt (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) μT (dx, dv, dw) ≤ lim j→+∞ T 0 T d ×R d ×R d 1 2 |w| 2 + F R j (x, v, m T t ) µ j t (dx, dv, dw)dt + T d ×R d ×R d g R j (x, v, m T T ) µ j T (dx, dv, dw) = lim inf j→+∞ inf µ∈C T (m 0 ) T 0 T d ×R d ×R d 1 2 |w| 2 + F R j (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g R j (x, v, m T T ) µ T (dx, dv, dw).
This completes the proof of equality (4.44). It remain to check the existence of a minimizer μT ∈ C T (m 0 ) of the problem in the left-hand side such that m T t = π♯μ T t . For this, let γ (x,v) denote the measurable selection of minimizers of u T (0, x, v) in (4.41) as in Lemma 4.28 below and define the measure

μT t = (x, v) → (γ (x,v) (t), γ(x,v) (t), D v u T (t, γ (x,v) (t), γ(x,v) (t))) ♯m 0 86CHAPTER 4.

ERGODIC BEHAVIOR OF ACCELERATION MODEL

for any t ∈ [0, T ]. Note that by [1, Lemma 3.5] μT t is well-defined since u(t, x, •) is differentiable along the optimal trajectory γ (x,v) with

γ(x,v) (t) = D v u T (t, γ (x,v) (t), γ(x,v) (t)), t ∈ [0, T ]
In particular, it is easy to see that μT ∈ C T (m 0 ) and moreover, by [1, Proposition 4.2] we have that m T t = π♯μ T t since m T t = ((x, v) → (γ (x,v) (t), γ(x,v) (t)))♯m 0 . By the representation formula of the value function we have that

u T (0, x, v) = T 0 1 2 |γ (x,v) (t)| 2 + F (γ (x,v) (t), γ(x,v) (t), m T t ) dt + g(γ (x,v) (T ), γ(x,v) (T ), m T T ) = T 0 1 2 |D v u T (t, γ (x,v) (t), γ(x,v) (t))| 2 + F (γ (x,v) (t), γ(x,v) (t), m T t ) dt + g(γ (x,v) (T ), γ(x,v) (T ), m T T ).
Integrating both side against the measure m 0 and using the definition of μT , we obtain that μT satisfies the equality in (4.44) and therefore is optimal.

Lemma 4.28. Assume that F satisfies (F1') and (F2') and g satisfies (G1).

For (x, v) ∈ T d × R d let Γ * (x, v) ⊂ Γ 0 (x, v
) be the set of minimizers of problem eq. (4.41) for t = 0. Then, the set-valued map

Γ * : (T d × R d , | • |) ⇒ (Γ, • ∞ ), (x, v) → Γ * (x, v)
has a measurable selection γ (x,v) , i.e. (x, v) → γ (x,v) is measurable and, for any

(x, v) ∈ T d × R d , γ (x,v) ∈ Γ * (x, v).
Proof. By using classical results from optimal control theory it is not difficult to see that Γ * has a closed graph, see for instance Lemma 3.9 in Chapter 3. Therefore, by [21, Proposition 9.5] the set-valued map (x, v) ⇒ Γ * (x, v) is measurable with closed values. This implies by [26, Theorem A 5.2] the existence of a measurable selection γ (x,v) ∈ Γ * (x, v).

Convergence of the solution of the time dependent MFG system

We now investigate the limit as the horizon T → +∞ of the time-dependent MFG problem. The main result of this subsection is the following proposition: Then lim

T →+∞ 1 T T d ×R d u T (0, x, v) m 0 (dx, dv) = λ.
Throughout the section, we assume that the assumption of Proposition 4.29 are in force. The proof of the proposition-given at the end of the subsection-is made at the level of the closed and T -closed measures. For this we first need to discuss how to manipulate them. The first lemma is a straightforward application of the definition of T -closed measures:

Lemma 4.30 (Concatenation of T -closed measure). Let T, T ′ > 0, m 0 ∈ P 2 (T d × R d ), µ 1 ∈ C T (m 0 ) and µ 2 ∈ C T ′ (m 1 ) with m 1 = π♯µ 1 (T ).
Then, the measure

µ t := µ 1 (t), t ∈ [0, T ] µ 2 (t -T ), t ∈ (T, T + T ′ ] belongs to C T +T ′ (m 0 ).
Next we explain how to link two measures by a T -closed measure:

∈ C T =1 (m 1 0 ) such that m 2 0 = π♯µ m 1 0 →m 2 0 1 and 1 0 T d ×R 2d ( 1 2 |w| 2 + c F (1 + |v| 2 )) µ m 1 0 →m 2 0 t (dx, dv, dw)dt ≤ C 2 (1 + M 2 (m 1 0 ) + M 2 (m 2 0 )), (4.52)
where

M 2 (m) = T d ×R d |v| 2 dm(x, v) (for m ∈ P 2 (T d × R d ))
and where C 2 depends only on α and c F .

Proof. Let (x 0 , v 0 ) ∈ spt(m 1 0 ) and let (x, v) ∈ spt(m 2 0 ). Then, following the proof of Lemma 4.8, there exists a curve σ

(x,v) (x 0 ,v 0 ) : [0, 1] → T d such that σ (x,v) (x 0 ,v 0 ) (0) = x 0 , σ(x,v) (x 0 ,v 0 ) (0) = v 0 and σ (x,v) (x 0 ,v 0 ) (1) = y, σ(x,v) (x 0 ,v 0 ) (1) = w with 1 0 ( 1 2 |σ (x,v) (x 0 ,v 0 ) (t)| 2 + c F (1 + | σ(x,v) (x 0 ,v 0 ) (t)| 2 ))dt ≤ C 2 (1 + |v| 2 + |v 0 | 2 ). (4.53)
Moreover, by construction, σ depends continuously on (x 0 , v 0 , x, v). Let λ ∈ Π(m 1 0 , m 2 0 ) be a transport plan between m 1 0 and m 2 0 (see (2.2)). We define the measure µ m 1 0 →m 2 0 ∈ C 1 (m 1 0 ) by

T d ×R d ×R d φ(x, v, w)µ m 1 0 →m 2 0 t (dx, dv, dw) = (T d ×R d ) 2 φ(σ (x,v) (x 0 ,v 0 ) (t), σ(x,v) (x 0 ,v 0 ) (t), σ(x,v) (x 0 ,v 0 ) (t)) λ(dx 0 , dv 0 , dx, dv) 88CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL for any φ ∈ C ∞ c (T d × R d × R d ).
Then, on easily checks that m 2 0 = π♯µ

m 1 0 →m 2 0 1
and that, by (4.53):

1 0 T d ×R 2d ( 1 2 |w| 2 + c F (1 + |v| 2 )) µ m 1 0 →m 2 0 t (dx, dv, dw)dt ≤ C 2 1 0 T d ×R 2d (1 + |v| 2 + |v 0 | 2 ) µ m 1 0 →m 2 0 t (dx, dv, dw)dt = C 2 (1 + M 2 (m 1 0 ) + M 2 (m 2 0 )).
Proposition 4.32 (Energy estimate). Under the notation and assumption of Proposition 4.29, there exists a constant C ≥ 0 (independent of T ) such that

T 0 sup (x,v)∈T d ×R d |F (x, v, m T t ) -F (x, v, m)| 2d+2 (1 + |v| 2 ) 2d dt ≤ CT 1 2 , ( 4.54)
where m = π♯μ, with π(x, v, w) = (x, v).

Proof. The proof consists in building from μ and µ T competitors in problems (4.35) and (4.44) respectively. Let us recall that µ T and μ are minimizers for these respective problems.

We start with problem (4.44). Fix T ≥ 2. We define the measure μT by

μT t = µ m 0 → m t , t ∈ [0, 1] μ, t ∈ (1, T ], (4.55)
where µ m 0 → m is the measure defined by Lemma 4.31. We know by Lemma 4.30 that μT belongs to C T (m 0 ). So we can use μT as a competitor in problem (4.44) to get

T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) ≤ 1 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ m 0 → m t (dx, dv, dw)dt + T 1 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) μ(dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) μ(dx, dv, dw).
(4.56)

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 89

Next we build from µ T a competitor for the minimization problem eq. (4.35) for which μ is a minimizer. In view of [1, Proposition 4.2] there exists a Borel measurable maps (x, v) → γ (x,v) such that, for each

(x, v) ∈ T d × R d , γ (x,v)
is a minimizer for u T (0, x, v) in (4.44) and satisfies

T 0 T d ×R 2d φ(x, v, w) µ T t (dx, dv, dw)dt = T d ×R d T 0 φ(γ (x,v) (t), γ(x,v) (t), γ(x,v) (t))dtm 0 (dx, dv) (4.57)
for any test function φ ∈ C 0 b (T d × R 2d ). By Lemma 4.10 and Remark 4.11, for any λ ≥ 2, there exist Borel measurable maps (x, v) → γ(x,v) and

(x, v) → τ (x,v) such that γ(x,v) (0) = γ(x,v) (T ) = x, γ(x,v) (0) = γ(x,v) (T ) = v and γ(x,v) = γ (x,v) on [0, τ (x,v) ] (4.58) and T τ (x,v) ( 1 2 | γ(x,v) (t)| 2 + c F (1 + | γ(x,v) (t)| 2 )dt ≤ C 3 (1 + |v|) 2 (λ 2 + λ -2 T ). (4.59)
Let us define μT by

T d ×R 2d φ(x, v, w) μT (dx, dv, dw) = T -1 T d ×R d T 0 φ(t, γ(x,v) (t), γ(x,v) (t), γ(x,v) (t))dtm 0 (dx, dv) (4.60) for any test function φ ∈ C 0 b (T d × R 2d
). Note that, by (4.58), μT belongs to C. So using the closed measure μT as a competitor in problem eq. ( 4. [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] we deduce that

T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) ≤ T d ×R 2d 1 2 |w| 2 + F (x, v,
T T d ×R 2d 1 2 |w| 2 + F (x, v, m) μT (dx, dv, dw) = T d ×R d T 0 ( 1 2 | γ(x,v) (t)| 2 + F (γ (x,v) (t), γ(x,v) (t), m))dt m 0 (dx, dv) ≤ T d ×R d τ (x,v) 0 ( 1 2 |γ (x,v) (t)| 2 + F (γ (x,v) (t), γ(x,v) (t), m))dt + T τ (x,v) ( 1 2 | γ(x,v) (t)| 2 + c F (1 + | γ(x,v) (t)| 2 ))dt m 0 (dx, dv) ≤ T d ×R d T 0 ( 1 2 |γ (x,v) (t)| 2 + F (γ (x,v) (t), γ(x,v) (t), m))dt + C 3 (1 + |v|) 2 (λ 2 + λ -2 T ) m 0 (dx, dv).
Plugging this inequality into (4.61) and using the representation of µ T in (4.57) then gives

T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) ≤ T d ×R 2d 1 2 |w| 2 + F (x, v, m) μT (dx, dv, dw) ≤ T -1 T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) µ T t (dx, dv, dw)dt + 2C 3 (1 + M 2 (m 0 ))(λ 2 T -1 + λ -2 ), (4.62)
where M 2 (m 0 ) = T d ×R d |v| 2 dm 0 (x, v). Putting together (4.56) and (4.62) (multiplied by T ) then implies that

T 0 T d ×R 2d ( 1 2 |w| 2 + F (x, v, m T t ))dµ T t (x, v, w) + T 0 T d ×R 2d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw)dt ≤ 1 0 T d ×R 2d 1 2 |w| 2 + F (x, v, m T t ) µ m 0 → m t (dx, dv, dw)dt + T 1 T d ×R 2d 1 2 |w| 2 + F (x, v, m T t ) μ(dx, dv, dw)dt + T d ×R 2d g(x, v, m T T ) μ(dx, dv, dw) - T d ×R 2d g(x, v, m T T ) µ T T (dx, dv, dw) + T 0 T d ×R 2d 1 2 |w| 2 + F (x, v, m) µ T t (dx, dv, dw)dt + 2C 3 (1 + M 2 (m 0 ))(λ 2 + λ -α T ).

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 91

Using (4.52) to bound the first term in the right-hand side (note that m belongs to

P α,2 (T d × R d × R d ) with α = 2, so that m ∈ P 2 (T d × R d )) we obtain therefore T 0 T d ×R 2d (F (x, v, m T t ) -F (x, v, m)) (µ T t (dx, dv, dw) -μ(dx, dv, dw))dt ≤ C 2 (1 + M 2 (m 0 ) + M 2 ( m)) + 2 g ∞ + 2C 3 (1 + M 2 (m 0 ))(λ 2 + λ -2 T ).
We now use the strong monotonicity condition eq. (4.6) and choose λ = T 1/4 to get

T 0 T d ×R 2d (F (x, v, m T t ) -F (x, v, m)) 2 dxdvdt ≤ CT 1 2
for a constant C independent of T . Recalling that F satisfies (F3'), we obtain (4.54) by the interpolation inequality Lemma 4.35 in the Appendix.

Proof of Proposition 4.29. Throughout the proof, C denotes a constant independent of T and which may change from line to line. Let µ T ∈ C T (m 0 ) be associated with a solution (u T , m T ) of the MFG system (4.4) as in Theorem 4.25. By Theorem 4.25 we have that

1 T T d ×R d u T (0, x, v) m 0 (dx, dv) = 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) = inf µ∈C T (m 0 ) 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) . (4.63)
We first claim that lim sup

T →+∞ inf µ∈C T (m 0 ) 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) ≤ inf μ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) .
(4.64) 92CHAPTER

ERGODIC BEHAVIOR OF ACCELERATION MODEL

In order to prove the claim, we first note that, by Young's inequality and Proposition 4.32, we have, for any µ ∈ C T (m 0 ),

T 0 T d ×R d ×R d F (x, v, m T t ) -F (x, v, m) µ t (dx, dv, dw)dt ≤ T 0 T d ×R d ×R d sup (x ′ ,v ′ )∈T d ×R d |F (x ′ , v ′ , m T t ) -F (x ′ , v ′ , m)| (1 + |v ′ | 2 ) d d+1 (1 + |v| 2 ) d d+1 µ t (dx, dv, dw)dt ≤ T 1 4 2d + 2 T 0 sup (x,v)∈T d ×R d |F (x, v, m T t ) -F (x, v, m)| 2d+2 (1 + |v| 2 ) 2d dt + (2d + 1)T - 1 4(2d+1) 2d + 2 T 0 T d ×R d ×R d (1 + |v| 2 ) 2d (2d+1) µ t (dx, dv, dw)dt ≤ CT 3 4 + T - 1 4(2d+1) T 0 T d ×R d ×R d (1 + |v| 2 ) µ t (dx, dv, dw)dt. (4.65)
As g is bounded, we have therefore, for any µ ∈ C T (m 0 ),

1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) ≤ 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) µ t (dx, dv, dw)dt + T - 1 4(2d+1) T 0 T d ×R d ×R d (1 + |v| 2 ) µ t (dx, dv, dw)dt + CT -1 4 + T -1 g ∞ . (4.66)
Given μ ∈ C, we know from Lemma 4.31 that there exists µ m 0 →π♯μ such that

1 0 T d ×R 2d ( 1 2 |w| 2 + c F (1 + |v| 2 ))µ m 0 →π♯μ t (dx, dv, dw)dt ≤ C 2 (1 + M 2 (m 0 ) + M 2 (π♯μ)). (4.67)
Let us then define μT by

μT t = µ m 0 →π♯μ t , t ∈ [0, 1] μ, t ∈ (1, T ],

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 93

By Lemma 4.31, μT belongs to C T (m 0 ) and we have, in view of (4.67),

T -1 T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μT t (dx, dv, dw)dt ≤ C 2 T -1 (1 + M 2 (m 0 ) + M 2 (π♯μ)) + T -1 (T -1) T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) while T 0 T d ×R d ×R d (1 + |v| 2 ) μT t (dx, dv, dw)dt ≤ C 2 (1 + M 2 (m 0 ) + M 2 (π♯μ)) + (T -1)M 2 (π♯μ).
Therefore, coming back to (4.66) and using the μT built as above from the μ ∈ C as competitors, we have

inf µ∈C T (m 0 ) 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T (dx, dv, dw) ≤ inf μ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) + CT - 1 4(2d+1) (1 + M 2 (m 0 ) + M 2 (π♯μ)) + CT -1 4 + T -1 g ∞ . (4.68)
Since, by assumption (F2'),

T d ×R d ×R d F (x, v, m) μ(dx, dv, dw) ≥ c -1 F M 2 (π♯μ) -c F ,
one easily checks that the limit of the right-hand side of (4.68) as

T → +∞ is inf μ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) .
This proves our claim (4.64).

Next we claim that there exists a closed measure μ ∈ C such that lim inf

T →+∞ 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) ≥ T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw).
(4.69) 94CHAPTER

ERGODIC BEHAVIOR OF ACCELERATION MODEL

For the proof of (4.69), we work with a subsequence of T → +∞ (still denoted by T ) along which the lower limit in the left-hand side is achieved.

Coming back to (4.65), we have

1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) ≥ 1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) µ T t (dx, dv, dw)dt -T - 1 4(2d+1) T 0 (1 + M 2 (π♯µ T t ))dt -CT -1 4 -g ∞ T -1 .
By the coercivity of F in assumption (F2'), we can absorb the second term in the right-hand side into the first one and obtain:

1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) ≥ 1 T (1 -C -1 T - 1 4(2d+1) ) T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) µ T t (dx, dv, dw)dt -CT - 1 4(2d+1) -CT -1 4 -g ∞ T -1 . (4.70)
As in the proof of Proposition 4.32 (see (4.62)), for any λ ≥ 1, we can find a closed measure μT ∈ C such that

T d ×R 2d 1 2 |w| 2 + F (x, v, m) μT (dx, dv, dw) ≤ T -1 T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) µ T t (dx, dv, dw)dt + 2C 3 (1 + M 2 (m 0 ))(λ 2 T -1 + λ -2 ).
Plugging this inequality into (4.70) we find therefore

1 T T 0 T d ×R d ×R d 1 2 |w| 2 + F (x, v, m T t ) µ T t (dx, dv, dw)dt + T d ×R d ×R d g(x, v, m T T ) µ T T (dx, dv, dw) ≥ (1 -C -1 T - 1 4(2d+1) ) T d ×R 2d 1 2 |w| 2 + F (x, v, m) μT (dx, dv, dw) -2C 3 (1 + M 2 (m 0 ))(λ 2 T -1 + λ -2 ) -CT - 1 4(2d+1) .

ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 95

By assumption (F2'), the functional in the right-hand side of the inequality is coercive for T large enough. So μT weakly- * converges (up to a subsequence) to a closed measure μ. Taking the lower-limit in the last inequality then implies (4.69).

Putting together (4.64) and (4.69), we find that μ is a minimizer in the right-hand side of (4.64) and that the semi-limits and the inequalities in (4.64) and (4.69) are in fact limits and equalities. So coming back to (4.63) we find that lim

T →+∞ 1 T T d ×R d u T (0, x, v) m 0 (dx, dv) = inf μ∈C T d ×R d ×R d 1 2 |w| 2 + F (x, v, m) μ(dx, dv, dw) .
The right-hand side of this equality is nothing than but λ since ( λ, μ) is a solution to the the ergodic MFG problem with m = π♯μ: this completes the proof of the proposition.

To complete the proof of Theorem 4.5, we need estimates on the oscillation of u T . This comes next: Lemma 4.33. For any R ≥ 1 and (x, v),

(x ′ , v ′ ) ∈ T d × B R , we have |u T (0, x, v) -u T (0, x ′ , v ′ )| ≤ CR 2 T 4d+3 4(d+1) ,
where C is independent of T and R.

Proof. Let γ ∈ Γ(x, v) be optimal for u T (0, x, v) in (4.41). We define γ ∈ Γ(x ′ , v ′ ) by

γ(t) = σ(t) if t ∈ [0, 1] γ(t -1) if t ∈ [1, T ].
where σ is as in Lemma 4.8 with σ(0

) = x ′ , σ(0) = v ′ , σ(1) = x, σ(1) = v and 1 0 1 2 |σ(t)| 2 + F (σ(t), σ(t), m T t ) dt ≤ 2C 2 R 2 .
Note that, as the problem for u T depends on time through (m T t ), the cost associated with γ could be quite far from the cost associated with γ. To overcome this issue, we use in a crucial way Proposition 4.32. Indeed, applying 96CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL (4.54) in Proposition 4.32, we have

T 0 F (γ(t), γ(t), m T t ) -F (γ(t), γ(t), m)) dt ≤ T 0 (1 + | γ(t)| 2 ) d d+1 sup (y,z)∈T d ×R d |F (y, z, m T t ) -F (y, z, m)| (1 + |v| 2 ) d d+1 dt ≤ T 0 (1 + | γ(t)| 2 ) 2d 2d+1 dt 2d+1 2d+2 T 0 sup (y,z)∈T d ×R d |F (y, z, m T t ) -F (y, z, m)| 2d+2 (1 + |v| 2 ) 2d dt 1 2d+2 ≤ CT 1 4(d+1) T 0 (1 + | γ(t)| 2 )dt 2d+1 2d+2
.

We have by assumption (F2') and Lemma 4.6 that

T 0 (c -1 F | γ(t)| 2 -c F )dt ≤ u T (0, x, v) ≤ c F T (1 + |v| 2 ). (4.71) Therefore T 0 F (γ(t), γ(t), m T t ) -F (γ(t), γ(t), m)) ≤ CT 4d+3 4(d+1) (1 + R 2 ) 2d+1 2d+2 . (4.72)
For the very same reason we also have

T 1 F (γ(t -1), γ(t -1), m T t ) -F (γ(t -1), γ(t -1), m)) ≤ CT 4d+3 4(d+1) (1 + R 2 ) 2d+1 2d+2 , (4.73)
because we only used the optimality of γ only in the estimate (4.71). So, by (4.72) and (4.73) we obtain

u T (0, x ′ , v ′ ) ≤ T 0 1 2 | γ(t)| 2 + F (γ(t), γ(t), m T t ) dt = 1 0 1 2 |σ(t)| 2 + F (σ(t), σ(t), m T t ) dt + T -1 0 1 2 |γ(t -1)| 2 + F (γ(t -1), γ(t -1), m T t ) dt ≤ 2C 2 R 2 + T 0 1 2 |γ(t)| 2 + F (γ(t), γ(t), m) dt + CT 4d+3 4(d+1) (1 + R 2 ) 2d+1 2d+2 ≤ 2C 2 R 2 + T 0 1 2 |γ(t)| 2 + F (γ(t), γ(t), m T t ) dt + 2CT 4d+3 4(d+1) (1 + R 2 ) 2d+1 2d+2 ≤ u T (0, x, v) + 2C 2 R 2 + 2CT 4d+3 4(d+1) (1 + R 2 ) 2d+1 2d+2 ,
from which the result derives easily.

APPENDIX 97

Proof of Theorem 4.5. Proposition 4.23 states the existence of a solution for the ergodic MFG system and its uniqueness under assumption (4.6). From Proposition 4.29 we know that lim

T →+∞ 1 T T d ×R d u T (0, x, v)m 0 (dx, dv) = λ.
It remains to prove the local uniform convergence of u T to λ. Fix R > 0 and ε > 0. We have by Lemma 4.6 that

0 ≤ u T (0, x, v) ≤ c F T (1 + |v| 2 ). (4.74) As m 0 ∈ P 2 (T d × R d ), there exists R ′ ≥ R such that T d ×(R d \B R ′ ) (1 + |v| 2 )m 0 (dx, dv) ≤ ε . (4.75)
Then, for any (x 0 , v 0 ) ∈ T d × B R , we have, by Lemma 4.33, (4.74) and (4.75),

| 1 T u T (0, x 0 , v 0 ) -λ | ≤ 1 T T d ×R d u T (0, x, v)m 0 (dx, dv) -λ + 1 T T d ×B R ′ u T (0, x, v) -u T (0, x 0 , v 0 ) m 0 (dx, dv) + 1 T T d ×(R d \B R ′ ) (|u T (0, x, v)| + |u T (0, x 0 , v 0 )|) m 0 (dx, dv) ≤ 1 T T d ×R d u T (0, x, v)m 0 (dx, dv) -λ + CT -1 (R ′ ) 2 T 4d+3 4(d+1) + c F ε(2 + R 2 ),
from which the local uniform convergence of u T (0, •, •)/T to λ can be obtained easily.

Appendix

Von Neumann minmax theorem

Let A, B be convex sets of some vector spaces and let us suppose that B is endowed with some Hausdorff topology. Let L : A × B → R be a saddle function satisfying

1. a → L(a, b) is concave in A for every b ∈ B, 2. b → L(a, b) is convex in B for every a ∈ A.

An interpolation inequality

Lemma 4.35. Assume that f :

T d × R d → R is locally Lipschitz continuous with |f (x, v)|+|D x f (x, v)|+|D v f (x, v)| ≤ c 0 (1+|v| α ) for a.e. (x, v) ∈ T d × R d (4.
76) for some constants c 0 > 0 and α ∈ (1,2]. There exists a constants C d > 0 (depending on dimension only) such that sup

(x,v)∈T d ×R d |f (x, v)| 2d+2 (1 + |v| α ) 2d ≤ C d c 2d 0 T d ×R d |f (x, v)| 2 dxdv. Proof. Let (x 0 , v 0 ) ∈ T d × R d be such that f (x 0 , v 0 ) = 0 and let R = |f (x 0 ,v 0 )| 2c 0 (3+2|v 0 | α )
. Note that, by our assumption on |f | in (4.76), R is less than 1. Then, for any (x, v) ∈ B R (x 0 , v 0 ), we have by assumption (4.76) that

|D x f (x, v)|+|D v f (x, v)| ≤ c 0 (1+(1+|v 0 |) α ) ≤ c 0 (1+2 α-1 +2 α-1 |v 0 | α ) ≤ c 0 (3+2|v 0 | α ),
(where we used the fact that R ≤ 1 and that (a + b) α ≤ 2 α-1 (a α + b α ) in the first inequality and the fact that α ≤ 2 in the second one). Therefore

|f (x, v)| ≥ |f (x 0 , v 0 )| -c 0 (3 + 2|v 0 | α )R = |f (x 0 , v 0 )| 2 .
Taking the square and integrating over B R (x 0 , v 0 ) gives

T d ×R d |f (x, v)| 2 dxdv ≥ |B 1 |R 2d |f (x 0 , v 0 )| 2 4 = |B 1 | |f (x 0 , v 0 )| 2d+2 2 2d+2 c 2d 0 (3 + 2|v 0 | α ) 2d ,
which implies the result.

Chapter 5

Singular limit problem for mean field control of acceleration

Assumptions and main results

In this following, we will use the same notation for similar objects for two problems. However, both the analysis are self contained and there are no intersections that might create ambiguity.

Control of acceleration

We begin with the analysis of the pure control problem of acceleration without mean field interaction. Assume that the Lagrangian L 0 : R 2d → R satisfy the following.

(L1) L 0 ∈ C 1 (R 2d ); (L2) there exists C 0 ≥ 0 such that for any (x, v) ∈ R 2d 1 C 0 |v| 2 -C 0 ≤ L 0 (x, v) ≤ C 0 (1 + |v| 2 ), (5.1
)

|D x L 0 (x, v)| ≤ C 0 1 + |v| 2 , (5.2) |D v L 0 (x, v)| ≤ C 0 1 + |v| , (5.3)
and, without loss of generality, we assume that L 0 (x, v) ≥ 0.

CHAPTER 5. SINGULAR LIMIT OF ACCELERATION

We consider the Hamilton-Jacobi equation

     -∂ t u ε (t, x, v) + 1 2 ε |D v u ε (t, x, v)| 2 -D x u ε (t, x, v), v -L 0 (x, v) = 0, (t, x, v) ∈ [0, T ] × R 2d u ε (T, x, v) = g(x), (x, v) ∈ R 2d . (5.4)
and assume the following on the function g : R d → R.

(TC) g(•) ∈ C 1 b (R d ) such that C 0 ≥ max 1 2 , 1 2 Dg(•) ∞,R d .
Let Γ be the set of C 1 curves γ : [0, T ] → R d , endowed with the local uniform convergence of the curve and its derivative, and given

(t, x, v) ∈ [0, T ] × R 2d let Γ t (x, v) be the subset of Γ such that γ(t) = x, γ(t) = v. Similarly, let Γ t (x) be the subset of Γ such that γ(t) = x. Define the functional J ε t,T : Γ → R J ε t,T (γ) = T t ε 2 |γ(s)| 2 + L 0 (γ(s), γ(s)) ds + g(γ(T )), if γ ∈ H 2 (0, T ; R d ) and set J ε t,T (γ) = +∞ if γ ∈ H 2 (0, T ; R d ).
Then, we know that the solution u ε of (5.4) can be represented as

u ε (t, x, v) = inf γ∈Γt(x,v) J ε t,T (γ), (t, x, v) ∈ [0, T ] × R 2d . (5.5)
Let H 0 : R 2d → R be the Hamiltonian associated with L 0 , i.e.,

H 0 (x, p) = sup v∈R d -p, v -L 0 (x, v) .
Theorem 5.1 (Main result 1). Assume (L1), (L2) and (TC). Let u ε be a solution to (5.4). Then, there exists a sequence {ε k } k∈N with ε k ↓ 0 as k → ∞, and a function

u 0 ∈ W 1,∞ loc ([0, T ] × R d ) such that for any R ≥ 0 lim k→∞ u ε k (t, x, v) = u 0 (t, x), uniformly on [0, T ] × B R × B R .
Moreover, u 0 satisfy

-∂ t u 0 (t, x) + H 0 (x, D x u 0 (t, x)) = 0, (t, x) ∈ [0, T ] × R d u 0 (T, x) = g(x), x ∈ R d
and, consequently, for any (t, x) ∈ [0, T ] × R d we have that

u 0 (t, x) = inf γ∈Γt(x) T t
L 0 (γ(s), γ(s)) ds + g(γ(T )) .

ASSUMPTIONS AND MAIN RESULTS

101

Mean field control of acceleration

We now list the main assumptions on the Lagrangian L 0 : R 2d × P 1 (R d ) → R.

(M1) L 0 is continuous w.r.t. all variables and for any m

∈ P 1 (R d ) the map (x, v) → L 0 (x, v, m) belongs to C 1 (R d ).
(M2) There exists M 0 ≥ 0 such that for any

(x, v, m) ∈ R 2d × P 1 (R d ) 1 M 0 |v| 2 -M 0 ≤ L 0 (x, v, m) ≤ M 0 (1 + |v| 2 ), (5.6
)

|D x L 0 (x, v, m)| ≤ M 0 1 + |v| 2 , ( 5.7
)

|D v L 0 (x, v, m)| ≤ M 0 1 + |v| , (5.8)
and, without loss of generality, L 0 (x, v, m) ≥ 0 for any

(x, v, m) ∈ R 2d × P 1 (R d ).
(M3) There exists two moduli θ : R + → R + and ω 0 : R + → R + such that

|L 0 (x, v, m 1 ) -L 0 (x, v, m 2 )| ≤ θ(|x|)ω 0 (d 1 (m 1 , m 2 )), for any (x, v) ∈ R 2d and m 1 , m 2 ∈ P 1 (R d ).
Let H 0 : R 2d × P 1 (R d ) → R be the Hamiltonian associated with L 0 , i.e.,

H 0 (x, p, m) = sup v∈R d -p, v -L 0 (x, v, m) .
We consider the following MFG system

     -∂ t u ε + 1 2 ε |D v u ε | 2 -D x u ε , v -L 0 (x, v, m ε t ) = 0, (t, x, v) ∈ [0, T ] × R 2d ∂ t µ ε t -D x µ ε t , v -1 ε div v (µ ε t D v u ε ) = 0, (t, x, v) ∈ [0, T ] × R 2d µ ε 0 = µ 0 , u ε (T, x, v) = g(x, m ε T ), (x, v) ∈ R 2d (5.9)
where m ε t = π 1 ♯µ ε t and π 1 : R 2d → R d denotes the projection onto the first factor, i.e., π 1 (x, v) = x. We assume the following on the boundary data of the system:

(BC1) µ 0 ∈ P(R 2d
) is absolutely continuous w.r.t. Lebesgue measure, we still denote by µ 0 its density, and it has compact support.

(BC2) g(•, m) belongs to C 1 b (R d ) such that M 0 ≥ max{ 1 2 , 1 2 Dg(•, m) ∞,R d } and g(x, •) uniformly continuous w.r.t. space.
Let Γ be the set of C 1 curves γ : [0, T ] → R d , endowed with the local uniform convergence of the curve and its derivative, and given

(t, x, v) ∈ [0, T ] × R 2d let Γ t (x, v) be the subset of Γ such that γ(t) = x, γ(t) = v. Similarly, let Γ t (x) be the subset of Γ such that γ(t) = x. Define the functional J ε t,T : Γ → R J ε t,T (γ) = T t ε 2 |γ(s)| 2 + L 0 (γ(s), γ(s), m ε s ) ds + g(γ(T ), m ε T ), if γ ∈ H 2 (0, T ; R d ) and set J ε t,T (γ) = +∞ if γ ∈ H 2 (0, T ; R d ).
Then, from Chapter 3 we know that there exist a solution

(u ε , µ ε ) ∈ W 1,∞ loc ([0, T ] × R 2d ) × C([0, T ]; P 1 (R 2d
)) to system (5.9) such that

u ε (t, x, v) = inf γ∈Γt(x,v) J ε t,T (γ) (5.10)
and for any t ∈ [0, T ] the probability measure µ ε t is the image of µ 0 under the flow

γ(t) = v(t) v(t) = -1 ε D v u ε (t, γ(t), v(t)).
(5.11)

That is, u ε solves the Hamilton-Jacobi equation in the viscosity sense and µ ε solves the continuity equation in the sense of distributions.

Remark 5.2. Note that for a.e. (x, v) ∈ R 2d there exists a unique solution to system (5.11), which we will denote by γ ε (x,v) , such that γ ε (x,v) (0) = x and γε (x,v) (0) = v. Moreover, such a curve γ ε (x,v) (•) is optimal for u ε (t, x, v) satisfying γ ε (x,v) (t) = x and γε (x,v) (t) = v as initial condition.

Theorem 5.3 (Main result 2). Assume (M1) -(M3) and (BC).

Let (u ε , µ ε ) be a solution to (5.9) and let

m ε t = π 1 ♯µ ε t for any t ∈ [0, T ]. That is, there exists a sequence {ε k } k∈N with ε k ↓ 0, as k → ∞, a function u 0 ∈ W 1,∞ loc ([0, T ] × R d ) and a flow of probability measures {m 0 t } t∈[0,T ] ∈ C([0, T ]; P 1 (R d )) such that for any R ≥ 0 lim k→∞ u ε k (t, x, v) = u 0 (t, x), uniformly on [0, T ] × B R × B R and lim k→∞ m ε k t = m 0 t , in C([0, T ]; P 1 (R d )).
Moreover, the following holds.

(i) (u 0 , m 0 ) ∈ W 1,∞ loc ([0, T ] × R d ) × C([0, T ]; P 1 (R d )) is a solution of        -∂ t u 0 (t, x) + H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d ∂ t m 0 t -div m 0 t D p H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d m 0 0 = m 0 , u 0 (T, x) = g(x, m 0 T ), x ∈ R d ,
(5.12)

PROOF OF THE MAIN RESULT

103 that is, u 0 solves the Hamilton-Jacobi equation in the viscosity sense and m 0 is a solution of the continuity equation in the sense of distributions.

(ii) For any t ∈ [0, T ] the probability measure m 0 t is the image of m 0 under the Euler flow associated with L 0 . Remark 5.4. Let (u ε , µ ε ) be a solution to (5.9). Assume that H 0 is of separated form, i.e., there exists a coupling function F :

R d × P 1 (R d ) → R such that H 0 (x, p, m) = H(x, p) -F (x, m), ∀(x, p, m) ∈ R 2d × P 1 (R d ).
Moreover, assume that F is continuous w.r.t. all variables, that the map x → F (x, m) belongs to C 1 b (R d ) and that the functions F , g are monotone in the sense of Lasry-Lions, i.e.

R d F (x, m 1 ) -F (x, m 2 ) (m 1 (dx) -m 2 (dx)) ≥ 0, ∀ m 1 , m 2 ∈ P 1 (R d ) R d g(x, m 1 ) -g(x, m 2 ) (m 1 (dx) -m 2 (dx)) ≥ 0, ∀ m 1 , m 2 ∈ P 1 (R d ).
Then, we know that that there exists a unique solution (5.12) and thus if (u ε , m ε ) is relatively compact then convergence of (u ε , m ε ) holds for the whole sequence.

(u 0 , m 0 ) ∈ W 1,∞ loc ([0, T ]× R d ) × C([0, T ]; P 1 (R d )) of

Proof of the main result

Proof for the control of acceleration

We start our analysis by considering the case of control of acceleration without mean field interaction and first we will show that the value function u ε is locally equibounded and locally equicontinuous.

Lemma 5.5. Assume (L1), (L2) and (TC). Then we have that

-C 0 T -g ∞,R d ≤ u ε (t, x, v)| ≤ C 0 T (1 + |v| 2 ) + g ∞,R d for any (t, x, v) ∈ [0, T ] × R 2d and any ε > 0.
Proof. On the one hand, by (5.5), (5.1) and (TC) we deduce that

u ε (t, x, v) ≥ -C 0 T -g ∞,R d .
On the other hand, the functions

ζ(t, x, v) = g(x) + C(1 + |v| 2 )(T -t), (t, x, v) ∈ [0, T ] × R 2d
is a supersolution to the equation satisfied by u ε for a suitable choice of the real constant C ≥ 0. Indeed, we have that

-∂ t ζ(t, x, v) + 1 2 ε |D v ζ(t, x, v)| 2 -D x ζ(t, x, v), v -L 0 (x, v) ≥ C(1 + |v| 2 ) + 2 (T -t) 2 C 2 ε |v| 2 -D x g(x), v -C 0 (1 + |v| 2 ) ≥ C(1 + |v| 2 ) - 1 2 Dg(•) ∞,R d - 1 2 |v| 2 -C 0 (1 + |v| 2 )
where the last inequality holds by Young's inequality. Thus, taking C = 2C 0 by (TC) we obtain

C 0 (1 + |v| 2 ) - 1 2 Dg(•) ∞,R d - 1 2 |v| 2 ≥ 0. □
An immediate consequence of Lemma 5.5 is the following uniform estimate on the velocity of minimizing trajectories for u ε . Corollary 5.6. Assume (L1), (L2) and (TC). Let (t, x, v) ∈ [0, T ] × R 2d and let γ ε be a minimizer for u ε (t, x, v). Then, there exists a constant S ≥ 0 such that

T t | γε (s)| 2 ds ≤ S(1 + |v| 2 )
where S is independent of ε, t, x and v.

Proof. From Lemma 5.5 we know that

u ε (t, x, v) ≤ C 0 T 1 + |v| 2 + g ∞,R d , ∀ (t, x, v) ∈ [0, T ] × R 2d .
On the other hand, let (t, x, v) ∈ [0, T ] × R 2d and let γ ε be a minimizer for u ε (t, x, v). Then, we have that

u ε (t, x, v) = T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )) ≥ T t L 0 (γ ε (s), γε (s)) ds -g ∞,R d ≥ T t 1 C 0 | γε (s)| 2 -C 0 ds -g ∞,R d .
Therefore, combining the above inequalities we get

T t | γε (s)| 2 ds ≤ 2C 0 g ∞,R d + C 0 T (1 + |v| 2 ) =: S(1 + |v| 2 ). □
We now provide uniform estimates estimates for the gradients of the value function u ε w.r.t. time and space, and we also show that the gradient w.r.t. the velocity variable decrease linearly in ε.

PROOF OF THE MAIN RESULT

105 Proposition 5.7. Assume (L1), (L2) and (TC). Then, there exists a constant C 1 ≥ 0 such that

|∂ t u ε (t, x, v)| ≤ C 1 T (1 + |v| 2 ),
(5.13)

|D x u ε (t, x, v)| ≤ C 1 T (1 + |v| 2 ), (5.14
)

|D v u ε (t, x, v)| 2 ≤ 2 ε C 1 (1 + |v| 2 + |v| 4 ) (5.15) for a.e. (t, x, v) ∈ [0, T ] × R 2d .
Proof. We start by proving (5.13). By similar arguments to the one in Lemma 5.5 we deduce that

u ε (t, •, •) -g(•) ∞,R 2d ≤ C 0 (T -t)(1 + |v| 2 ), ∀ t ∈ (0, T ). (5.16)
Moreover, the functions

f ± : [0, T ] × R 2d × → R defined by f ± (t, x, v) = u ε (t -h, x, v) ± u ε (T -h, •, •) -g(•) ∞,R 2d + h(T -t)(1 + |v| 2 )
are, respectively, supersolution and subsolution to the equation satisfied by u ε . Therefore, by Comparison Theorem [12, Theorem 3.7] we get

|u ε (t, x, v) -u ε (t -h, x, v)| ≤ u ε (T -h, •, •) -g(•) ∞,R d ×B R + h(T -t)
which yields to (5.13).

Next we show (5.14). Let (t, x, v) ∈ [0, T ]×R 2d and let γ ε be a minimizer for u ε (t, x, v). Then, by (5.2) we get

u ε (t, x + h, v) ≤ T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s) + h, γε (s)) ds + g(γ ε (T ) + h) = u ε (t, x, v) + T t L 0 (γ * (s) + h, γ * (s)) -L 0 (γ * (s), γ * (s)) ds + g(γ ε (T ) + h) -g(γ ε (T )) ≤ u ε (t, x, v) + T t C 0 |h|(1 + | γ * (s)| 2 ) ds + Dg(•) ∞ |h|.
Moreover, from (5.1) we obtain

T t -C 0 + 1 C 0 | γ * (s)| 2 ds ≤ u ε (t, x, v) ≤ C 0 T (1 + |v| 2 ) + g ∞,R d
which in turn completes the proof of (5.14). We finally proceed with the proof of (5.15). To do so, we first show the result assuming that u ε belongs to C 1 ([0, T ] × R 2d ) and then we treat the general case with an approximation argument.

Now, assuming that u

ε ∈ C 1 ([0, T ] × R 2d ) we have that -∂ t u ε + 1 2 ε |D v u ε | 2 -D x u ε , v -L 0 (x, v) = 0
in the classical sense. Thus, by (5.1) we obtain

1 2 ε |D v u ε | 2 ≤ |∂ t u ε | + |D x u ε ||v| -C 0 (|v| 2 + 1) ≤ |∂ t u ε | + 1 2 |D x u ε | 2 + 1 2 |v| 2 -C 0 (|v| 2 + 1).
(5.17)

Hence, combining (5.17) with (5.13) and (5.14) we get (5.15). Let us consider now the general case. Take δ > 0 and let ξ 1,δ ∈ C ∞ c ([0, T ]) be a smooth mollifier w.r.t. time. Then define the function

u ε 1,δ (t, x, v) = u ε ⋆ ξ 1,δ (t, x, v), (t, x, v) ∈ [0, T ] × R 2d . Then, u ε
1,δ satisfy the following inequality in the viscosity sense

-∂ t u ε 1,δ + 1 2 ε |D v u ε 1,δ | 2 -D x u ε 1,δ , v ≤ L 0 (x, v). Let ξ 2,δ ∈ C ∞ c (R d
) be a smooth mollifier w.r.t. space and define the function

u ε 2,δ (t, x, v) = u ε 1,δ ⋆ ξ 2,δ (t, x, v), (t, x, v) ∈ [0, T ] × R 2d
. Then, we have that u ε 2 δ satisfy the following inequality in the viscosity sense

-∂ t u ε 2,δ + 1 2 ε |D v u ε 2,δ | 2 -D x u ε 2,δ , v ≤ L 0 ⋆ξ 2,δ (x, v) ≤ L 0 (x, v)+C 0 δ(1+|v| 2 ). Let ξ 3,δ ∈ C ∞ c (R d
) be a smooth mollifier w.r.t. velocity variable and define the function

u ε 3,δ (t, x, v) = u ε 2,δ ⋆ ξ 3,δ (t, x, v), (t, x, v) ∈ [0, T ] × R 2d .
Then, by Jensen's inequality we deduce that u ε 3,δ satisfy

-∂ t u ε 3,δ + 1 2 ε |D v u ε 3,δ | 2 -D x u ε 3,δ , v ≤ L 0 ⋆ ξ 3,δ (x, v) + C ε δ(1 + |v| 2 ) ≤ L 0 (x, v) + C 0,ε δ(1 + |v| 2 ) (5.18)
in the classical sense. Therefore, applying the argument in (5.17) to the function u ε 3,δ , which solves (5.18), we get the result as δ ↓ 0. Now, define the function

u 0 : [0, T ] × R 2d as u 0 (t, x) = inf γ∈Γt(x) T t L 0 (γ(s), γ(s)) ds + g(γ(T )) . (5.19)
Then, by standard arguments in control theory it is easy to prove the following result.

Lemma 5.8. Assume (L1), (L2) and (TC). Let (t, x) ∈ [0, T ] × R d and let γ 0 ∈ Γ t (x) be a minimizer for u 0 (t, x). Then, we have that

T t | γ0 (s)| 2 ds ≤ C T for some constant C T ≥ 0.
We are now in the position to prove Theorem 5.1. To do so, we first show in Proposition 5.9 that u ε locally uniformly converges to u 0 and then in Proposition 5.10 we prove that any minimizers of u ε converges to a minimizer of u 0 at any point of differentiability of u 0 . Proposition 5.9. Assume (L1), (L2) and (TC). Then, there exists a sequence ε k → 0 such that u ε k locally uniformly converges to u 0 . Proof. It is enough to show that u ε converges to u 0 pointwise. Indeed, if this holds then by Lemma 5.5 and by Proposition 5.7 we can apply Ascoli-Arzela Theorem to obtain that there exists {ε k } k∈N such that u ε k converges to u 0 locally uniformly.

Let

R ≥ 0, let (t, x, v) ∈ [0, T ] × R d × B R
and let γ ε be a minimizer for u ε (t, x, v). Then, we have that

u ε (t, x, v) = T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )) ≥ T t L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )) ≥ inf γ∈Γt(x) T t L 0 (γ(s), γ(s)) ds + g(γ(T )) = u 0 (t, x).
On the other hand, for any

R ≥ 0 et (t, x, v) ∈ [0, T ] × R d × B R
and let γ 0 ∈ Γ t (x) be a minimizer for u 0 (t, x). If γ0 (t) = v, by the Euler equation and the regularity of L 0 we have that γ ∈ C 2 ([0, T ]) and thus we can use γ 0 to estimate u ε (t, x, v) from above. So, we get 1).

u ε (t, x, v) ≤ T t ε 2 |γ 0 (s)| 2 + L 0 (γ 0 (s), γ0 (s)) ds + g(γ 0 (T )) ≤ u 0 (t, x) + o(
(5.20)

If this is not the case, we observe that

u ε (t, x, v) = u ε (t, x, v) -u ε (t, x, γ0 (t)) + u ε (t, x, γ0 (t)) ≤ o(1) + u ε (t, x, γ0 (t))
where the last inequality holds by (5.15). Thus, in order to conclude it is enough to estimate u ε (t, x, γ0 (t)) as in (5.20). Therefore, we obtain

u 0 (t, x) ≤ u ε (t, x, v) ≤ u 0 (t, x) + o(1)
which implies that u ε converges to (5.19) pointwise.

CHAPTER 5. SINGULAR LIMIT OF ACCELERATION

Proposition 5.10. Assume (L1), (L2) and (TC). Let (t, x, v) ∈ [0, T ] × R 2d be a point of differentiability for u 0 (t, x) and let γ ε be a minimizer for u ε (t, x, v). Then, γ ε uniformly converges to a curve γ 0 ∈ AC([0, T ]; R d ) and γ 0 minimize u 0 at (t, x).

Proof. Let us start by proving that γ ε uniformly converges, up to a subsequence. By Corollary 5.6 we know that

T t | γε (s)| 2 ds ≤ S(1 + |v| 2 ).
Thus, for any s ∈ [t, T ], by Hölder's inequality we have that

|γ ε (s)| ≤ |x| + √ T √ S(1 + |v| 2 ) 1 2 .
Therefore, γ ε is bounded in H 1 (0, T ; R d ) which implies that by Ascoli-Arzela Theorem there exists a sequence {ε k } k∈N and a curve γ 0 ∈ AC([0, T ]; R d ) such that γ ε k converges uniformly to γ 0 . We proceed now that γ 0 is a minimizer for u 0 (t, x). First, we observe that

lim inf ε→0 T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )) ≥ lim inf ε→0 T t
L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )).

(5.21)

Since γ ε is uniformly bounded in H 1 (0, T ), by lower-semicontinuity of the functional we deduce that

lim inf ε→0 T t L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )) ≥ T t L 0 (γ 0 (s), γ0 (s)) ds + g(γ 0 (T )).
Moreover, given (t, x, v) ∈ [0, T ]×R d ×B R , for any R ≥ 0, by Proposition 5.9 we have that

u ε (t, x, v) ≤ u 0 (t, x) + o(1)
and by definition

u ε (t, x, v) = T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s)) ds + g(γ ε (T )).
Hence, we have that

o(1) + u 0 (t, x) ≥ T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s)) ds + g(γ ε (T ))
which implies the result passing to the limit as ε ↓ 0 in (5.21) since

u 0 (t, x) ≥ T t
L 0 (γ 0 (s), γ0 (s)) ds + g(γ 0 (T )). □

Proof of Theorem 5.1. The result follows by Proposition 5.9 and Proposition 5.10.

Proof for mean field game of acceleration

In order to prove the main result we proceed by steps analyzing the behavior of the value function u ε and of the flow of probability measures m ε separately. First, we show that u ε is equibounded and we prove that, up to a subsequence, m ε converges to a flow of probability measure in C([0, T ];

P 1 (R d )).
Then, we address the convergence of the value function, up to a subsequence, to a solution of a suitable Hamilton-Jacobi equation and we study the limit of its minimizing trajectories. Finally, we are able to characterize the limit flow of measures as solution of a suitable continuity equation which coupled with the Hamilton-Jacobi equation, previously found, define the MFG system (5.12).

Lemma 5.11. Assume (M1) -(M3) and (BC). Then we have that

-T M 0 -g(•, m ε T ) ∞,R d ≤ u ε (t, x, v) ≤ M 0 T (1 + |v| 2 ) + g(•, m ε T ) ∞,R d ,
for any (t, x, v) ∈ [0, T ] × R 2d and for any ε > 0.

Proof. First, since u ε satisfy (5.10), from (5.6) and (BC) follows that for any (t, x, v) ∈ [0, T ] × R 2d there holds

u ε (t, x, v) ≥ -C 0 T -g(•, m ε T ) ∞,R d .
On the other hand, the function

ζ(t, x, v) = g(x, m ε T ) + C(1 + |v| 2 )(T -t), (t, x, v) ∈ [0, T ] × R 2d
is a supersolution to the equation satisfied by u ε for a suitable choice of the real constant C ≥ 0. Indeed, we have that

-∂ t ζ(t, x, v) + 1 2 ε |D v ζ(t, x, v)| 2 -D x ζ(t, x, v), v -L 0 (x, v) ≥ C(1 + |v| 2 ) + 2 (T -t) 2 C 2 ε |v| 2 -D x g(x, m ε T ), v -M 0 (1 + |v| 2 ) ≥ C(1 + |v| 2 ) - 1 2 Dg(•, m ε T ) ∞,R d - 1 2 |v| 2 -M 0 (1 + |v| 2 )
where the last inequality holds by Young's inequality. Thus, taking C = 2M 0 by (BC) we obtain

M 0 (1 + |v| 2 ) - 1 2 Dg(•, m ε T ) ∞,R d - 1 2 |v| 2 ≥ 0. □ 110 CHAPTER 5.

SINGULAR LIMIT OF ACCELERATION

Corollary 5.12. Assume (M1) -(M3) and (BC). Let (t, x, v) ∈ [0, T ] × R 2d and let γ ε be a minimizer for u ε (t, x, v). Then, there exists a constant

Q 1 ≥ 0 such that T t | γε (s)| 2 ds ≤ Q 1 (1 + |v| 2 ), ∀ ε > 0.
where Q 1 is independent of ε, t, x and v.

Proof. On the one hand, from Lemma 5.11 we know that

u ε (t, x, v) ≤ M 0 T 1 + |v| 2 + g(•, m ε T ) ∞,R d , ∀ (t, x, v) ∈ [0, T ] × R 2d .
On the other hand, let (t, x, v) ∈ [0, T ] × R 2d and let γ ε be a minimizer for u ε (t, x, v). Then, by (5.6) we have that

u ε (t, x, v) = T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s), m ε s ) ds + g(γ ε (T ), m ε T ) ≥ T t L 0 (γ ε (s), γε (s), m ε s ) ds -g(•, m ε T ) ∞,R d ≥ T t 1 M 0 | γε (s)| 2 -M 0 ds -g(•, m ε T ) ∞,R d .
Therefore, combining the above inequalities we get

T t | γε (s)| 2 ds ≤ 2M 0 g(•, m ε T ) ∞,R d + M 0 T (1 + |v| 2 ) =: Q 1 (1 + |v| 2 )
where Q 1 depends only on M 0 , T and g(•, m ε T ) ∞,R d which is bounded uniformly in m ε T .

Corollary 5.13. Assume (M1) -(M3) and (BC).

Then, there exists a constant Q 2 ≥ 0 such that for any s 1 , s 2 ∈ [0, T ] with s 1 ≤ s 2 there holds

d 1 (m ε s 2 , m ε s 1 ) ≤ Q 2 |s 1 -s 2 | 1 2 , ∀ ε > 0 where Q 2 is independent of ε.
Proof. We first recall that for any t ∈ [0, T ] we know that m ε t = π 1 ♯µ ε t where µ ε t is the image of µ 0 under the flow (5.11) whose space marginal we denote by

γ ε (x,v) for (x, v) ∈ R 2d . Let s 1 , s 2 ∈ [0, T ] be such that s 1 ≤ s 2 .
Then, by (2.3) we have that

d 1 (m ε s 1 , m ε s 2 ) ≤ R d |γ ε (x,v) (s 1 ) -γ ε (x,v) (s 2 )| µ 0 (dx, dv)
and thus, appealing to Corollary 5.12 and the Hölder inequality we obtain

d 1 (m ε s 1 , m ε s 2 ) ≤ |s 1 -s 2 | 1 2 R 2d Q 1 (1 + |v| 2 ) µ 0 (dx, dv) 1 2
.

So, since µ 0 has compact support we get the result setting

Q 2 = R 2d Q 1 (1 + |v| 2 ) µ 0 (dx, dv) 1 2
. □

We are now ready to prove that the flow of probability measures m ε converges in C([0, T ]; P 1 (R d )), up to a subsequence. First, we recall that for any t ∈ [0, T ] the measure m ε t is the space marginal of µ ε t which is given by the push-forward of the initial distribution µ 0 under the optimal flow (5.11), that is

γ(x,v) (t) = v(t), γ (x,v) (0) = x v(t) = -1 ε D v u ε (t, γ (x,v) (t), v(t)), v(0) = v.
Theorem 5.14. Assume (M1) -(M3) and (BC). Then, the flow of measures {m ε t } t∈[0,T ] is tight and there exists a sequence {ε k } k∈N such that m ε k converges to some probability measure m 0 in C([0, T ]; P 1 (R d )).

Proof. Since m ε t = π 1 ♯µ ε t , for any t ∈ [0, T ], where µ ε t is given by pushforward of µ 0 under the flow (5.11), we know that

R d |x| 2 m ε t (dx) = R 2d |γ ε (x,v) (t)| 2 µ 0 (dx, dv).
So, we are interested in estimating the curve γ ε (x,v) for any (x, v), uniformly in ε > 0. In order to get it, from Corollary 5.12 we immediately deduce that

|γ ε (x,v) (s)| ≤ |x| + √ T Q 1 (1 + |v| 2 ) 1 2 , ∀ s ∈ [0, T ].
Hence, for any t ≥ 0 we have that

R d |x| 2 m ε t (dx) = R d |γ ε (x,v) (t)| 2 µ 0 (dx, dv) ≤ R 2d C 0 |x| 2 + T Q 1 (1 + |v| 2 ) µ 0 (dx, dv)
for some constant C 0 ≥ 0. Thus, since µ 0 has compact support we deduce that {m ε t } t∈[0,T ] has bounded second-order momentum, uniformly in ε > 0 and, consequently, {m ε t } t∈[0,T ] is tight. Therefore, by Prokhorov Theorem and Ascoli-Arzela Theorem there exists a sequence {ε k } k∈N and measure

m 0 ∈ C([0, T ]; P 1 (R d )) such that m ε k → m 0 in C([0, T ]; P 1 (R d )).

CHAPTER 5. SINGULAR LIMIT OF ACCELERATION

Next, we turn to the convergence of u ε . Before proving it, we need preliminary estimates on the oscillation of the value function w.r.t. velocity variable and then w.r.t. time and space variable. In particular, we will show that the function u ε (t, x, •) has uniformly decreasing oscillation which will allowed us to conclude that the limit function does not depend on v.

(x, v 0 ), (x, v) ∈ R d × B R . Then, there exists C R ≥ 0 and a parametric curve σ : [0, √ ε] → R d such that σ(0) = σ( √ ε) = x, σ(0) = v 0 , σ( √ ε) = v and 1 √ ε √ ε 0 ε 2 |σ(s)| 2 + L 0 (σ(s), σ(s), m ε s ) ds ≤ C R where C R is independent of ε, x, v and v 0 . Proof. Let R ≥ 0 and let (x, v 0 ), (x, v) ∈ R d × B R . Define the curve σ : [0, √ ε] → R d by σ(t) = x + v 0 t + Bt 2 + At 3
with A, B ∈ R satisfying the following conditions

σ(0) = σ( √ ε) = x, σ(0) = v 0 , σ( √ ε) = v.
Thus, we obtain

B = -(2v 0 + v) ε -1 2 A = (v + v 0 ) ε -1 .
Hence, we get

√ ε 0 ε 2 |σ(s)| 2 + L 0 (σ(s), σ(s), m ε s ) ds ≤ √ ε 0 ε 2 |2B + 6At| 2 + M 0 (1 + |v + 2tB + 3t 2 A| 2 ) ds ≤ C √ εR 2
for some positive constant C and the proof is thus complete.

Lemma 5.16. Assume (M1), (M2) and (BC). Let R ≥ 0, let T > 1 and ε > 0. Then, there exists C R (ε) ≥ 0 such that for any t ∈ [0, T ], any x ∈ R d , and any v, w in B R there holds

|u ε (t, x, v) -u ε (t, x, w)| ≤ C R (ε)
and C R (ε) → 0 as ε ↓ 0.

PROOF OF THE MAIN RESULT 113

Proof. Fix R ≥ 0 and take (x, v), (x, w) ∈ R d × B R . Let γ ε be a minimizer for u ε (t, x, v) and define the curve

γ(s) = σ(s -t), s ∈ [t, t + √ ε] γ ε (s - √ ε), s ∈ [t + √ ε, T ]
where σ : [0, √ ε] → R 2d connects, in the sense of Lemma 5.15, (x, w) with (x, v). Then, we obtain

u ε (t, x, w) -u ε (t, x, v) ≤ t+ √ ε t ε 2 |σ(s -t)| 2 + L 0 (σ(s -t), σ(s -t), m ε s ) ds + T t+ √ ε ε 2 |γ ε (s - √ ε)| 2 + L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s ) ds + g(γ ε (T - √ ε), m ε T ) -u ε (t, x, v) = t+ √ ε t ε 2 |σ(s -t)| 2 + L 0 (σ(s -t), σ(s -t), m ε s-t ) ds + T t+ √ ε ε 2 |γ ε (s - √ ε)| 2 + L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s- √ ε ) ds + g(γ ε (T ), m ε T ) + g(γ ε (T - √ ε), m ε T ) -g(γ ε (T ), m ε T ) -u ε (t, x, v) + t+ √ ε t L 0 (σ(s -t), σ(s -t), m ε s ) -L 0 (σ(s -t), σ(s -t), m ε s-t ) ds + T t+ √ ε L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s ) -L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s- √ ε ds.
Now, from Lemma 5.15 we know that

t+ √ ε t ε 2 |σ(s -t)| 2 + L 0 (σ(s -t), σ(s -t), m ε s-t ) ds ≤ C R √ ε, ( 5.22)
and, moreover, from the optimality of γ ε we get

T t+ √ ε ε 2 |γ ε (s - √ ε)| 2 + L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s- √ ε ) ds -u ε (t, x, v) ≤ - T T - √ ε ε 2 |γ ε (s)| 2 + L 0 (γ ε (s), γε (s), m ε s ) ds ≤ 0. (5.23)
Then, as observed before from Corollary 5.12 we obtain that

|γ ε (s)| ≤ |x| + √ T Q 1 (1 + |v| 2 ) 1 2 , ∀ s ∈ [0, T ]
and we also know that the curve σ is bounded. Hence, by (M3) and Corollary 5.13 we deduce that there exists P (ε) ≥ 0, with P (ε) → 0 as ε ↓ 0, such that

t+ √ ε t L 0 (σ(s -t), σ(s -t), m ε s ) -L 0 (σ(s -t), σ(s -t), m ε s-t ) ds + T t+ √ ε L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s ) -L 0 (γ ε (s - √ ε), γε (s - √ ε), m ε s- √ ε ds + g(γ ε (T - √ ε), m ε T ) -g(γ ε (T ), m ε T ) ≤ P (ε) (5.24)
where we have used that the modulus θ in (M3) is bounded from the boundedness of γ ε and σ. Therefore, combining (5.22), (5.23) and (5.24) we get the result.

Proposition 5.17. Assume (M1) -(M3) and (BC). Then, for any R ≥ 0 there exists a modulus ω R : R + → R + and a constant C 1 ≥ 0, independent of R, such that for any ε > 0 the following holds:

|u ε (t, x, v) -u ε (s, x, v)| ≤ ω R (|t -s|), ∀ (t, s, x, v) ∈ [0, T ] × [0, T ] × B R × B R (5.25) |D x u ε (t, x, v)| ≤ C 1 T (1 + |v| 2 ), a.e. (t, x, v) ∈ [0, T ] × R d × R d . ( 5.26)
Proof. We begin by proving (5.26). Let (t, x, v) ∈ [0, T ] × R d × R d and let γ ε be a minimizer for u ε (t, x, v). Then, from (5.7) we get

u ε (t, x + h, v) ≤ T t ε 2 |γ ε (s)| 2 + L 0 (γ ε (s) + h, γε (s), m ε s ) ds + g(γ ε (T ) + h, m ε T ) = u ε (t, x, v) + T t L 0 (γ ε (s) + h, γε (s), m ε s ) -L 0 (γ ε (s), γε (s), m ε s ) ds + g(γ ε (T ) + h, m ε T ) -g(γ ε (T ), m ε T ) ≤ u ε (t, x, v) + T t M 0 |h|(1 + | γε (s)| 2 ) ds + Dg(•, m ε T ) ∞,R d |h|.
By (5.6) and by Lemma 5.11 we obtain

T t 1 M 0 | γ * (s)| 2 -M 0 ds ≤ u ε (t, x, v) ≤ M 0 T (1 + |v| 2 ) + g(•, m ε T ) ∞,R d
which, in turns, yields to the conclusion. Next, we proceed to show (5.25). Let R ≥ 0 and take

(t, x, v) ∈ [0, T ] × B R × B R . Let γ ε be a minimizer for u ε (t, x, v) and let h ∈ [0, T -t]. Then, we have that u ε (t + h, x, v) ≤ T t+h ε 2 |γ ε (s -h)| 2 + L 0 (γ ε (s -h), γε (s -h), m ε s ) ds + g(γ(T -h), m ε T ) = T t+h ε 2 |γ ε (s -h)| 2 + L 0 (γ ε (s -h), γε (s -h), m ε s-h ) ds + g(γ ε (T ), m ε T ) + T t+h L 0 (γ ε (s -h), γε (s -h), m ε s ) -L 0 (γ ε (s -h), γε (s -h), m ε s-h ) ds + g(γ ε (T -h), m ε T ) -g(γ ε (T ), m ε T ) ≤ u ε (t, x, v) + T t+h θ(|γ ε (s -h)|)ω 0 (d 1 (m ε s , m ε s-h )) ds + Dg(•, m ε T ) ∞,R d |h|
where the last inequality holds by (M3). Hence, from Corollary 5.12 we know that

|γ ε (s)| ≤ |x| + √ T Q 1 (1 + |v| 2 ) 1 2 , ∀ s ∈ [0, T ]
and thus θ(•) turns out to be bounded. Therefore, appealing to Corollary 5.13 we obtain

u ε (t + h, x, v) -u ε (t, x, v) ≤ T θ(R)ω 0 (|h| 1 2 ) + Dg(•, m ε T ) ∞,R d |h|. (5.27)
On the other hand, let R ≥ 0 and let

(t, x, v) ∈ [0, T ] × B R × B R . For h ∈ [0, T -t], define the curve γ : [t, t + h] → R d by γ(s) = x + (s -t)v.
Then, by Dynamic Programming Principle we deduce that

u ε (t, x, v) ≤ t+h t ε 2 |γ(s)| 2 + L 0 (γ(s), γ(s), m ε s ) ds + u ε (t + h, γ(t + h), γ(t + h)) = t+h t L 0 (x + (s -t)v, v, m ε s ) ds + u ε (t + h, x + hv, v) ≤ M 0 (1 + R 2 )|h| + u ε (t + h, x, v) + C 1 T (1 + R 2 )|h| (5.28)
where we applied (5.6) and Equation (5.26) to get the last inequality. Therefore, combining (5.27) and (5.28) the proof is complete.

Remark 5.18. Next, we study the behavior of the value function u ε as ε → 0 and before doing that we recall the following argument needed to get uniform convergence from point-wise convergence.

Assume that there exists a nonnegative function Θ(δ 0 , ε 0 , R 0 ) such that

Θ(δ 0 , ε 0 , R 0 ) → 0, as ε 0 , δ 0 ↓ 0,
and assume that for any |t 1 -

t 2 | + |x 1 -x 2 | ≤ δ 0 , any ε ≤ ε 0 and any |x i |, |v i | ≤ R 0 (i = 1, 2) there holds |u ε (t 1 , x 1 , v 1 ) -u ε (t 2 , x 2 , v 2 )| ≤ Θ(δ 0 , ε 0 , R 0 ).
Then: if u ε converge point-wise then u ε converges locally uniformly and the limit function does not depend on v.

Let m 0 ∈ C([0, T ]; P 1 (R d ))
be the flow of measures obtained in Theorem 5.14 as limit of the flow

m ε k in C([0, T ]; P 1 (R d )) for some subsequence ε k ↓ 0. Define the function u 0 : [0, T ] × R 2d → R by u 0 (t, x) = inf γ∈Γt(x) T t L 0 (γ(s), γ(s), m 0 s ) ds + g(γ(T ), m 0 T ) . ( 5.29)
We will prove now that for the subsequence ε k the sequence of value functions u ε k locally uniformly converge to u 0 .

Theorem 5.19. Assume (M1) -(M3) and (BC).

Then, there exists a subsequence ε k ↓ 0 such that u ε k locally uniformly converges to u 0 .

Proof. We proceed to show first the point-wise convergence of u ε k to u 0 , for some subsequence ε k ↓ 0, and then, using Remark 5.18, i.e., constructing such a modulus Θ, we deduce that the convergence is locally uniform. From Theorem 5.14, let ε k be the subsequence such that m

ε k → m 0 in C([0, T ]; P 1 (R d )) as k → ∞. Let R ≥ 0, let (t, x, v) ∈ [0, T ] × R d × B R
and let γ ε k be a minimizer for u ε k (t, x, v). Then, we have that

u ε k (t, x, v) = T t ε k 2 |γ ε k (s)| 2 + L 0 (γ ε k (s), γε k (s), m ε k s ) ds + g(γ ε k (T ), m ε k T ) ≥ T t L 0 (γ ε k (s), γε k (s), m ε k s ) ds + g(γ ε k (T ), m ε k T ) ≥ inf γ∈Γt(x) T t L 0 (γ(s), γ(s), m 0 s ) ds + g(γ(T ), m 0 T ) + g(γ ε k (T ), m ε T ) -g(γ ε k (T ), m 0 T ) + T t L 0 (γ ε k (s), γε k (s), m ε k s ) -L 0 (γ ε k (s), γε k (s), m 0 s ) ds ≥ u 0 (t, x) -o(1)
where the last inequality holds by (M1) and the convergence of

m ε k in C([0, T ]; P 1 (R d )).
On the other hand, let R ≥ 0 and take

(t, x, v) ∈ [0, T ] × R d × B R . Let γ 0 ∈ Γ t (x) be a solution of inf γ∈Γt(x) T t L 0 (γ(s), γ(s), m 0 s ) ds + g(γ(T ), m 0 T ) .
Next, we distinguish two cases: first, when γ0 (t) = v and then when γ0 (t) = v. Indeed, if γ0 (t) = v, by the Euler equation and the C 2 -regularity of L 0 we have that γ ∈ C 2 ([0, T ]). Hence, we can use γ 0 as a competitor for u ε k (t, x, v) and we get

u ε k (t, x, v) ≤ T t ε 2 |γ 0 (s)| 2 + L 0 (γ 0 (s), γ0 (s), m ε k s ) ds + g(γ 0 (T ), m ε k (T )) ≤ T t ε 2 |γ 0 (s)| 2 + L 0 (γ 0 (s), γ0 (s), m 0 s ) ds + g(γ 0 (T ), m 0 (T )) + T t L 0 (γ 0 (s), γ0 (s), m ε k s ) -L 0 (γ 0 (s), γ0 (s), m 0 s ) ds + g(γ 0 (T ), m ε k (T )) -g(γ 0 (T ), m 0 (T )) ≤ u 0 (t, x) + o(1) (5.30)
where the last inequality again follows from the convergence of

m ε k in C([0, T ]; P 1 (R d ))
. If this is not the case, i.e., γ0 (t) = v, from Lemma 5.16 we deduce that

u ε k (t, x, v) = u ε k (t, x, v) -u ε k (t, x, γ0 (t)) + u ε k (t, x, γ0 (t)) ≤ o(1) + u ε k (t, x, γ0 (t)).
Thus, in order to conclude it is enough to estimate u ε k (t, x, γ0 (t)) as in (5.30). Therefore, we obtain

u 0 (t, x) -o(1) ≤ u ε k (t, x, v) ≤ u 0 (t, x) + o(1)
which implies that u ε k point-wise converges to u 0 . Finally, in order to conclude we need to show that the convergence is locally uniform. From (5.25), (5.26) and Lemma 5.16 we have that for any R ≥ 0 and any

(t 1 , x 1 , v 1 ), (t 2 , x 2 , v 2 ) ∈ [0, T ] × B R × B R there holds |u ε (t 1 , x 1 , v 1 ) -u ε (t 2 , x 2 , v 2 )| ≤ ω R (|t 1 -t 2 |) + C 1 |x 1 -x 2 | + C R √ ε.
Therefore, setting

Θ(δ 0 , ε 0 , R 0 ) = ω R 0 (δ 0 ) + C 1 δ 0 + C R 0 √ ε 0
by Remark 5.18 we deduce that the convergence is locally uniform and the proof is thus complete. After proving the convergence of u ε , we go back to the analysis of the flow of measures and in particular we will characterize it in terms of the limit function u 0 . In order to do so, we study the convergence of minimizers for u ε and appealing to such a result we will show that m 0 ∈ C([0, T ]; P 1 (R d )) solves a continuity equation with vector field D p H 0 (x, D x u 0 ), in the sense of distribution. Proposition 5.20. Assume (M1) -(M3) and (BC). Let (t, x, v) ∈ [0, T ]× R 2d be such that u 0 is differentiable at (t, x) and let γ ε be a minimizer for u ε (t, x, v). Then, γ ε uniformly converges to a curve γ 0 ∈ AC([0, T ]; R d ) and γ 0 is the unique minimizer for u 0 (t, x) in (5.29).

Proof. Let us start by proving that γ ε uniformly converges, up to a subsequence. By Corollary 5.12 we know that

T t | γε (s)| 2 ds ≤ Q 1 (1 + |v| 2 ).
Thus, for any s ∈ [t, T ], by Hölder inequality we have that

|γ ε (s)| ≤ |x| + √ T Q 1 (1 + |v| 2 ) 1 2 .
Therefore, γ ε is bounded in H 1 (0, T ; R d ) which implies that by Ascoli-Arzela Theorem there exists a sequence {ε k } k∈N and a curve γ 0 ∈ AC([0, T ]; R d ) such that γ ε k converges uniformly to γ 0 . We show now that such a limit γ 0 is a minimizer for u 0 (t, x). First, we observe that

lim inf k→∞ T t ε k 2 |γ ε k (s)| 2 + L 0 (γ ε k (s), γε k (s), m ε s ) ds + g(γ ε k (T ), m ε k T ) ≥ lim inf k→∞ T t L 0 (γ ε k (s), γε k (s), m ε k s ) ds + g(γ ε k (T ), m ε k T ) .
Then, as observed at the beginning of this proof γ ε is uniformly bounded in H 1 (0, T ). So by lower-semicontinuity of L and Theorem 5.14 we deduce that

lim inf k→∞ T t L 0 (γ ε k (s), γε k (s)) ds + g(γ ε k (T ), m ε k T ) ≥ T t L 0 (γ 0 (s), γ0 (s), m 0 s ) ds + g(γ 0 (T ), m 0 T ). (5.31)
Moreover, for any

R ≥ 0 taking (t, x, v) ∈ [0, T ] × R d × B R , from Theo- rem 5.19 we obtain u ε k (t, x, v) ≤ u 0 (t, x) + o(1)
and we recall that

u ε k (t, x, v) = T t ε k 2 |γ ε k (s)| 2 + L 0 (γ ε k (s), γε k (s), m ε k s ) ds+g(γ ε k (T ), m ε k T ).
Hence, we get

o(1) + u 0 (t, x) ≥ T t ε k 2 |γ ε k (s)| 2 + L 0 (γ ε k (s), γε k (s), m ε k s ) ds + g(γ ε k (T ), m ε k T ).
Therefore, passing to the limit as ε ↓ 0 from (5.31) we obtain

u 0 (t, x) ≥ T t L 0 (γ 0 (s), γ0 (s), m 0 s ) ds + g(γ 0 (T ), m 0 T )
which proves that γ 0 is a minimizer for u 0 (t, x). Since u 0 is differentiable at (t, x) ∈ R d there exists a unique minimizing trajectory and thus we have that the uniform convergence of γ ε holds for the whole sequence.

∈ [0, T ] × R d .
Let u 0 be as in (5.29) and let γ 0 t (•) be the flow associated with the vector field

x → D p H 0 (x, D x u 0 (t, x), m 0 t ),
that is,

γ0 t (x) = D p H 0 (γ 0 t (x), D x u 0 (t, γ 0 t (x)), m 0 t ), t ∈ [0, T ] γ 0 (x) = x.
Note that such a flow exists since the vector field x → D p H 0 (x, D x u 0 (t, x), m 0 t ) is Lipschitz continuous by the Lipschitz continuity of the value function u 0 and by the regularity of the Hamiltonian H 0 . We also recall that the measure µ ε is the image of µ 0 under the flow (5.11), which is optimal as observed in Remark 5.2 for u ε (0, x, v) for a.e. (x, v) ∈ R 2d , and thus, for any function

φ ∈ C ∞ c (R d ) the measure m ε t is given by R d φ(x) m ε t (dx) = R 2d φ(γ ε (x,v) (t)) µ 0 (dx, dv). ( 5.32)
We finally recall that by assumption µ 0 is absolutely continuous w.r.t. Lebesgue measure.

Corollary 5.22. Assume (M1) -(M3) and (BC). Then, we have that

m 0 t = γ 0 t (•)♯m 0 , ∀ t ∈ [0, T ]. (5.33
)

Moreover, m 0 ∈ C([0, T ]; P 1 (R d )) solves ∂ t m 0 t -div m 0 t D p H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d m 0 0 = m 0 , x ∈ R d ,
in the sense of distributions.

Proof. From Theorem 5.14 let

ε k ↓ 0 be such that m ε k → m 0 in C([0, T ]; P 1 (R d )).
Then, since µ 0 is absolutely continuous w.r.t. Lebesgue measure by Proposition 5.20 we have that

γ ε k (x,v) (t) → γ 0 t (x), µ 0 -a.e. (x, v), ∀t ∈ [0, T ].
Therefore, from (5.32), for ε = ε k , as k → ∞ we get

R d φ(x) m 0 t (dx) = R d φ(γ 0 t (x)) m 0 (dx), ∀ t ∈ [0, T ]
which proves (5.33). Moreover, again by Proposition 5.20 we have that γ 0 t is a minimiser for u 0 (0, x) since it is the limit of γ ε (x,v) which is optimal u ε (0, x, v) and we are taking (x, v) in a subset of full measure w.r.t. µ 0 . Therefore, from the optimality of γ 0 we get

γ0 t (x) = D p H 0 (γ 0 t (x), Du 0 (t, γ 0 t (x)), m 0 t ), t ∈ (0, T ] γ 0 0 (x) = x.
Hence, for any

ψ ∈ C ∞ c ([0, T ) × R d ) we obtain d dt R d ψ(t, x) m 0 t (dx) = d dt R d ψ(t, γ 0 t (x)) m 0 (dx) = R d ∂ t ψ(t, γ 0 t (x)) + D x ψ(t, γ 0 t (x)), D p H 0 (γ 0 t (x), D x ψ(t, γ 0 t (x)), m 0 t ) m 0 (dx) = R d ∂ t ψ(t, x) + D x ψ(t, x), D p H 0 (x, D x ψ(t, x), m 0 t ) m 0 t (dx)
and integrating, in time, over [0, T ] we get the result.

We are now ready to prove the main result. Proof of Theorem 5.3. Let {ε k } k∈N be such that m ε k → m 0 in C([0, T ]; P 1 (R d ) and u ε k → u 0 locally uniformly on [0, T ] × R 2d . Then, appealing to Theorem 5.19 and Corollary 5.22 we deduce that (u 0 , m 0 ) is a solution to the MFG system

     -∂ t u 0 (t, x) + H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d ∂ t m 0 t -div m 0 t D p H 0 (x, D x u 0 (t, x), m 0 t ) = 0, (t, x) ∈ [0, T ] × R d m 0 0 = m 0 , u 0 (T, x) = g(x, m 0 T ) x ∈ R d
which completes the proof.

Chapter 6

Asymptotic analysis for Hamilton-Jacobi equations associated with sub-Riemannian control systems

Settings and assumptions

For m ∈ N and i = 1, . . . , m, let

f i : R d → R d and u i : [0, ∞) → R
be smooth vector fields and measurable controls, respectively, and consider the following controlled dynamics of sub-Riemannian type

γ(t) = m i=1 f i (γ(t))u i (t) = F (γ(t))U (t), t ∈ [0, +∞) (6.1)
where

F (x) = [f 1 (x)| . . . |f m (x)
] is an d×m real matrix and

U (t) = (u 1 (t), . . . , u m (t)) ⋆1 .
For any s 0 , s 1 ∈ R such that s 0 < s 1 and x, y ∈ R d we set

Γ x→ s 0 ,s 1 = {(γ, u) ∈ AC([s 0 , s 1 ]; R d ) × L 2 (s 0 , s 1 ; R m ) : γ(t) = F (γ(t))u(t), γ(s 0 ) = x}, Γ →y s 0 ,s 1 = {(γ, u) ∈ AC([s 0 , s 1 ]; R d ) × L 2 (s 0 , s 1 ; R m ) : γ(t) = F (γ(t))u(t), γ(s 1 ) = y}, Γ x→y s 0 ,s 1 = Γ x→ s 0 ,s 1 ∩ Γ →y s 0 ,s 1 .
1 (u1, . . . , um) ⋆ denotes the transpose of (u1, . . . , um)

122CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

Throughout the Chapter we assume the vector fields f i to satisfy the following:

(F0) there exists a constant c f ≥ 1 such that for any i = 1, . . . , m

|f i (x)| ≤ c f (1 + |x|), ∀x ∈ R d . ( 6.2)
In literature, under assumption (F0) the distribution {f i } i=1,...,m is called regular. By (6.2) and Gronwall's inequality we get the following estimate for the trajectories of (6.1).

Lemma 6.1. Let x ∈ R d , t ≥ 0 and (γ, u) ∈ Γ x→ 0,t . If u ∈ L ∞ (0, t; R m ) then we have that |γ u (s)| ≤ (|x| + c f u ∞ s)e c f ∥u∥∞s , ∀ s ∈ [0, t].
Moreover, still from (6.2) we obtain the following.

Lemma 6.2. Let x ∈ R d , t ≥ 0 and (γ, u) ∈ Γ x→ 0,t . Then there exists a constant κ( u 2 , t) ≥ 0 such that |γ(s)| ≤ κ( u 2 , t)(1 + |x|), ∀ s ∈ [0, t] (6.3)
and

|γ(t 2 ) -γ(t 1 )| ≤ c f κ( u 2 , t)(1 + |x|) u 2 |t 2 -t 1 | 1 2 , 0 ≤ t 1 ≤ t 2 ≤ t. (6.4)
Proof. We begin by proving (6.3). For any s ∈ [0, t] we have that

|γ(s)| ≤ |x| + t 0 |F (γ(s))||u(s)| ds ≤ |x| + t 0 c f (1 + |γ(s)|)|u(s)| ds ≤ |x| + c f t 0 1 + |γ(s)| 2 ds 1 2 u 2 .
Thus, we get

|γ(s)| 2 ≤ C |x| 2 + c 2 f t u 2 2 + c 2 f u 2 2 t 0 |γ(s)| 2 ds
which implies the (6.3) by Gronwall's inequality.

SETTINGS AND ASSUMPTIONS

123

We now proceed to show (6.4). Fix t 1 , t 2 such that 0 ≤ t 1 ≤ t 2 ≤ t. Then, we have that

|γ(t 2 ) -γ(t 1 )| ≤ t 2 t 1 |F (γ(s))||u(s)| ds ≤ c f t 2 t 1 (1 + |γ(s)|)|u(s)| ds ≤ c f κ( u 2 , t)(1 + |x|) t 2 t 1 |u(s)| ds
where the last inequality holds by (6.3). Hence, by Hölder's inequality we obtain

|γ(t 2 ) -γ(t 1 )| ≤ c f κ( u 2 , t)(1 + |x|) u 2 |t 2 -t 1 | 1 2 .
This completes the proof of the lemma.

Let the Lagrangian L :

R d × R m → R be such that (L0) L ∈ C 2 (R d × R m ) and L(x, u) ≥ L(x, 0) for any (x, u) ∈ R d × R m . (L1) There exist a non-decreasing function β : [0, ∞) → R and a constant ℓ 1 ≥ 0 such that L(x, u) ≤ β(|x|)(1 + |u| 2 ), ∀ (x, u) ∈ R d × R m |D x L(x, u)| ≤ ℓ 1 (1 + |u| 2 ), ∀ (x, u) ∈ R d × R m D 2 u L(x, u) ≥ 1 ℓ 1 , ∀ (x, u) ∈ R d × R m .
(L2) There exists a compact set K L ⊂ R d and a constant δ L > 0 such that inf

x∈R d \K L L(x, 0) ≥ δ L + min x∈K L L(x, 0). (6.5)
Observe that a special class of functions L which satisfy

(L0) is the class of Lagrangians L ∈ C 2 (R d × R m ) which are convex w.r.t. u ∈ R d and reversible, that is, L(x, u) = L(x, -u) for any (x, u) ∈ R d × R m .
Moreover, note that by (L0), (L1) and (L2) we obtain

L(x, u) ≥ 1 2ℓ 1 |u| 2 + L(x * , 0), ∀ (x, u) ∈ R d × R m (6.6) where x * ∈ K L is such that L(x * , 0) = min x∈K L L(x, 0). Furthermore, set δ * (x) = d SR (x, x * ), ∀ x ∈ R d
and observe that, by Corollary 2.7, there exists a nondecreasing function

D : R + → R + with δ * (x) ≤ D(|x|), ∀ x ∈ R d . ( 6.7)
124CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

Boundedness of optimal trajectories

We consider the following minimisation problem: for any T > 0 and x ∈ R d to minimize

T 0 L(γ(s), u(s)) ds over all (γ, u) ∈ Γ x→ 0,T (6.8)
and we set

V T (x) = inf (γ,u)∈Γ x→ 0,T T 0 L(γ(s), u(s)) ds, ∀ x ∈ R d . ( 6.9)
For any x ∈ R d we say that a trajectory-control pair (γ, u) ∈ Γ x→ 0,T is optimal if it solves (6.8). Remark 6.3. We observe that by using classical technics from optimal control theory one can easily obtain the existence of optimal trajectory-control pairs for (6.8) (see, for instance, [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.4]).

In this section, we prove the uniform boundedness of optimal trajectories for (6.8) starting from a given compact set. We begin by deriving a uniform bound for the Lebsegue measure of all times at which an optimal trajectory may lie outside the compact set K L of assumption (L3). Proposition 6.4. Assume (F0), and (L0) -(L2). For any R ≥ 0 there exists a constant M R ≥ 0 such that for any x ∈ B R , any T ≥ δ * (x), and any optimal pair (γ x , u x ) ∈ Γ x→ 0,T for (6.8) we have that

L 1 ({t ∈ [0, T ] : γ x (t) ∈ K L }) ≤ M R . (6.10) Proof. Fix R ≥ 0 and let x ∈ B R . Let (γ x , ūx ) ∈ Γ x→x * 0,δ * (x)
be a solution of (2.5) and recall that, as observed in (6.7), δ * (x) ≤ D(R). Define the control

u x (t) = ūx (t), t ∈ [0, δ * (x)] 0, t ∈ (δ * (x), T ]. Then, ( γ x , u x ) ∈ Γ x→x * 0,T
and we obtain

V T (x) ≤ δ * (x) 0 L(γ x (t), ūx (t)) dt + (T -δ * (x))L(x * , 0) =: c 1 (x, δ * (x)) + (T -δ * (x))L(x * , 0). (6.11)
Invoking Corollary 2.7 once again, we have that c 1 (x, δ * (x)) ≤ C 1 (R) for some positive constant C 1 (R). Now, let (γ x , u x ) ∈ Γ x→ 0,T be optimal for (6.8).

Then, we have that

V T (x) = T 0 L(γ x (t), u x (t)) dt ≥ T 0 L(γ x (t), 0) dt ≥ c 2 (x, δ * (x)) + T δ * (x) L(γ x (t), 0)1 K L (γ x (t)) dt + T δ * (x) L(γ x (t), 0)1 K c L (γ x (t)) dt ≥ c 2 (x, δ * (x)) + L(x * , 0)L 1 {t ∈ [δ * (x), T ] : γ x (t) ∈ K L } + inf x∈R d \K L L(x, 0) L 1 {t ∈ [δ * (x), T ] : γ x (t) ∈ K L } = c 2 (x, δ * (x)) + L(x * , 0)L 1 {t ∈ [δ * (x), T ] : γ x (t) ∈ K L } + inf x∈R d \K L L(x, 0) T -δ * (x) -L 1 {t ∈ [δ * (x), T ] : γ x (t) ∈ K L } (6.12)
where

c 2 (x, δ * (x)) := δ * (x) 0 L(γ x (t), 0) dt.
From (6.12) it also follows that

V T (x) ≥ c 2 (x, δ * (x)) + L(x * , 0)(T -δ * (x)),
which, together with (6.11), yields

c 2 (x, δ * (x)) ≤ c 1 (x, δ * (x)) ≤ C 1 (R). (6.13)
Hence, we have that

c 3 (δ * (x)) := c 1 (δ * (x)) -c 2 (δ * (x)) ≥ 0, c 3 (δ * (x)) ≤ C 1 (R).
So, combining (6.11) and (6.12) and recalling (L2), we deduce that

c 3 (δ * (x)) ≥ inf x∈R d \K L L(x, 0) -L(x * , 0) T -δ * (x) -L 1 {t ∈ [δ * (x), T ] : γ u (t) ∈ K L } ≥ δ L L 1 {t ∈ [δ * (x), T ] : γ x (t) ∈ K L } .
Therefore,

L 1 {t ∈ [0, T ] : γ x (t) ∈ K L } ≤ c 3 (δ * (x)) δ L + δ * (x).
Recalling that δ * (x) ≤ D(R) and setting

M R := C 1 (R) δ L + D(R)
we obtain the conclusion. 126CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS Theorem 6.5. Assume (F0), and (L0) -(L2). For any R ≥ 0 there exist two constants P R , Q R ≥ 0 such that for any x ∈ B R , any T ≥ δ * (x), and any optimal pair (γ x , u x ) ∈ Γ x→ 0,T for (6.8) we have that

T 0 |u x (t)| 2 dt ≤ P R (6.14) and |γ x (t)| ≤ Q R , ∀ t ∈ [0, T ]. (6.15)
Proof. We begin with the proof of (6.14). Since (γ x , u x ) ∈ Γ x→ 0,T is optimal for (6.8) we have that

V T (x) = T 0 L(γ x (t), u x (t)) dt ≥ 1 2ℓ 1 T 0 |u x (t)| 2 dt + T L(x * , 0). (6.16)
On the other hand, let (γ x , ūx ) ∈ Γ x→x * 0,δ * (x) be a solution of (2.5) and define the control

u x (t) = ūx (t), t ∈ [0, δ * (x)] 0, t ∈ (δ * (x), T ],
that is, ( γ x , u x ) ∈ Γ x→x * 0,T . By the definition of V T we deduce that

V T (x) ≤ δ * (x) 0 L(γ x (t), ūx (t)) dt + (T -δ * (x))L(x * , 0). (6.17)
Combining (6.16) and (6.17) we obtain

1 2ℓ 1 T 0 |u x (t)| 2 dt ≤ δ * (x) 0 L(γ x (t), ūx (t)) dt -δ * (x)L(x * , 0).
In order to prove (6.14), we need an upper bound for the term

δ * (x) 0 L(γ x (t), ūx (t)) dt.
Observe that, since ūx ∞,[0,δ * (x)] ≤ 1 and δ * (x) ≤ D(R), Lemma 6.1 ensures that

|γ x (t)| ≤ (R + c f D(R))e c f D(R) =: Λ(R), ∀ t ∈ [0, δ * (x)]. (6.18)
Therefore, by assumption (L1) we deduce that δ * (x) 0 L(γ x (t), ūx (t)) dt ≤ 2β(Λ(R)).

LONG-TIME AVERAGE AND ERGODIC CONSTANT 127

Hence, (6.14) follows taking

P R = 2ℓ 1 (2β(Λ(R)) + D(R)L(x * , 0)).
We now proceed to prove (6.15). Let (γ x , u x ) ∈ Γ x→ 0,T be a solution of (6.8). Clearly, we just need to estimate |γ x (t)| for all times at which the optimal trajectory lies outside the compact set K L . Let t ∈ [0, T ] be such that γ x ( t ) ∈ K L and set

t 0 = sup{t ∈ [0, t ] : γ x (t) ∈ K L }, if {t ∈ [0, t ] : γ x (t) ∈ K L } = ∅ 0, if {t ∈ [0, t ] : γ x (t) ∈ K L } = ∅
We only consider the case of t 0 = 0 since the reasoning is similar when t 0 = 0. Since γ x is a solution of (6.1) we deduce that for any t ∈ [t 0 , t]

|γ x (t)| ≤ |γ x (t 0 )| + c f t t 0 1 + |γ x (s)|)|u x (s)| ds ≤|γ x (t 0 )| + c f u x 2,[t 0 , t] t t 0 1 + |γ x (s)| 2 ds 1 2
.

Hence, appealing to (6.10) and (6.14) we deduce that

|γ x (t)| 2 ≤ C |γ x (t 0 )| 2 + u x 2 2,[t 0 , t] M R + u x 2 2,[t 0 , t] t t 0 |γ x (s)| 2 ds ≤ C |γ x (t 0 )| 2 + P R M R + P R t t 0 |γ x (s)| 2 ds
for some constant C ≥ 0. Thus, recalling that |t -t 0 | ≤ M R by Proposition 6.4, the Gronwall inequality yields

|γ x (t)| ≤ C|γ x (t 0 )| 2 P R M R e P R M R , ∀ t ∈ [t 0 , t] and we set Q R := C|γ x (t 0 )| 2 P R M R e P R M R . Since |γ x (t 0 )| ≤ max{|y| : y ∈ K L } and |γ x (t)| ≤ max{|y| : y ∈ K L }
for all times t at which γ x (t) ∈ K L , we get the conclusion.

Long-time average and ergodic constant

In this section, we investigate the existence of the limit

lim T →∞ 1 T V T (x) (x ∈ R d ),
where V T (x) is defined in (6.9), as well as the related problem of the existence of solutions to the ergodic Hamilton-Jacobi equation

c + H(x, Dχ(x)) = 0 (x ∈ R d ) (6.19) 128CHAPTER 6

. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

for some c ∈ R, where H : R d × R d → R is the Hamiltonian associated with L, that is,

H(x, p) = sup u∈R m m i=1 u i p, f i (x) -L(x, u) , ∀ (x, p) ∈ R d × R d . (6.20)

Long-time average

In order to prove the main result of this section, that is, Theorem 6.8, we need to show, first, that the value function V T is locally equicontinuous uniformly in T .

Lemma 6.6. Assume (F0), and (L0) -(L2). For any R ≥ 0 there exist two constants T R ≥ 0 and K(R) ≥ 0 such that

|V T (x) -V T (y)| ≤ K(R)d SR (x, y), ∀ T ≥ T R ∀ x, y ∈ B R . Proof. Let x, y ∈ B R , set δ = d SR (x, y) and let T ≥ d SR (x, y) =: T R . Let (γ y , ūy ) ∈ Γ y→x 0,δ
be a solution of (2.5), let (γ x , u x ) ∈ Γ x→ 0,T be a solution of (6.8), and define the control

u y (t) = ūy (t), t ∈ [0, δ] u x (t -δ), t ∈ (δ, T ].
Then, we have that Hence, going back to (6.21), from Corollary 2.7 it follows that

V T (y) -V T (x) ≤ T 0 L( γ y (t), u y (t)) dt - T 0 L(γ x (t), u x (t)) dt ≤ δ 0 L(γ y (t), ūy (t)) dt + T δ L(γ x (t -δ), u x (t -δ)) dt - T 0 L(γ x (t), u x (t)) dt = δ 0 L(γ y (t), ūy (t)) dt + T -δ 0 L(γ x (s), u x (s)) ds - T 0 L(γ x (s), u x (s)) ds = δ 0 L(γ y (t), ūy (t)) dt - T T -δ L(γ x (s), u x (s)) ds.
V T (y) -V T (x) ≤ δ 0 L(γ y (s), ūy (s)) ds - T T -δ L(γ x (s), u x (s)) ds ≤δ(2β(Λ(R)) -L(x * , 0)) = K(R)d SR (x, y)
where K(R) := c2 (2β(Λ(R)) -L(x * , 0)). Switching x and y in the above reasoning completes the proof. Lemma 6.7. Assume (F0), and (L0) -(L2). For any R ≥ 0 there exists a constant C R ≥ 0 such that for any x ∈ B R , any T > 0, and any optimal pair (γ x , u x ) ∈ Γ x→ 0,T of (6.8) there exists a pair

(γ T , u T ) ∈ Γ x→x 0,T such that T 0 L(γ T (t), u T (t)) dt ≤ T 0 L(γ x (t), u x (t)) dt + C R .
Proof. Fix R ≥ 0, x ∈ B R , and take an optimal pair (γ x , u x ) ∈ Γ x→ 0,T . If γ x (T ) = x then it is enough to take C R = 0 and (γ T , u T ) = (γ x , u x ). If this is not the case, let δ 0 ∈ (0, T ) be such that

δ 0 = d SR (γ x (T -δ 0 ), x).
Note that such a number δ 0 exists since g(δ) := d SR (γ x (T -δ), x) -δ, for δ ∈ [0, T ], is a continuous function satisfying

g(T ) = -T < 0 g(0) = d SR (γ x (T ), x) > 0.
For simplicity of notation set y = γ x (T -δ 0 ) and observe that |y| ≤ Q R by Theorem 6.5. Let (γ y , ūy ) ∈ Γ y→x 0,δ 0 be a solution of (2.5) and define the control

u T (t) = u x (t), t ∈ [0, T -δ 0 ] ūy (t + δ 0 -T ), t ∈ (T -δ 0 , T ]. Then T 0 L(γ T (t), u T (t)) dt = T -δ 0 0 L(γ x (t), u x (t)) dt + δ 0 0 L(γ y (t), ūy (t)) dt = T 0 L(γ x (t), u x (t)) dt - T T -δ 0 L(γ x (t), u x (t)) dt + δ 0 0 L(γ y (t), ūy (t)) dt.
130CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS By (6.6) we obtain

T T -δ 0 L(γ x (t), u x (t)) dt ≥ δ 0 L(x * , 0). Let R ′ = max{R, Q R }.
Then, since ūy ∞,[0,δ 0 ] ≤ 1 and |y| ≤ Q R , by Lemma 6.1 we also have that

|γ y (t)| ≤ (Q R + c f R ′ )e c f R ′ =: Λ(R)
for any t ∈ [0, δ 0 ]. So, we obtain Proof. Let R ≥ 0. By Lemma 6.6, for all x ∈ B R we deduce that

- T T -δ 0 L(γ x (t), u x (t)) dt + δ 0 0 L(γ y (t), ūy (t)) dt ≤ -δ 0 L(x * , 0) + δ 0 0 L(γ y (t), ūy (t)) dt ≤ δ 0 2β(Λ(R)) -L(x * , 0) .

The conclusion follows taking

C R = R ′ 2β(Λ(R)) -L(x * , 0) . □
|V T (x) -V T (0)| ≤ K ′ (R). (6.24)
Hence, to obtain the conclusion it suffices to prove the existence of the limit lim

T →∞ 1 T V T (0) =: α(L) . ( 6.25)
For this purpose let {T n } n∈N and {(γ n , u n )} n∈N ⊂ Γ 0→ 0,Tn be such that

lim inf T →∞ 1 T V T (0) = lim n→∞ 1 T n inf (γ,u)∈Γ 0→ 0,Tn
T →∞ 1 T V T (0) ≤ lim sup T →∞ 1 T T 0 L( γ 0 n (t), u 0 n (t)) dt, ∀ n ∈ N (6.28)
by using u 0 n as a competitor in (6.9). Then, by periodicity and (6.27) we obtain lim sup

T →∞ 1 T T 0 L( γ 0 n (t), u 0 n (t)) dt = 1 T n Tn 0 L(γ 0 n (t), u 0 n (t)) dt ≤ 1 T n Tn 0 L(γ n (t), u n (t)) dt + C 0 T n .
Therefore, recalling (6.28) and (6.26) we conclude that lim sup

T →∞ 1 T V T (0) ≤ lim n→∞ 1 T n Tn 0 L(γ n (t), u n (t)) dt + C 0 T n = lim inf T →∞ 1 T V T (0).
This yields (6.25), thus completing the proof.

Corollary 6.9. Assume (F0), and (L0) -(L2). Then, we have that

α(L) = L(x * , 0).
Proof. First, we recall that

α(L) = lim T →∞ 1 T V T (0) = lim T →∞ 1 T inf (γ,u)∈Γ 0→ 0,T T 0 L(γ(s), u(s)) ds.
So, taking (γ x , u x ) ∈ Γ 0→ 0,T optimal for V T (0) we obtain

α(L) = lim T →∞ 1 T T 0 L(γ x (s), u x (s)) ds ≥ lim T →∞ T 0 L(x * , 0) ds = L(x * , 0)
since, by assumption (L0) and (L2), we have that L(x, u) ≥ L(x * , 0) for any

(x, u) ∈ R d × R m .
On the other hand, we observe that, owing to Theorem 6.8, the value of α(L) could be computed replacing 0 in (6.25) with any other point of R d . So,

α(L) = lim T →∞ 1 T V T (x * ).
This implies that

α(L) = lim T →∞ 1 T inf (γ,u)∈Γ x * → 0,T T 0 L(γ(s), u(s)) ds ≤ lim T →∞ 1 T T 0 L(x * , 0) ds = L(x * , 0)
which yields the conclusion. 132CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS Remark 6.10. Note that in view of Theorem 6.8 we have that lim

T →∞ sup x∈B R 1 T inf (γ,u)∈Γ x→ 0,T T 0 L(γ(s), u(s)) -α(L) ds = 0, ∀ R ≥ 0.
Moreover, from Corollary 6.9 we deduce that min

(x,u)∈R d ×R m L(x, u) -α(L) = 0.
Therefore, by replacing L with L(x, u) := L(x, u) -α(L) one can reduce the analysis to the case of α( L) = min (x,u)∈R d ×R m L(x, u) = 0.

Application to Abel means

Now, we move to the analysis of the ergodic equation

c + H(x, Dχ) = 0, x ∈ R d
showing the existence of viscosity solutions to such an equation by studying the limit behavior of solutions to the discounted problem

λv λ (x) + H(x, Dv λ (x)) = 0, x ∈ R d (6.29)
as λ ↓ 0. To do so, define the function

v λ (x) = inf (γ,u)∈Γ x→ 0,∞ (e -λt dt)
+∞ 0 e -λt L(γ(t), u(t)) dt , (6.30) where

Γ x→ 0,∞ (e -λt dt) := (γ, u) ∈ L ∞ loc (0, ∞; R d ) × L 2 loc (0, ∞; R m ) : (γ, u) ∈ Γ x→ 0,T ∀ T > 0, and ∞ 0 e -λt |u(t)| 2 dt < ∞ .
Hereafter, we assume the following.

(L2') There exists a compact set K ⊂ R d such that min x∈K L(x, 0) = 0, and inf

x∈R d \K L(x, 0) > 0.
We recall that in view of Remark 5.10 assumption (L2') is not restrictive and, moreover, by Corollary 6.9 we have that α(L) = 0. Furthermore, (L2') stands for the corresponding of (L2) given so far. Note that v λ (x) ≥ 0 for any x ∈ R d . Then v λ is the continuous viscosity solution of (6.29). Proposition 6.11. Assume (F0), and (L0) -(L2'). Then, for any R ≥ 0 we have that:

(i) {λv λ } λ>0 is equibounded on B R ; (ii) there exists a constant C R ≥ 0 such that |v λ (x) -v λ (y)| ≤ C R d SR (x, y), ∀ x, y ∈ B R . ( 6
|v λ (x) -v λ (y)| ≤ C R c 2 |x -y| 1 r R ∀ x, y ∈ B R .
Proof of Proposition 6.11: Let R ≥ 0 and let x ∈ B R . Taking (γ, ū) ∈ Γ x→ 0,∞ (e -λt dt) such that (γ(t), ū(t)) ≡ (x, 0), by (L1) we get

λv λ (x) ≤ λ +∞ 0 e -λt L(x, 0) dt ≤ β(R) +∞ 0 λe -λt dt = β(R).
On the other hand, by (L2') we have that

λv λ (x) ≥ 0.
Thus, for any λ > 0 we conclude that

λ|v λ (x)| ≤ β(R), ∀ x ∈ B R .
In order to prove (ii), for any fixed x, y ∈ B R set δ = d SR (x, y). Let (γ y , ūy ) ∈ Γ y→x 0,δ be a solution of (2.5). Let (γ x , u x ) ∈ Γ x→ 0,+∞ (e -λt dt) be such that

∞ 0 e -λt L(γ x (t), u x (t)) dt ≤ v λ (x) + λ. Define a new control u y (t) = ūy (t), t ∈ [0, δ] u x (t -δ), t ∈ (δ, +∞), 134CHAPTER 6

. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

and so ( γ y , u y ) ∈ Γ y→ 0,∞ (e -λt dt). Then, we have that Therefore, setting C R = 2β(Λ(R)) we obtain (6.31) recalling that δ = d SR (x, y). Note that, the above proof fails for general control systems, i.e., of the form (6.1), under the assumption (LUGC) since, a priori, T R might not be of the order of |x -y|. Proof. First, we observe that, by an adaptation of [8, Theorem 5] (see Theorem 6.21 in Section 6.5.1), we have that

v λ (y) -v λ (x)
0 = lim T →+∞ 1 T V T (x) = lim λ→0 λv λ (x) (6.34)
locally uniformly in space. We recall that v λ (x) is a continuous viscosity solution of

λv λ (x) + H(x, Dv λ (x)) = 0 (x ∈ R d )
Since v λ (x * ) = 0, by Proposition 6.11 we deduce that {v λ } λ>0 is equibounded and equicontinuous. So, applying the Ascoli-Arzelá Theorem and a diagonal argument we deduce that there exists a sequence λ n ↓ 0 such that {v λn (x)} n∈N is locally uniformly convergent, i.e., for any R ≥ 0

lim n→∞ v λn (x) =: χ(x) uniformly on B R .
Hence, from (L2') we immediately deduce that χ(x) ≥ 0 and, again, since v λ (x * ) = 0 we get χ(x * ) = 0. Furthermore, from (6.31) we get (6.32). Finally, the stability of viscosity solutions ensures that χ is a solution of (6.33), which proves (ii). is called the critical (or, ergodic) Hamilton-Jacobi equation. A continuous function χ is called a critical subsolution (resp. supersolution) if it is a viscosity subsolution (resp. supersolution) of (6.35) and a critical solution if it is both a subsolution and a supersolution.

Representation formula

In this last section, we construct a critical solution that can be represented as the value function of a sub-Riemannian optimal control problem. Such a solution, which is useful to develop the Aubry-Mather theory in the sub-Riemannian case, will be obtained as the asymptotic limit as t → ∞ of the Lax-Oleinik semigroup, applied to χ given by Theorem 6.13. We begin by giving the definition of dominated functions.

136CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

Let us introduce, now, the following class of functions

S = φ ∈ C(R d ) : φ(x) ≥ 0 ∀ x ∈ R d , φ ≺ L
endowed with the topology induced by the uniform convergence on compact sets. Then, for any x ∈ R d , any t ≥ 0 and any φ ∈ S define the functional

F φ : Γ →x 0,t → R as F φ (γ, u) = φ(γ(0)) + t 0 L(γ(s), u(s)) ds
and

T t φ(x) = inf (γ,u)∈Γ →x 0,t φ(γ(0)) + t 0
L(γ(s), u(s)) ds . (6.36) Before proceeding to derive several properties of T t φ, including the fact that T t φ(x) ≥ 0, we first show that the class S is non-empty. Lemma 6.16. Assume (F0), and (L0) -(L2'). Then, the function χ constructed in Theorem 6.13 belongs to S.

Proof. Let χ be the critical solution given in Theorem 6.13, i.e.,

χ(x) = lim n→∞ v λn (x)
where the limit is uniform on compact subsets of R d . Recall that

v λ (x) = inf (γ,u)∈Γ x→ 0,∞ (e λt dt) ∞ 0 e -λt L(γ(t), u(t)) dt.
Next, we show that χ ∈ S. From Theorem 6.13 we know that χ(x) ≥ 0 for any x ∈ R d . Hence, we only need to prove that χ ≺ L. To do so, let R ≥ 0 and let x, y

∈ B R . Fix a, b ∈ R and let (γ, u) ∈ Γ x→y a,b . Let (γ y , u y ) ∈ Γ y→ 0,∞ (e -λt dt) be λ-optimal for v λ (y), that is, ∞ 0 e -λt L(γ y (t), u y (t)) dt ≤ v λ (y) + λ.
and define the control

u(t) = u(t + a), t ∈ [0, b -a] u y (t -a + b), t ∈ (b -a, ∞). 6.4. REPRESENTATION FORMULA 137 Then, ( γ, u) ∈ Γ x→ 0,∞ v λ (x) -v λ (y) ≤ ∞ 0 e -λt L( γ(t), u(t)) dt - ∞ 0 e -λt L(γ y (t), u y (t)) dt + λ = b-a 0 e -λt L(γ(t + a), u(t + a)) dt + ∞ b-a e -λt L(γ y (t), u y (t)) dt - ∞ 0 e -λt L(γ y (t), u y (t)) dt + λ ≤ b a L(γ(t), u(t)) dt + e -λ(b-a) -1 ∞ 0 e -λt L(γ y (t), u y (t)) dt + λ = b a L(γ(t), u(t)) dt -(b -a)λv λ (y)(1 + o(1)) + λ.
Therefore, since λ = λ n as n → ∞ from the previous estimate we get

χ(x) -χ(y) ≤ b a L(γ(t), u(t)) dt
which completes the proof. Theorem 6.17 (Lax-Oleinik semigroup). Assume (F0), and (L0) -(L2'). The following holds.

1. For any φ ∈ S there exists a function N φ : R d → R, which is bounded on compact sets, such that for any (t, x) ∈ [0, ∞) × R d there exists a trajectory-control pair (γ x , u x ) ∈ Γ →x 0,t such that F φ (γ x , u x ) ≤ N φ (x). (6.37)

2. For any φ ∈ S the following holds. For any R ≥ 0 there exists a nondecreasing function C φ : [0, ∞) → [0, ∞) such that for any (t, x) ∈ [0, ∞) × B R and any (γ, u) ∈ Γ →x 0,t satisfying (6.37) we have that

d SR (x, γ(0)) ≤ C φ (R) := β(R)D(R) + max x∈B R φ(x). ( 6
N φ (x) = φ(x * ) + D(|x|)β(|x|), (t, x) ∈ [D(|x|), ∞) × R d φ(x) + D(|x|)β(|x|), (t, x) ∈ [0, D(|x|)) × R d .
Note that, since φ ∈ S we deduce that N φ is bounded on any compact subset of R d . We now proceed with the first part of the proof, i.e., we show that for any

(t, x) ∈ [D(|x|), ∞) × R d there exists (γ x , u x ) ∈ Γ →x 0,t such that F φ (γ x , u x ) ≤ N φ (x).
Let (γ 0 , u 0 ) ∈ Γ x * →x 0,δ * (x) be optimal for (2.5) and define the control

u x (s) = 0, s ∈ [0, t -δ * (x)) u 0 (s -t + δ * (x)), s ∈ [t -δ * (x), t]. so that (γ x , u x ) ∈ Γ x * →x 0,t . Then F φ (γ x , u x ) = φ(x * ) + t t-δ * (x) L(γ 0 (s -t + δ * (x)), u 0 (s -t + δ * (x))) ds = φ(x * ) + δ * (x) 0 L(γ 0 (s), u 0 (s)) ds ≤ φ(x * ) + δ * (x) 0 L(γ 0 (s), u 0 (s)) ds
Let us estimate the rightmost term above. Recalling that |u 0 (s)| ≤ 1 for any s ∈ [0, δ * (x)] we have that

|γ 0 (t)| ≤ (|x * | + c f δ * (x))e c f δ * (x) =: Λ(|x|), ∀ t ∈ [0, δ * ].
Thus, we get

δ * (x) 0 L(γ 0 (s), u 0 (s)) ds ≤ δ * (x)β(Λ(|x|)) ≤ D(|x|)β(Λ(|x|)).
Hence, we obtain

F φ (γ, u) ≤ φ(x * ) + D(|x|)β(Λ(|x|)) (6.39)
which completes the proof of (1) for (t, x) ∈ [D(|x|), ∞) × R d . We now consider the case

(t, x) ∈ [0, D(|x|)) × R d . Let (γ x , u x ) ∈ Γ →x 0,t
be defined as

u x (s) = 0, γ x (s) ≡ x, s ∈ [0, D(|x|)).
Then

F φ (x, 0) ≤ φ(x) + tL(x, 0) ≤ φ(x) + D(|x|)β(|x|). (6.40)
This completes the proof of (1).

We proceed now with the proof of (2). In order to prove (6.38) we estimate from below F φ (γ, u), for any (γ, u) ∈ Γ →x 0,t satisfying (6.37), and then we combine such estimate with the definition of N φ (•). In view of (1) we also analyze two cases: first, we show that the conclusion holds for any (t, x) ∈ [D(|x|), ∞)×R d and then we do the same for

(t, x) ∈ [0, D(|x|))×R d . Let (t, x) ∈ [D(|x|), ∞) × R d
and let (γ, u) ∈ Γ →x 0,t satisfy (6.37). Then, by (6.6) we have that

F φ (γ, u) ≥ φ(γ(0)) + 1 2ℓ 1 t 0 |u(s)| 2 ds ≥ 1 2ℓ 1 d SR (x, γ(0)) 2 . ( 6
d SR (x, γ(0)) ≤ C φ (R) := β(R)D(R) + max x∈B R φ(x).
Now, given (t, x) ∈ [0, ∞)×R d , let φ ∈ S and let (γ, u) ∈ Γ →x 0,t be optimal for T t φ(x). Then, by definition we have that

T t φ(x) ≥ φ(γ(0)) + 1 2ℓ 1 t 0 |u(s)| 2 ds ≥ 0.
The existence of minimizing pairs (γ, u) ∈ Γ →x 0,t follows by classical results in optimal control theory (see, for instance, [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.4]). This 140CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS completes the proof of (3). Then, (4) is a direct consequence of the definition of T t φ.

In order to prove (5), from the previous point we know that for any (t, x) ∈ [0, ∞) × R d and any φ ∈ S we have that T t φ(x) ≥ 0. Moreover, the proof of the fact that T t φ(x) ≺ L and of the semigroup property, the latter being based on the dynamic programming principle, is similar to the proof of [42, (1) of Proposition 3.3] and [41, Proposition 4.6.2.] and will be omitted here.

We finally show [START_REF] Alvarez | Ergodicity, stabilization, and singular perturbations for bellman-isaacs equations[END_REF]. Let R ≥ 0, let x ∈ B R and let t ≥ 0. Let {φ n } n∈N ∈ S and let φ ∈ S be such that φ n → φ locally uniformly. Then, on the one hand, taking (γ φ x , u φ x ) ∈ Γ →x 0,t optimal for T t φ(x) we obtain

T t φ n (x) -T t φ(x) ≤ φ n (γ φ x (0)) -φ(γ φ x (0)).
Hence, from (2) we deduce that there exists a constant C φ (R) ≥ 0 such that

T t φ n (x) -T t φ(x) ≤ φ n (•) -φ(•) ∞,B Cφ(R) . (6.44)
Similarly, on the other hand, let (γ n x , u n x ) ∈ Γ →x 0,t optimal for T t φ n (x). Then, from (2) there exists a constant C φn (R) ≥ 0 such that

T t φ(x)-T t φ n (x) ≤ φ(γ n x (0))-φ n (γ n x (0)) ≤ φ n (•)-φ(•) ∞,B Cφ n (R) . (6.45)
Again in view of (2) by the locally uniform convergence of the sequence φ n the constant C φn (R) can be chosen uniform w.r.t. n ∈ N. Therefore, combining (6.44) and (6.45) the proof of ( 6) is complete. We call T t the Lax-Oleinik semigroup, adapted to the sub-Riemannian systems. Now, recall that we are interested in finding a critical solution χ such that

χ(x) = T t χ(x), ∀ t ≥ 0, ∀ x ∈ R d .
Hereafter, we take χ a critical solution in S (the existence of which is guaranteed by Theorem 6.13). We will show that T t χ(x) converges as the t → ∞ to the function χ we are looking for. Proof. In order to prove(i) we argue as in Theorem 6.17. Let R ≥ 0, let t ≥ 0 let x ∈ B R . Let (γ x , u x ) ∈ Γ →x 0,t be optimal for T t χ(x). Since χ is Lipschitz continuous w.r.t. d SR the following holds

χ(x) ≤ ℓ R d SR (x, 0), ∀ x ∈ B R .
(6.46)

REPRESENTATION FORMULA 141

Then, since χ ≥ 0 from (6.6) we obtain

T t χ(x) ≥ χ(γ(0)) + 1 2ℓ 1 t 0 |u x (s)| 2 ds = 1 2ℓ 1 d SR (x, γ x (0)) 2 (6.47)
which is bounded by (2) in Theorem 6.17. Then, on the one hand, if t ∈ [D(R), ∞) by (6.39) and (6.46) we obtain

T t χ(x) ≤ χ(x * ) + D(R)β(R) ≤ ℓ R K d SR (x * , 0) + D(R)β(R) (6.48)
where R K stands for the diameter of K. On the other hand, if t ∈ [0, D(R)) by (6.40) and (6.46) we get

T t χ(x) ≤ χ(x) + D(R)β(R) ≤ ℓ R d SR (x, 0) + D(R)β(R). (6.49)
Hence, combining (6.47) with (6.48) and, also, (6.47) with (6.49) the proof of (i) is complete. We proceed to show (ii), that is, the equicontinuity of T t χ(x) for t ≥ 1. Let R ≥ 0, let x, y ∈ B R and let t ≥ 1.

To begin with, assume that d SR (x, y) > 1. Then, we have that

|T t χ(x) -T t χ(y)| ≤ 2 T t χ ∞,B R ≤ 2 T t χ ∞,B R d SR (x, y).
We now consider the other case, i.e., d SR (x, y) ≤ 1. Let (γ 0 , u 0 ) ∈ Γ y→x 0,d SR (x,y) be optimal for (2.5) and let (γ y , u y ) ∈ Γ →y 0,t be optimal for T t χ(y). Then, define the control

u(s) = u y (s + d SR (x, y)), s ∈ [0, t -d SR (x, y)] u 0 (s -t + d SR (x, y)), s ∈ (t -d SR (x, y), t]
and call γ the corresponding trajectory, that is, ( γ, u) ∈ Γ →x 0,t . Note that u can be used to estimate T t χ(x) from above. We have that

T t χ(x) -T t χ(y) ≤ χ( γ(0)) -χ(γ y (0)) + t 0 L( γ(s), u(s)) ds - t 0 L(γ y (s), u y (s)) ds = χ(γ y (d SR (x, y))) -χ(γ y (0)) + t 0 L(γ y (s), u y (s)) ds - d SR (x,y) 0 L(γ y (s), u y (s)) ds + d SR (x,y) 0 L(γ 0 (s), u 0 (s)) ds - t 0 L(γ y (s), u y (s)) ds = χ(γ y (d SR (x, y))) -χ(γ y (0)) - d SR (x,y) 0 L(γ y (s), u y (s)) ds + d SR (x,y) 0 L(γ 0 (s), u 0 (s)) ds. (6.50)
We estimate first the integral terms. By (6.6) we immediately obtain

d SR (x,y) 0 L(γ y (s), u y (s)) ds ≥ 0. 142CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS Moreover, since u 0 ∞,[0,d SR (x,y)] ≤ 1 we have that |γ 0 (t)| ≤ (|x| + c f d SR (x, y))e c f d SR (x,y) , ∀ t ∈ [0, d SR (x, y)].
So, we get

d SR (x,y) 0 L(γ 0 (s), u 0 (s)) ds ≤ d SR (x, y) |x| + c f d SR (x, y) e c f d SR (x,y) .
Combining both inequalities we have that

- d SR (x,y) 0 L(γ y (s), u y (s)) ds + d SR (x,y) 0 L(γ 0 (s), u 0 (s)) ds ≤ d SR (x, y) |x| + c f d SR (x, y) e c f d SR (x,y) . (6.51)
Therefore, in order to obtain the results we need to estimate χ(γ y (d SR (x, y))) -χ(γ y (0)).

(6.52)

First, we claim that |γ y (0)| and |γ y (d SR (x, y))| are bounded. Indeed, observe that from (2) in Theorem 6.17 and the equivalence of the sub-Riemannian topology with the Euclidean one we deduce that

|γ y (0)| ≤ 2 max{R, C χ (R)}.
Moreover, by Lemma 6.2 we know that

|γ y (s)| ≤ κ( u y 2 , 1)(1 + |γ y (0)|), ∀ s ∈ [0, d SR (x, y)]. So, in particular, |γ y (d SR (x, y))| ≤ κ( u y 2 , 1)(1 + |γ y (0)|).
We claim that u y 2,[0,d SR (x,y)] is bounded by a constant that only depends on R. Indeed, from (i) we know that T t χ(y) is locally uniformly bounded and by (6.6) we know that

T t χ(y) ≥ χ(γ y (0)) + 1 2ℓ 1 t 0 |u y (s)| 2 ds ≥ 1 2ℓ 1 t 0 |u y (s)| 2 ds.
Thus, we obtain

1 2ℓ 1 t 0 |u y (s)| 2 ds ≤ T t χ(y) ∞,B R
and this completes the proof of the claim since T t χ is locally equibounded by (i). For simplicity of notation, let R y ≥ 0 be such that Then, by Corollary 2.7 we have that

|γ y (d SR (x, y))| ≤ R y , |γ y (0)| ≤ R y .
χ(γ y (d SR (x, y))) -χ(γ y (0)) ≤ c 2 |γ y (d SR (x, y)) -γ y (0)| 1 ry (6.53)
where c 2 depends only on R y . Next, from Lemma 6.2 we have that

|γ y (d SR (x, y))-γ y (0)| ≤ κ u y 2,[0,d SR (x,y)] , d SR (x, y) (1+|γ y (0)|)d SR (x, y) 1 2 .
Hence, we get that there exists a constant

C ′ R ≥ 0 such that χ(γ y (d SR (x, y)) -χ(γ y (0)) ≤ C ′ R d SR (x, y) 1 2ry . (6.54)
Therefore, combining (6.50), (6.51) and (6.54) we obtain

T t χ(x) -T t χ(y) ≤ d SR (x, y) |x| + c f d SR (x, y) e c f d SR (x,y) + C ′ R d SR (x, y) 1 2ry .
Finally, exchanging the role of x and y the proof of the equicontinuity is complete. uniformly on B R for any R ≥ 0. Moreover, we have that

χ(x) = T t χ(x), t ≥ 0, x ∈ R d and χ satisfies H(x, Dχ(x)) = 0, (x ∈ R d ) (6.56)
in the viscosity sense.

Proof. In order to prove the existence of the limit in (6.55), we first show that the map t → T t χ(x) is nondecreasing for any x ∈ R d . Indeed, we have that

T t χ(x) ≤ T t (T s χ(x)) = T t+s χ(x)
where the inequality holds since χ ≺ L. This implies that

T t χ(x) ≤ T t+s χ(x) 144CHAPTER 6

. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

and so we have that

T t χ(x) ≤ T t ′ χ(x), ∀ t ≤ t ′ .
Therefore, since by Proposition 6.19 we have that T t χ is locally equibounded it follows that the pointwise limit

lim t→∞ T t χ(x)
exists for all x ∈ R d . Moreover, again by Proposition 6.19 we know that the family T t χ is locally equicontinuous. Thus the above limit is locally uniform. Let us set

χ(x) = lim t→∞ T t χ(x), ∀ x ∈ R d
Next, we show that χ(x) = T t χ(x) for any x ∈ R d and any t ≥ 0. Indeed, let s ≥ 0. Then

T s χ(x) = lim t→∞ T s T t χ(x) = lim t→∞ T s+t χ(x)
where we have used the continuity of the semigroup T t and property (4) in Theorem 6.17. Hence, we get

T s χ(x) = lim t→∞ T s+t χ(x) = χ(x).
So, we have that

χ(x) = T t χ(x) = inf (γ,u)∈Γ →x 0,t χ(x) + t 0 L(γ(s), u(s)) ds (6.57)
The proof of the fact that from (6.57) the function χ solves (6.56) in the viscosity sense is similar to the proof of [42, Proposition 5.1, Proposition 5.2].

Appendix

Abelian-Tauberian Theorem

In this appendix, we give a new formulation of the Abelian-Tauberian Theorem, stated in [8, Theorem 5], tailored for the proof of Theorem 6.13. Theorem 6.21. Let ψ(t, x) be the solution of

∂ t ψ(t, x) + H(x, Dψ(t, x)) = 0, (t, x) ∈ [0, T ] × R d ψ(T, x) = 0, x ∈ R d .
For any λ > 0, let ψ λ (x) be the solution of

λψ(x) + H(x, Dψ(x)) = 0, x ∈ R d .
Then:

6.5. APPENDIX 145 (i) if {λψ λ (• )} λ>0 locally uniformly converges to a constant d ∈ R as λ ↓ 0, then { 1 T ψ(0, • )} T >0 locally uniformly converges to d as T → ∞; (ii) if { 1 T ψ(0, • )} T >0 locally uniformly converges to a constant d ∈ R as T → ∞, then {λψ λ (•)} λ>0 locally uniformly converges to d as λ ↓ 0.
This result can be proved arguing as in [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation[END_REF]Theorem 5] keeping in mind the following differences:

1. the uniform convergence on the full space Ω is replaced by the locally uniform convergence on R d ;

2. whenever the boundedness assumption on L is used in [START_REF] Arisawa | Ergodic problem for the Hamilton-Jacobi-Bellman equation[END_REF] one here has to invoke the boundedness of optimal pairs

(γ, u) in L ∞ (0, T ; R d ) × L 2 (0, T ; R m ). 146CHAPTER 6

. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS

Chapter 7

Aubry-Mather Theory for sub-Riemannian control systems

Settings and assumptions

For m ∈ N and i = 1, . . . , m, let

f i : R d → R d and u i : [0, ∞) → R
be smooth vector fields and measurable controls, respectively, and consider the following controlled dynamics of sub-Riemannian type

γ(t) = m i=1 f i (γ(t))u i (t) = F (γ(t))U (t), t ∈ [0, +∞) (7.1)
where

F (x) = [f 1 (x)| . . . |f m (x)
] is an d×m real matrix and U (t) = (u 1 (t), . . . , u m (t)) ⋆1 . For any s 0 , s 1 ∈ R such that s 0 < s 1 and x, y ∈ R d we set

Γ x→ s 0 ,s 1 = {(γ, u) ∈ AC([s 0 , s 1 ]; R d ) × L 2 (s 0 , s 1 ; R m ) : γ(t) = F (γ(t))u(t), γ(s 0 ) = x}, Γ →y s 0 ,s 1 = {(γ, u) ∈ AC([s 0 , s 1 ]; R d ) × L 2 (s 0 , s 1 ; R m ) : γ(t) = F (γ(t))u(t), γ(s 1 ) = y}, Γ x→y s 0 ,s 1 = Γ x→ s 0 ,s 1 ∩ Γ →y s 0 ,s 1 .
Throughout the paper we assume the vector fields f i to satisfy the following.

CHAPTER 7. ON THE AUBRY SET

(F0) There exists an integer r 0 ≥ 1 such that f i ∈ C r 0 -1 (R d ), for any i = 1, . . . , m, and

∆ r 0 (x) = R d , ∀x ∈ R d ;
(F1) There exists a constant c f ≥ 1 such that for any i = 1, . . . , m

|f i (x)| ≤ c f (1 + |x|), ∀ x ∈ R d ; (7.2) (F2) m ≤ d, F ∈ C 1,1 loc (R d ) and for any x ∈ R d the matrix F (x) has full rank m.
Observe that, from (F2) we have that the vector fields f i are linearly independent.

By (F1) and Gronwall inequality we get the following estimate on solutions of (7.1).

Lemma 7.1. Let x ∈ R d , t ≥ 0 and (γ, u) ∈ Γ x→ 0,t . If u ∈ L ∞ (0, t; R m ) then we have that |γ(s)| ≤ (|x| + c f u ∞ t)e c f ∥u∥∞t , ∀ s ∈ [0, t].
We now state the assumptions on the Lagrangian L :

R d × R m → R. (L0) L ∈ C 2 (R d × R m ) is reversible, that is L(x, u) = L(x, -u) for any (x, u) ∈ R d × R m ;
(L1) There exists a positive constant ℓ 1 , C 1 such that

D 2 u L(x, u) ≥ 1 ℓ 1 , (x, u) ∈ R d × R m |D x L(x, u)| ≤ C 1 (1 + |u| 2 ), (x, u) ∈ R d × R m ;
and L is locally semiconcave in space uniformly w.r.t. u ∈ R m .

(L2) There exists ℓ 2 ≥ 0 such that

L(x, u) ≤ ℓ 1 |u| 2 + ℓ 2 , (x, u) ∈ R d × R m (L3) There exists a compact set K L ⊂ R d and a constant δ L > 0 such that inf x∈R d \K L L(x, 0) ≥ δ L + min x∈K L L(x, 0); (7.3)
Note that by (L0), (L1) and (L3) we obtain

L(x, u) ≥ 1 2ℓ 1 |u| 2 + L(x * , 0), ∀ (x, u) ∈ R d × R m (7.4)

SETTINGS AND ASSUMPTIONS

149 where x * ∈ K L is such that

L(x * , 0) = min x∈K L L(x, 0).
Let H : R d × R d → R be the Hamiltonian associated with L, that is,

H(x, p) = sup u∈R m m i=1 u i p, f i (x) -L(x, u) , ∀ (x, p) ∈ R d × R d . (7.5)
Since the subject of this Chapter is deeply connected with the problem studied in Chapter 6, we first recall the main results of such Chapter. Consider the following minimization problem: for any T > 0 and any x ∈ R d to minimize T 0 L(γ(s), u(s)) ds over all (γ, u) ∈ Γ x→ 0,T (7.6) and define the function

V T : R d → R by V T (x) = inf (γ,u)∈Γ x→ 0,T T 0 L(γ(s), u(s)) ds, ∀ x ∈ R d . ( 7.7)
For any x ∈ R d we say that a trajectory-control pair (γ, u) ∈ Γ x→ 0,T is optimal if it solves (7.6). Note that, the existence of optimal trajectory-control pairs for (7.6) is a well-known result (see, e.g., [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]Theorem 7.4.4]).

Then, we know that for any R ≥ 0 there exist two constants P R , Q R ≥ 0 such that for any x ∈ B R , any T ≥ d SR (x, x * ), and any optimal pair (γ x , u x ) ∈ Γ x→ 0,T for (7.6) the following holds:

T 0 |u x (t)| 2 dt ≤ P R (7.8) and |γ x (t)| ≤ Q R , ∀ t ∈ [0, T ]. (7.9)
Moreover, there exists α(L) ∈ R such that for any R ≥ 0 Furthermore, such a solution che ne represented as

lim T →+∞ 1 T V T (x) = α(L), uniformly on B R ( 7
χ(x) = inf (γ,u)∈Γ →x 0,t χ(γ(0)) + t 0 L(γ(s), u(s)) ds -α(L) t (7.12) for any (t, x) ∈ [0, T ] × R d .
We say that α(L) is the critical constant for our problem and χ is any critical solution to the ergodic equation (7.11). Hereafter, we will always work with critical solutions that satisfy (7.12).

Differently from Chapter 6 here we also need to assume the following on the sub-Riemannian system: (S) there are no singular minimizing controls of problem (7.6).

The above extra assumption is needed for the critical solution χ to be more regular. Indeed, under (S) any critical solution is locally semiconcave and consequently locally Lipschitz continuous, see [ (ii) In view of the assumptions on L, we deduce that for any R ≥ 0 there exists a constant C R ≥ 0 such that

|H(x, p) -H(y, p)| ≤ C R (1 + |p|)|x -y|, ∀ x, y ∈ B R . (7.13)

Characterization of the ergodic constant

We begin by introducing a class of probability measures that adapts the notion of closed measures to sub-Riemannian control systems. Set

P 2 c (R d ×R m ) = µ ∈ P(R d × R m ) : R d ×R m |u| 2 µ(dx, du) < +∞, spt(π 1 ♯µ) compact
where π 1 : R d × R m → R d denotes the projection onto the first factor, i.e.

π 1 (x, u) = x. Recall that F (x) = [f 1 (x)| . . . |f m (x)]
is the real d × m matrix in (7.1).

Definition 7.3 (F -closed measure). We say that µ

∈ P 2 c (R d × R m ) is an F-closed measure if R d ×R m F ⋆ (x)Dφ(x), u µ(dx, du) = 0, ∀ φ ∈ C 1 (R d ).
We denote by C F the set of all F -closed measures.

Closed measures were first introduced in [START_REF] Fathi | Existence of c 1 critical subsolutions of the hamilton-jacobi equation[END_REF] in order to overcome the lack of regularity of the Lagrangian L. Indeed, if L is merely continuous, then there is no Euler flow and, consequently, it makes no sense to introduce invariant measure as in [START_REF] Fathi | Weak KAM Theorem and Lagrangian Dynamics[END_REF]. Similarly, in our setting such a flow does not exists and for this reason the use of closed measures turns out to be necessary. Moreover, as we will show in the next result, such measures collect the behavior of minimizing trajectories for (7.7) as the time horizon T goes to infinity. We now proceed to construct one closed measure that will be particularly useful to study the Aubry set.

Fix x 0 ∈ R d and for any T > 0 let the pair (γ x 0 , u x 0 ) ∈ Γ x 0 → 0,T be optimal for (7.6). Define the probability measure µ T x 0 by x 0 } T >0 is tight and there exists a sequence T n → ∞ such that µ Tn x 0 weakly- * converges to an F -closed measure µ ∞ x 0 .

R d ×R m φ(x, u) µ T x 0 (dx, du) = 1 T T 0 φ(γ x 0 (t), u x 0 (t)) dt, ∀ φ ∈ C b (R d ×R m ). ( 7
Proof. First, from (7.9) it follows that {π 1 ♯µ T x 0 } T >0 has compact support, uniformly in T . Thus, such a family of measures is tight. Let us prove that {π 2 ♯µ T } T >0 is also tight.

On the one hand, taking the null control we have that

1 T v T (x 0 ) ≤ 1 T T 0 L(x 0 , 0) ds ≤ ℓ 2 .
On the other hand, since (γ x 0 , u x 0 ) is a minimizing pair for V T (x 0 ), from (7.4) we get

1 T v T (x 0 ) = 1 T T 0 L(γ x 0 (t), u x 0 (t)) dt = R d ×R m L(x, u) µ T x 0 (dx, du) ≥ R d ×R m 1 2ℓ 1 |u| 2 + L(x * , 0) µ T x 0 (dx, du) which implies that 1 2ℓ 1 R d ×R m |u| 2 µ T x 0 (dx, du) ≤ ℓ 2 -L(x * , 0).
Consequently, the family of probability measures {π 2 ♯µ T x 0 } T >0 has bounded second order moment (w.r.t. T ). So, {π 2 ♯µ T x 0 } T >0 is tight. Since {π 1 ♯µ T x 0 } T >0 and {π 2 ♯µ T x 0 } T >0 are tight, so is {µ T x 0 } T >0 by [7, Theorem 5.2.2]. Therefore, by Prokhorov's Theorem there exists {T n } n∈N , with T n → ∞, and µ

∞ x 0 ∈ P 2 c (R d × R m ) such that µ Tn x 0 ⇀ * µ ∞ x 0 . We now show that µ ∞ x 0 is an F -closed measure, that is R d ×R m F ⋆ (x)Dψ(x), u µ ∞ x 0 (dx, du) = 0, ∀ ψ ∈ C 1 (R d ).

CHAPTER 7. ON THE AUBRY SET

By definition we have that

R d ×R m F ⋆ (x)Dψ(x), u µ Tn x 0 (dx, du) = 1 T n Tn 0 F ⋆ (γ x 0 (t)Dψ(γ x 0 (t)), u x 0 (t) dt = 1 T n Tn 0 Dψ(γ x 0 (t), γx 0 (t) dt = ψ(γ x 0 (T n )) -ψ(x 0 ) T n .
Then, from (7.9) we know that

γ x 0 (T n ) ∈ B Q |x 0 | . So, we get lim n→∞ ψ(γ x 0 (T n )) -ψ(x 0 ) T n = 0
and, consequently,

R d ×R m F ⋆ (x)Dψ(x), u µ ∞ x 0 (dx, du) = 0. □ Set P 2 R (R d × R m ) = µ ∈ P 2 c (R d × R m ) : spt(π 1 ♯µ) ⊂ B R .
The following property, which is interesting in its own right, will be crucial for the characterization of the critical constant derived in Theorem 7.8 below.

Proposition 7.5. Assume (F0) -(F2), (L0) -(L3) and (S).

Then, for any R ≥ Q 0 , where Q 0 is given in (7.9), we have that

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = -inf ψ∈C 1 (R d ) sup x∈B R H(x, Dψ(x)).
(7.15) Lemma 7.6. Assume (F0) -(F2), (L0) -(L3) and (S). Then, for any R ≥ Q 0 , where Q 0 is given in (7.9), we have that

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = inf µ∈P 2 R (R d ×R m ) sup ψ∈C 1 (R d ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du). (7.16)
The proof of the above lemma is based on an argument which is quite common in optimal transport theory see, for instance, [START_REF] Villani | Topics in optimal transportation[END_REF]Theorem 1.3]. We give the reasoning for the reader's convenience.

Proof. Since L is bounded below we have that

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = inf µ∈P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) + ω(µ)
where

ω(µ) = 0, µ ∈ C F +∞, µ ∈ C F .
So, observing that

ω(µ) = sup ψ∈C 1 (R d ) - R d ×R m F ⋆ (x)Dψ(x), u µ(dx, du)
we obtain (7.16).

Lemma 7.7. Let ϕ ∈ C(R d × R m ) be such that ϕ 0 ≤ ϕ(x, u) ≤ C ϕ (1 + |u| 2 ), ∀ (x, u) ∈ R d × R m for some constants ϕ 0 ∈ R and C ϕ ≥ 0. Let {µ j } j∈N ∈ P 2 (R d × R m ) and let µ ∈ P 2 (R d × R m ) be such that µ j ⇀ * µ as j → ∞. Then, we have that lim inf j→∞ R d ×R m ϕ(x, u) µ j (dx, du) ≥ R d ×R m ϕ(x, u) µ(dx, du). ( 7

.17)

Proof. We first prove (7.17) assuming that ϕ 0 = 0 and then we remove such a constraint. For any ε > 0 we have that

R d ×R m ϕ(x, u) µ T j x 0 (dx, du) = R d ×R m ϕ(x, u) 1 + ε |u| 2 (1 + ε |u| 2 ) µ T j x 0 (dx, du) ≥ R d ×R m ϕ(x, u) 1 + ε |u| 2 µ T j x 0 (dx, du).
From the growth assumption on ϕ we deduce that the function ϕ(x,u) 1+ε |u| 2 is bounded and so by weak- * convergence we get

lim inf j→+∞ R d ×R m ϕ(x, u) µ T j x 0 (dx, du) ≥ R d ×R m ϕ(x, u) 1 + ε |u| 2 µ ∞ x 0 (dx, du).
Therefore, as ε ↓ 0 we obtain (7.17).

CHAPTER 7. ON THE AUBRY SET

For ϕ 0 = 0, we have that

R d ×R m [(ϕ(x, u) -ϕ 0 ) + ϕ 0 ] µ T j x 0 (dx, du) = R d ×R m ϕ(x, u) -ϕ 0 1 + ε |u| 2 (1 + ε |u| 2 ) µ T j x 0 (dx, du) + ϕ 0 ≥ R d ×R m ϕ(x, u) -ϕ 0 1 + ε |u| 2 µ T j x 0 (dx, du) + ϕ 0 .
Thus, we obtain

lim inf j→+∞ R d ×R m ϕ(x, u) µ T j x 0 (dx, du) ≥ R d ×R m ϕ(x, u) -ϕ 0 1 + ε |u| 2 µ ∞ x 0 (dx, du) + ϕ 0
which in turn yields the result as ε ↓ 0.

Proof of Proposition 7.5. We divide the proof into two steps.

(

): Define F : C 1 (R d ) × P 2 R (R d × R m ) → R by F(ψ, µ) = R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du). 1
We will apply the Minimax Theorem ([64, Theorem A.1]) to prove that

inf µ∈P 2 R (R d ×R m ) sup ψ∈C 1 (R d ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du) = sup ψ∈C 1 (R d ) inf µ∈P 2 R (R d ×R m ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du).
In order to check that the hypothesis of such a theorem are satisfied, let us define

c * = 1 + L(x * , 0).
We claim that the level set

E := µ ∈ P 2 R (R d × R m ) : F(0, µ) ≤ c * is compact in P 2 R (R d × R m ), d 1 .
Indeed, for any given µ ∈ P 2 R (R d × R m ) we know that π 1 ♯µ has compact support contained in B R . Moreover, the coercivity of L implies that for any given µ ∈ E we have that π 2 ♯µ has bounded second moment which in turn yields the tightness of the family π 2 ♯µ for any

µ ∈ P 2 R (R d × R m ).
2 R (R d ×R m ) sup ψ∈C 1 (R d ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du) = sup ψ∈C 1 (R d ) inf µ∈P 2 R (R d ×R m ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du).

CHARACTERIZATION OF THE ERGODIC CONSTANT

155

(2): Proof of (7.15). By (7.16) we get

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = inf µ∈P 2 R (R d ×R m ) sup ψ∈C 1 (R d ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du) = sup ψ∈C 1 (R d ) inf µ∈P 2 R (R d ×R m ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du).
Now, the coercivity of L ensures the existence of the min

(x,u)∈B R ×R m L(x, u) -F ⋆ (x)Dψ(x), u .
Therefore, by taking a Dirac mass centered at any minimizer of the above function, one deduce that

sup ψ∈C 1 (R d ) inf µ∈P 2 R (R d ×R m ) R d ×R m L(x, u) -F ⋆ (x)Dψ(x), u µ(dx, du) sup ψ∈C 1 (R d ) min (x,u)∈B R ×R m L(x, u) -F ⋆ (x)Dψ(x), u . = sup ψ∈C 1 (R d ) - max (x,u)∈B R ×R m L(x, u) -F ⋆ (x)Dψ(x), u = -inf ψ∈C 1 (R d ) max (x,u)∈B R ×R m L(x, u) -F ⋆ (x)Dψ(x), u = -inf ψ∈C 1 (R d ) max x∈B R H(x, Dψ(x))
where the last equality holds true observing that max

(x,u)∈B R ×R m F ⋆ (x)Dψ(x), u -L(x, u) = max x∈B R sup u∈R m F ⋆ (x)Dψ(x), u -L(x, u) = sup x∈B R H(x, Dψ(x)).
This completes the proof.

The following characterization of the critical value is essential for the analysis in Section 7.3. Theorem 7.8. Assume (F0) -(F2), (L0) -(L3) and (S). Then, for any R ≥ Q 0 , where Q 0 is given in (7.9), we have that

α(L) = inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = inf µ∈C F R d ×R m L(x, u) µ(dx, du). ( 7
α(L) +H(x, Dχ ε (x)) ≤ κ R ε, ∀ x ∈ B R (7.19)
where χ ε (x) = χ ⋆ ξ ε (x) and ξ ε is a smooth mollifier.

Proof. From (S) we have that

χ belongs to W 1,∞ loc (R d ). So, α(L) +H(x, Dχ(x)) = 0, a.e. x ∈ R d . (7.20)
Let R ≥ 0 and let x 0 ∈ B R . Then, by Jensen's inequality we get

α(L) +H(x 0 , Dχ ε (x 0 )) = α(L) +H x 0 , R d Dχ(x 0 -y)ξ ε (y) dy ≤ R d α(L) +H(x 0 , Dχ(x 0 -y)) ξ ε (y) dy.
Moreover, writing

R d α(L) +H(x 0 , Dχ(x 0 -y)) ξ ε (y) dy = R d α(L) +H(x 0 -y, Dχ(x 0 -y)) ξ ε (y) dy I + R d H(x 0 , Dχ(x 0 -y)) -H(x 0 -y, Dχ(x 0 -y)) ξ ε (y) dy II , by (7
.20) we deduce that I = 0 and by (7.13) we get II ≤ κ R ε.

Proof of Theorem 7.8. We divide the proof into two steps.

Step 1: We first show that for any R ≥ Q 0 , where Q 0 is given in (7.9),

α(L) = inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du).
Indeed, by (7.10) we know that

α(L) = lim T →+∞ 1 T v T (0).
Hence, appealing to Lemma 7.7 and recalling that L(x, u) ≥ L(x * , 0) we obtain

α(L) = lim T →∞ R d ×R m L(x, u) µ T 0 (dx, du) ≥ R d ×R m L(x, u) µ ∞ 0 (dx, du).
(7.21)

CHARACTERIZATION OF THE ERGODIC CONSTANT

157 Recalling that µ ∞ 0 ∈ C F ∩ P 2 R (R d × R m ) for any R ≥ Q 0 , we deduce that α(L) ≥ inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du).
Next, by Proposition 7.5 we have that for any

ψ ∈ C 1 c (R d ) inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = -inf ψ∈C 1 (R d ) sup x∈B R H(x, Dψ(x)). (7.22)
Let χ be a critical solution. For ε ≥ 0 let χ ε (x) = χ ⋆ ξ ε (x), where ξ ε is a smooth mollifier. From Lemma 7.9 we know that for any R ≥ 0

α(L) +H(x, Dχ ε (x)) ≤ κ R ε, x ∈ B R .
Then, using χ ε in (7.22) we obtain

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) ≥ -sup x∈B R H(x, Dχ ε (x)) ≥ α(L) -κ R ε .
Hence, as ε ↓ 0 we get

inf µ∈C F ∩P 2 R (R d ×R m ) R d ×R m L(x, u) µ(dx, du) ≥ α(L)
and this completes the first step.

Step 2: Now we prove that

α(L) = inf µ∈C F R d ×R m L(x, u) µ(dx, du), that is, we remove the constraint µ ∈ P 2 R (R d × R m ). Let {µ j } j∈N ⊂ C F be such that lim j→∞ R d ×R m L(x, u) µ j (dx, du) = inf µ∈C F R d ×R m L(x, u) µ(dx, du). (7.23) Since µ j ∈ C F ⊂ P 2 c (R d × R m )
we deduce that there exists {R j } j∈N such that spt(µ j ) ⊂ B R j .

Moreover, without loss of generality, we can assume that for any j ∈ N where the inequality holds true by (7.4).

On the other hand we observe that the Dirac measure δ (x * ,0) is F -closed. So,

α(L) = inf µ∈C F R d ×R m L(x, u) µ(dx, du) ≤ R d ×R m
L(x, u) δ (x * ,0) (dx, du) = L(x * , 0).

(ii) Recalling Lemma 7.7 we obtain

α(L) = lim T →∞ R d ×R m L(x, u) µ T 0 (dx, du) ≥ R d ×R m L(x, u) µ ∞ 0 (dx, du).
Thus, the conclusion follows from Theorem 7.8 recalling that µ ∞ x 0 is F -closed by Proposition 7.4.

Aubry set

We denote by L * the Legendre Transform of L, that is, We denote by Cal(φ) the set of all calibrated curves for φ.

L * (x, p) = sup
For any t ≥ 0 and for any x, y ∈ R d we denote by A t (x, y) the action functional, also called fundamental solution of the critical equation, i.e., We note that φ ≺ L -α(L) if and only if for any x, y in R d and for any t ≥ 0 we have that φ(y) -φ(x) ≤ A t (x, y) -α(L) t. (7.25) Then, Peierls's barrier is defined as (i) For any x, y ∈ R d we have that 0 ≤ A ∞ (x, y) < ∞.

A ∞ (x, y) = lim inf
(ii) For any x, y, z ∈ R d we have that

A ∞ (x, z) ≤ A ∞ (x, y) + A ∞ (y, z) (7.27)
and, for any t ≥ 0 we have that A ∞ (x, z) ≤ A ∞ (x, y) + A t (y, z) -α(L) t. (7.28) Proof. Point (i) follow by (L2) and the reversibility of L, respectively. Point (ii) follows by similar arguments as in [START_REF] Fathi | Existence of c 1 critical subsolutions of the hamilton-jacobi equation[END_REF]. Then, there exists a subsequence, still denoted by (γ n , u n ), and a trajectorycontrol pair (γ, ū) ∈ Γ x→y 0,∞ such that (i) {u n } n∈N weakly converges to ū in L 2 on any compact subset of [0, ∞);

(ii) {γ n } n∈N uniformly converges to γ on every compact subset of [0, ∞).

Proof. From (7.26) it follows that there exists n ∈ N such that for any n ≥ n we have that Therefore, there exists a subsequence, still denoted by {u n }, that weakly converges to an admissible control ū in L 2 on any compact subset of [0, +∞). Moreover, let R ≥ 0 be such that |x| ≤ R. Then, by (7.9) for any t > 0 we have that Then, there exists a subsequence, still denoted by (γ n , u n ), and a trajectorycontrol pair (γ, ū) such that (i) {u n } n∈N weakly converges to ū in L 2 on any compact subset of (-∞, 0];

|γ n (s)| 2 ≤ Q R , ∀ s ∈ [0,
(ii) {γ n } n∈N uniformly converges to γ on every compact subset of (-∞, 0]. Then, from Remark 6.16 there exists (γ, ū) such that u n weakly converges to ū and γ n uniformly converges to γ, on every compact subset of (-∞, 0]. Let R ≥ 0 be such that |x| ≤ R. Fix t ∈ [0, ∞), take n ∈ N such that d n = d SR (γ(-t), γ n (-t)) ≤ 1 and t n > t + 1. Let (γ 0 , u 0 ) ∈ Γ γn(-t)→γ(-t) -t,-t+dn be optimal for (2.5) and let u n ∈ L 2 (-t n , -t + d n ) be given by

u n (s) = u n (s), s ∈ [-t n , -t] u 0 (s), s ∈ (-t, -t + d n ].
We denote by γ n the associated trajectory, that is, (

γ n , u n ) ∈ Γ x→γ(-t)
-tn,-t+dn . Then, defining the control u n (s) = u n (s -t n ), denoting by γ n the associated trajectory by (L2) and the fact that u 0 ∞ ≤ 1 we get By combining (7.31) and (7.32) we obtain (7.30). The fact that y → A ∞ (x, y) is a critical solution for any x ∈ R d can be proved by a standard argument which uses the dynamic programming principle.

Compactness of the Aubry set

In this section, we prove that the projected Aubry set A is a compact subset of R d . We begin with some preliminaries. Proof. We proceed by contradiction,. Suppose that there exist x 0 ∈ R d , {t k } k∈N with t k → ∞, and a sequence of optimal pairs (γ k , u k ) ∈ Γ x 0 → 0,t k of (7.6) such that L 1 ({s ∈ [0, t k ] : γ k (s) ∈ K L }) = 0.

On the one hand, we have that On the other hand, having fixed any optimal pair (γ 0 , u 0 ) ∈ Γ x 0 →x * 0,δ(x 0 ) for (2.5) and for any k ∈ N such that t k > δ(x 0 ) define the control With δ L > 0. Taking the limit as k → ∞ in the above inequalities yields δ L ≤ 0 which is a contradiction.

u k (s) = u 0 (
In view of the reversibility of L, the above Lemma implies the following. We observe that since calibrated curves are, in particular, minimizing trajectories for (7.6) then Corollary 7.19 can be applied to such curves. This is a key point to deduce that the projected Aubry set is bounded, as we show below. Proof. Let x 0 ∈ R d be such that A ∞ (x 0 , x 0 ) = 0. By Proposition 7.17 there exists (γ, ū) ∈ Γ →x 0 -∞,0 such that γ is a calibrated curve for A ∞ (x 0 , • ) and, by Corollary 7.19, we know that there exists t 0 ∈ (-∞, 0] such that γ(t 0 ) ∈ K L .

Thus, the the trajectory γ associated with the control u(s) = ū(s + t 0 ), for s ∈ [0, -t 0 ], with γ(-t 0 ) = x 0 , is a calibrated curve for A ∞ (x 0 , • ) such that γ(0) ∈ K L . Then, from (7.9) this implies that there exists R L ≥ 0 such that x 0 ∈ B R L .

Next, we show that the projected Aubry set is closed.

Horizontal regularity of critical solutions

In this section we show that any critical solution is differentiable along the range of F , see the definition below, at any point lying on the projected Aubry set. Clearly, if ψ is Frechét differentiable at x, than ψ is differentiable along the range of F (x) and q x = F ⋆ (x)Dψ(x). Proof. We first prove that if D + F ψ(x) is a singleton then ψ is differentiable at x along the range of F (x). Let {q x } = D + F ψ(x) and take p x ∈ D + ψ(x). Then

ψ(x + F (x)v) -ψ(x) -p x , F (x)v ≤ o(|F (x)v|) ≤ o(|v|).
Therefore, we deduce that lim sup v→0 ψ(x + F (x)v) -ψ(x) -q x , v |v| ≤ 0.

In order to prove the reverse inequality for the lim inf, let {v k } k∈N be any sequence such that v k = 0, v k → 0 as k → +∞ and let

p k ∈ D + ψ(x + F (x)v k ).
Then

1 |v k | ψ(x + F (x)v k ) -ψ(x) -p x , F (x)v k = 1 |v k | ψ(x + F (x)v k ) -ψ(x) -p k , F (x)v k + p k -p x , F (x)v k ≥ 1 |v k | o(|F (x)v k |) -|F * (x)p k -q x ||v k |.
By the upper-semicontinuity of D + ψ we have that |F * (x)p k -q x | → 0 as k ↑ ∞. Since since this is true for any sequence v k → 0, we conclude that lim inf v→0 ψ(x + F (x)v) -ψ(x) -q x , v |v| ≥ 0.

We now prove that, if ψ is differentiable along the range of F (x), then D + F ψ(x) is a singleton. To do so, let p ∈ D + ψ(x) and let q x ∈ R m be as in (7.35). Then, we know that lim h↓0 ψ(x + hF (x)θ) -ψ(x) h ≥ q x , θ . Therefore, q x , θ ≤ F * p, θ , ∀ θ ∈ R d .

Thus F * (x)p = q x .

Hereafter, the vector q x given in Definition 7.23 will be called the horizontal differential of ψ at x ∈ R d and will be denoted by D F ψ(x).

The next two propositions ensure that any critical solution χ is differentiable along the range of F at any point lying on a calibrated curve γ. The proof consists of showing that D + F χ is a singleton on γ. We recall that

L * (x, p) = sup v∈R d p, v -L(x, v)
is the Legendre Transform of L. We will rather write the critical equation using L * , instead of the Hamiltonian H, to underline the role of horizontal differentiability. for all τ > 0.

Proof. On the one hand, since χ is a subsolution of (7.11) we have that α(L) +H(γ(τ ), p) ≤ 0, ∀ p ∈ D + χ(γ(τ )).

So, from (7.24) and recalling that D + F χ(x) = F * (x)D + χ(x) for any x ∈ R d we get α(L) +L * (γ(τ ), p) ≤ 0, ∀ p ∈ D + F χ(γ(τ )). Thus, it is enough to prove the reverse inequality.

Let h ≥ 0, then since γ is a calibrated curve for χ we have that Proof. We recall that from [25, Theorem 1] we have that χ is semiconcave. By Proposition 7.25 we know that α(L) +L * (γ(τ ), p) = 0 for any p ∈ D + F χ(γ(τ )). Moreover, we have that L * (x, •) is strictly convex and the set D + F χ(x) is convex. Therefore, the above equality implies that D + F χ(γ(τ )) is a singleton. Consequently, Lemma 7.24 ensure that χ is differentiable at γ(τ ) along the range of F (γ(τ )).

χ(γ(τ )) -χ(γ(τ -h)) =
We are now ready to prove the differentiability of any critical solution on the Aubry set. Then, by Remark 6.16 there exists (γ - x , u - x ) such that u - n weakly converges to u - x and γ - n uniformly converges to γ - x , on every compact subset of [0, ∞), respectively. Therefore, defining the control

u x (s) = u + x (s), s ∈ [0, ∞) u - x (s), s ∈ (-∞, 0]
and denoting by γ x the associated trajectory the proof of (i) is complete. Next, we prove (ii), that is, γ x is a calibrated curve for χ. From (7.12) we know that χ ≺ L -α(L) and thus for any t ≥ 0 the following hold χ(γ x (t)) -χ(x) ≤ t 0 L(γ x (s), u x (s)) ds -α(L) t.

Moreover, again from (7.12) we deduce that χ(x) -χ(γ x (t)) ≤ A s (γ x (t), x) -α(L) s for any s ≥ 0. Thus, we get χ(x) -χ(γ x (t)) ≤ A ∞ (γ x (t), x) = -t 0 L(γ x (s), u x (s)) ds + α(L) t. This proves that γ x is a calibrated curve for χ on [0, ∞). Similarly, one can prove that the same holds on (-∞, 0]. Moreover, if we consider -s < 0 < t we can write Finally, by (ii) and Proposition 7.26 we deduce that χ is differentiable at x ∈ A along the range of F (x). Proposition 7.28. Assume (F0) -(F2), (L0) -(L3) and (S). Let χ be a critical solution. Let x ∈ A and let (γ x , u x ) be such that γ x is a calibrated curve for χ on R with γ x (0) = x. Then, we have that γ x (t) ∈ A, t ≥ 0.

(7.40)

Proof. In order to prove (7.40)

Example 2 . 5 .

 25 The following are two well-known examples of sub-Riemannian systems for which Chow's condition holds true.

Proposition 3 . 22 .

 322 Assume (L1) -(L3). Let x ∈ R d and η ∈ P Lip(P 1 ) m 0

  satisfies the following assumptions: (F1') F is globally continuous with respect to all the variables;

66CHAPTER 4 .

 4 ERGODIC BEHAVIOR OF ACCELERATION MODEL Proposition 4.16. Assume that F satisfies (F1) -(F3). Then, we have: (i) {δV δ (x, v)} δ>0 is locally uniformly bounded;

(4. 31 )

 31 By Remark 4.20 above, we can test the fact that a measure is closed by smooth and globally Lipschitz continuous maps. Let E(T d × R d ) be such a 72CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL set. Then

  (4.34) Passing to the limit as R → +∞ in (4.27) proves the Lemma thanks to (4.33) and (4.34).

(4. 41 )78CHAPTER 4 .

 414 ERGODIC BEHAVIOR OF ACCELERATION MODELIn order to compare the solution of this time-dependent problem with the solution of the ergodic MFG problem, which is written in terms of closed measures, we need to rewrite the time-dependent problem in term of flows of Borel probability measures onT d × R d × R d. The following definition mirrors the definition of closed measure in the ergodic setting: Definition 4.24 (T-Closed measures). Let T be a finite time horizon and let m 0

4. 3 .

 3 ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 79 Lemma 4.26 (Approximate problem 2). Let f : Td × R d × [0, T ] → Rbe a continuous map with at most a polynomial growth and which is locally Lipschitz continuous in space locally uniformly in time and g : T d × R d → R be a locally Lipschitz continuous map with at most a polynomial growth. Let R > 0 and let ξ R be a smooth cut-off function such that

  we have, by the definition of a T -closed measure in eq. (4.42),

Proposition 4 . 29 (

 429 Convergence of MFG solution). Assume that F satisfies (F1'), (F2'), (F3') with α = 2 and the monotonicity condition (4.6), that g satisfies (G1) and that the initial distribution m 0 in eq. (4.4) belongs to P 2 (T d × R d ). Let (u T , m T ) be a solution of the MFG system4.3. ASYMPTOTIC BEHAVIOR OF MFG WITH ACCELERATION 87eq. (4.4) and let ( λ, μ) be the solution of the ergodic MFG problem eq. (4.35).

Lemma 4 . 31 ( 1 0 and m 2 0

 43112 Linking two measures by a T -closed measure). Let m belong to P 2 (T d × R d ). Then, there exists µ m 1 0 →m 2 0

It is always true that inf b∈B sup a∈A L

 a∈A (a, b) ≥ sup a∈A inf b∈B L(a, b).

98CHAPTER 4 .

 4 ERGODIC BEHAVIOR OF ACCELERATION MODEL Theorem 4.34 ([64]). Assume that there exists a * ∈ A and c * > sup a∈A inf b∈B L(a, b) such that B * := {b ∈ B : L(a * , b) ≤ c * } is not empty and compact in B, and that b → L(a, b) is lower semicontinuous in B * for every a ∈ A. Then min b∈B sup a∈A L(a, b) = sup a∈A inf b∈B L(a, b).

Lemma 5 . 15 .

 515 Assume (M1) -(M3) and (BC). Let R ≥ 0 and let

0 L

 0 x (s), u x (s)) ds ≥ δL(x * , 0).Then, since ūy ∞ ≤ 1 and δ ≤ c(R), by Lemma 6.1 we have that|γ y (t)| ≤ (R + c f c(R))e c f c(R) =: Λ(R), ∀ t ∈ [0, δ].(6.22)6.3. LONG-TIME AVERAGE AND ERGODIC CONSTANT129 Thus, by (L1) we obtain δ (γ y (s), ūy (s)) ds ≤ 2δβ(Λ(R)).

Theorem 6 . 8 (

 68 Existence of the critical constant). Assume (F0), and (L0) -(L2).There exists a constant α(L) ∈ R, called the critical constant (or Mañé's critical value), such that lim x) -α(L) = 0, ∀ R > 0.(6.23)

≤ δ 0 e 0 e 0 e 0 e 0 e

 00000 -λt L(γ y (t), ūy (t)) dt+ +∞ δ e -λt L(γ x (t -δ), u x (t -δ)) dt -+∞ -λt L(γ x (t), u x (t)) dt + λ = δ -λt L(γ y (t), ūy (t)) dt + (e -λδ -1) +∞ 0 e -λs L(γ x (s), u x (s)) ds + λ = δ -λt L(γ y (t), ūy (t)) dt + -δλ + o(δλ) (v λ (x) + λ) + λ where lim q→0 o(q) q = 0.By point (i) we have that δλv λ (x) ≤ δβ(R) and for λ ≤ 1 we obtain o(δλ) ≤ δ. Moreover, by Lemma 6.1 we know that|γ y (t)| ≤ (|y| + c f δ)e c f δ =: Λ(R), ∀ t ∈ [0, δ] since ūy ∞,[0,δ] ≤ 1. Thus, by(L1) we deduce that δ -λt L(γ y (t), ūy (t)) dt ≤ δ 0 β(|γ y (t)|)(1 + |ū y (t)| 2 ) dt ≤ 2δβ(Λ(R)).

Theorem 6 . 13 (

 613 Existence of correctors). Assume (F0), and (L0) -(L2'). Then there exists a continuous function χ : R d → R and a sequence λ n ↓ 0 such that, for any R ≥ 0,lim n→∞ v λn (x) = χ(x), uniformly on B R .Moreover, we have that:(i) χ(x) ≥ 0, χ(x * ) = 0 and χ is locally Lipschitz continuous w.r.t. d SR ,that is, for any R ≥ 0 there exists a constant ℓ R ≥ 0 such that|χ(x) -χ(y)| ≤ ℓ R d SR (x, y), ∀ x, y ∈ B R . (6.32) (ii) χ is a viscosity solution of the ergodic Hamilton-Jacobi equation H(x, Dχ(x)) = 0 (x ∈ R d ). (6.33) 6.4. REPRESENTATION FORMULA 135

Definition 6 . 14 (

 614 Critical equation and critical solutions). The equationH(x, Dχ(x)) = 0 (x ∈ R d ) (6.35)

Definition 6 . 15 (

 615 Dominated functions). Let a, b ∈ R such that a < b and let x, y ∈ R d . Let ϕ be a continuous function on R d . We say that ϕ is dominated by L -c, and we denote this by ϕ ≺ L -c, if for any trajectorycontrol pair (γ, u) ∈ Γ x→y a,b we have that ϕ(y) -ϕ(x) ≤ b a L(γ(s), u(s)) ds -c (b -a).

Moreover, we denote by r y ≥ 1

 1 the degree of nonholonomy associated with the compact set B Ry . Hence, going back to by the Lipschitz continuity of χ w.r.t. d SR we get χ(γ y (d SR (x, y))) -χ(γ y (0)) ≤ ℓ Ry d SR (γ y (d SR (x, y)), γ y (0)).

Theorem 6 . 20 .

 620 Assume (F0), and (L0) -(L2'). Then, there exists a continuous function χ such that lim t→∞ T t χ(x) = χ(x) (6.55)

. 10 )

 10 and a continuous viscosity solution χ : R d → R of the ergodic Hamilton-Jacobi equationα(L) +H(x, Dχ(x)) = 0, x ∈ R d . (7.11)

25 ,

 25 Theorem 1]. Remark 7.2. (i) Observe that the sub-Riemannian systems in Example 2.5 fit assumption (S), see for instance [25, Theorem 5.1].

. 14 )Proposition 7 . 4 .

 1474 Then, we have the following. Assume (F0) -(F2), (L0) -(L3) and (S). Then, {µ T

. 18 ) 7 . 9 .

 1879 Lemma Assume (F0) -(F2), (L0) -(L3) and (S).Let χ be a critical solution. Then, for any R ≥ 0 there exists a constant κ R ≥ 0 such that for any ε > 0

inf µ∈C F ∩P 2 Corollary 7 . 10 .

 2710 R j (R d ×R m ) R d ×R m L(x, u) µ(dx, du) = R d ×R m L(x, u) µ j (dx, du).158 CHAPTER 7. ON THE AUBRY SETSince, for j sufficiently large, we have proved thatα(L) = R d ×R m L(x, u) µ j (dx, du)the conclusion follows from(7.23). Assume (F0) -(F2), (L0) -(L3) and (S). Then the following holds.(i) α(L) = L(x * , 0) = min x∈K L L(x, 0). (ii) For any x 0 ∈ R d we have that α(L) = R d ×R m L(x, u) µ ∞ x 0 (dx, du)where µ ∞ x 0 is given in Proposition 7.4.Remark 7.11. Note that point (i) of the conclusion has beed already proved in[START_REF] Cannarsa | Asymptotic analysis for hamiltonjacobi equations associated with sub-riemannian control systems[END_REF] Corollary 5.4]. Here we propose a different approach which relies on (7.18).Proof. (i)On the one hand, by Theorem 7.8, we have thatα(L) = inf µ∈C F R d ×R mL(x, u) µ(dx, du) ≥ L(x * , 0)

Definition 7 . 12 ( 1 .

 7121 u∈R m p, v -L(x, u) ,7.3. AUBRY SET159 and we observe thatH(x, p) = L * (x, F * (x)p), (x, p) ∈ R d × R d . (7.24) Moreover, since L satisfies (L0) -(L2) we know that L * is coercive and strictly convex in p. Dominated functions and calibrated curves). Let c ∈ R and let φ be a continuous function on R d . We say that φ is dominated by L -c and we denote this by φ ≺ L -c, if for any a, b ∈ R, with a < b, and any trajectory-control pair (γ, u) ∈ Γ γ(a)→γ(b) a,b we have that φ(γ(b)) -φ(γ(a)) ≤ b a L(γ(s), u(s)) ds -c (b -a). 2. We say that the first component γ : [a, b] → R d , with a, b ∈ R and a < b, of a trajectory-control pair (γ, u) ∈ Γ γ(a)→γ(b) a,b is a calibrated curve for φ if φ(γ(b)) -φ(γ(a)) = b a L(γ(s), u(s)) ds -c (b -a).

A,t t 0 L

 0 t (x, y) = inf (γ,u)∈Γ x→y 0(γ(s), u(s)) ds .

. 26 ) 7 . 13 .

 26713 t→∞ A t (x, y) -α(L) t , x, y ∈ R d . (7LemmaThe following properties hold.

Definition 7 . 14 (Lemma 7 . 15 .

 714715 Projected Aubry set). The projected Aubry set A is defined byA = x ∈ R d : A ∞ (x, x) = 0 . Assume (F0) -(F2), (L0) -(L3) and (S). Let (x, y) ∈ R 2d be such that h := A ∞ (x, y) ∈ R. Let {t n } n∈N ∈ R and (γ n , u n ) ∈ Γ x→y 0,tn be such that t n → +∞ and lim n→+∞ tn 0 L(γ un (s), u n (s)) ds -α(L) t n = h. (7.29)

tn 0 L

 0 (γ n (s), u n (s)) ds -α(L) t n ≤ h + 1.On the other hand, by (L2) we obtaintn 0 L(γ n (s), u n (s)) ds -α(L) t n ≥ 1 2ℓ 1 tn 0 |u n (s)| 2 ds -(L(x * , 0) + α(L))t n .Appealing to (i) in Corollary 7.10 we have that L(x * , 0) + α(L) = 0. So, tn 0 |u n (s)| 2 ds ≤ 2ℓ 1 (h + 1), ∀n ≥ n.

Proposition 7 . 17 .

 717 Assume (F0) -(F2), (L0) -(L3) and (S). For eachx, y ∈ R d there exists (γ, ū) ∈ Γ →y -∞,0 such that A ∞ (x, y) -A ∞ (x, γ(-t)) = 0 -t L(γ(s), ū(s)) ds -α(L) t, ∀ t ≥ 0. (7.30) Moreover, for each x ∈ R d the map y → A ∞ (x,y) is a critical solution on R d . Proof. Fix x, y ∈ R d and let {t n } n∈N , (γ n , u n ) ∈ Γ x→y -tn,0 be such that t n → ∞, and lim n→∞ 0 -tn L(γ n (s), u n (s)) ds -α(L) t n = A ∞ (x, y).

A 0 L

 0 tn-t+dn (x, γ(-t)) -α(L)(t n -t + d n ) ≤ tn-t+dn 0 L( γ n (s), u n (s)) ds -α(L)(t n -t + d n ) n (s), u n (s)) ds + -t+dn -t L(γ 0 (s), u 0 (s)) ds -α(L)(t n -t + d n ) n (s), u n (s)) ds + (ℓ 2 + ℓ 1 -α(L))d n -α(L)(t n -t).Hence, from the lower-semicontinuity of the action we obtainA ∞ (x, γ(-t)) + 0 -t L(γ(s), ū(s)) ds -α(L) t ≤ lim inf n→+∞ A tn-t+dn (x, γ(-t)) -α(L)(t n -t -d n ) + 0 -t L(γ(s), ū(s)) ds -α(L) t ≤ lim inf n→+∞ (ℓ 2 + ℓ 1 -α(L))d n + -t -tn L(γ n (s), u n (s)) ds -α(L)(t n -t) n (s), u n (s)) ds -α(L) t .By combining together the terms inside the brackets we getA ∞ (x, γ(-t)) + 0 -t L(γ(s), ū(s)) ds -α(L) t ≤ lim inf n→+∞ (ℓ 2 + ℓ 1α(L))d n + 0 -tn L(γ n (s), u n (s)) ds -α(L) t n = A ∞ (x, y).Therefore, we obtainA ∞ (x, y) -A ∞ (x, γ(-t)) ≥ 0 -t L(γ(s), ū(s)) ds -α(L) t.(7.31)7.3. AUBRY SET 163Next, we claim thatA ∞ (x, y) -A ∞ (x, γ(-t)) ≤ 0 -t L(γ(s), ū(s)) ds -α(L) t. (7.32)Indeed, by(7.28) we have thatA ∞ (x, y) -A ∞ (x, γ(-t)) ≤ A t (γ(-t), y) -α(L) t.Hence, defining the control u(s) = ū(s -t), s ≥ 0 and denoting by γ the associated trajectory, we deduce thatA ∞ (x, y) -A ∞ (x,γ(-t)) ≤ A t (γ(-t), y) -α(L) t ≤ t ( γ(s), u(s)) ds -α(L) t = 0 -t L(γ(s), ū(s)) ds -α(L) t.

Proposition 7 . 18 .

 718 Assume (F0) -(F2), (L0) -(L3) and (S). For anyx ∈ R d there exists T x ≥ 0 such that, for any t ≥ T x , any optimal pair (γ x , u x ) ∈ Γ x→ 0,t for (7.6) satisfiesL 1 ({s ∈ [0, t] : γ x (s) ∈ K L }) > 0.

t k 0 L

 0 (γ k (s), u k (s)) ds > t k inf

Corollary 7 . 19 .

 719 Assume (F0) -(F2), (L0) -(L3) and (S). For anyx ∈ R d there exists T x ≥ 0 such that for any t ≥ T x , any optimal pair (γ x , u x ) ∈ Γ →x -t,0 for problem (7.6) we have thatL 1 ({s ∈ [-t, 0] : γ x (s) ∈ K L }) > 0.

Proposition 7 . 20 .

 720 Assume (F0) -(F2), (L0) -(L3), and (S).Then A is bounded.

Definition 7 . 23 (

 723 Horizontal differentiability). We say that a continuous function ψ on R d is differentiable at x ∈ R d along the range of F (x) (or, horizontally differentiable at x) if there exists q x ∈ R m such thatlim v→0 ψ(x + F (x)v) -ψ(x) -q x , v |v| = 0. (7.35)

  For any ψ ∈ C(R d ) we set D + F ψ(x) = F * (x)D + ψ(x). Lemma 7.24. Assume (F0) -(F2), (L0) -(L3) and (S). Let ψ ∈ C(R d ) be locally semiconcave. Then, ψ is differentiable at x ∈ R d along the range of F (x) if and only if D + F ψ(x) = {q x }.

7. 4 .

 4 HORIZONTAL REGULARITY OF CRITICAL SOLUTIONS 167 Moreover, by definition we have that for any θ ∈ R d lim h↓0 ψ(x + hF (x)θ) -ψ(x) h ≤ F * p, θ .

Proposition 7 . 25 .

 725 Assume (F0) -(F2), (L0) -(L3) and (S). Let χ be a critical subsolution and let (γ, u) be such that γ : [0, ∞) → R d is calibrated for χ. Then we have that α(L) +L * (γ(τ ), p) = 0, ∀ p ∈ D + F χ(γ(τ ))

FL

  s), u(s)) ds -α(L) h.Then, by the definition of super-differential we getχ(γ(τ )) -χ(γ(τ -h)) ≤ p, γ(τ ) -γ(τ -h) + o(h) = p, τ τ -h γ(s) ds + o(h) = τ τ -h F * (γ(s))p, u(s) ds + o(h) 168 CHAPTER 7. ON THE AUBRY SET Therefore, we conclude that τ τ -h L(γ(s), u(s)) ds -α(L) h ≤ τ τ -h F * (γ(s))p, u(s) ds + o(h) * (γ(s))p, u(s) -L(γ(s), u(s)) ds + o(1) * (γ(s), F * (γ(s))p) ds + o(1).Thus, for h → 0 we obtain the conclusion.Proposition 7.26. Assume (F0) -(F2), (L0) -(L3) and (S).Let χ be a critical solution and let (γ, u) be such that γ : [0, ∞) → R d is calibrated for χ. Then, for any τ > 0 we have that χ is differentiable at γ(τ ) along the range of F (γ(τ )).

Theorem 7 . 27 ( 0 L

 7270 Horizontal differentiability on the Aubry set). Assume (F0) -(F2), (L0) -(L3) and (S). Let χ be a critical solution. Then, the following holds.(I) For any x ∈ A there exists a trajectory-control pair(γ x , u x ) ∈ Γ →x -∞,0 ∩ Γ x→ 0,∞ such that A ∞ (γ x (t), x) = -t (γ x (s), u x (s)) ds + α(L) t (7.36)andA ∞ (x, γ x (-t)) = -0 -t L(γ x (s), u x (s)) ds + α(L) t (7.37)(ii) γ x : R → R d is calibrated for χ.

0 L

 0 χ(γ x (t)) -χ(γ x (-s)) = χ(γ x (t)) -χ(x) + χ(x) -χ(γ x (-s)) = t (γ x (τ ), u x (τ )) dτ -α(L) t + 0 -s L(γ x (τ ), u x (τ )) dτ -α(L) s = t -s L(γ x (τ ), u x (τ )) dτ -α(L)(t + s),and this completes the proof of (ii).

Theorem 2.2 (Compactness and convergence). A

  set K ⊂ P p (X) is relatively compact if and only if it is p-uniformly integrable and tight. Moreover, for a given sequence {µ i } i∈N ⊂ P p (X) we have that if and only if µ i narrowly converge to µ and {µ i } i∈N has uniformly integral p-moment.

	Remark 2.1. Notice that, if
	0 < p < p 1 , and sup
	µ∈K X
	lim i→∞	d p (µ i , µ) = 0
	Theorem 2.3. Let r ≥ p > 0 and let K

d(x, x

) p 1 µ(dx) < +∞, then K has uniformly integrable p-moment.

  projected Mather set. See [41, Theorem 4.12.3] for the following result. If u is dominated by L + c(H), then it is differentiable at every point of the projected Mather set M

	Theorem 2.16.

Lemma 3.4. Assume (L1) -(L3). Then

	Corollary 3.3. Assume (L1) -(L3). Then, there exists a constant C 1 ≥ 0
	such that for any x ∈ R d , any m ∈ C([0, T ]; P 1 (R d )) and any optimal
	control u * of (3.2) we have that
		γ *	∞ ≤ C 1 (1 + |x|)
	where γ Hence,
			t
		|γ * (t)| ≤ e T ∥A∥ |x| + B	|u * (s)| ds
			0
	and by Hölder's inequality
		|γ * (t)| ≤ e T ∥A∥ |x| + B T	1 2 u *	2 .	□
			, there exists a constant C 2 > 0
	such that for any x ∈ R d , any m ∈ C([0, T ]; P 1 (R d )) and any u * optimal
	control for (3.2) we have that
		γ *	2 ≤ C 2 (1 + |x|)
	where γ * is the trajectory associated with u * . Moreover, the family of mini-
	mizing trajectories Γ * (x) is uniformly Hölder continuous.
	Proof. From Proposition 3.2 and Corollary 3.3 there holds
	γ *	2

* is the trajectory associated with u * . Proof. Since γ * is a solution of (3.1) associated with u * , we know that γ * (t) = e tA x + t 0 e (t-s)A Bu * (s) ds.

Definition 3.6 (Mean Field Games equilibrium). Given

  m 0 ∈ P α (R d ),Consequently, the family of measures {e t ♯η} t∈[0,T ] is tight.2. For any {ηi } i∈N ⊂ P m 0 (Γ T , R) and η ∈ P m 0 (Γ T , R) such that η i ⇀ * ηwe have that d 1 (e t ♯η i , e t ♯η) → 0

	32	CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG
	we say that η ∈ P m 0 (Γ T , R) is a Mean Field Games equilibrium for m 0 if
		spt(η) ⊂	Γ * η (x).
			x∈R d
	Proposition 3.7. Assume (L1) -(L3).
		1. For any η ∈ P m 0 (Γ T , R) we have that
		sup	|x| α e t ♯η(dx) ≤ R.	(3.5)
		t∈[0,T ] R d
		for every t ∈ [0, T ].

3. For any η

∈ P m 0 (Γ T , R) we have that the map t ∈ [0, T ] → e t ♯η is continuous.

Lemma 3.11. Assume (L1) -(L3). Then

  ∞ ∈ X λ . Hence, since the sets X λ are convex and strongly closed it implies that they are closed also in the L 2 weak topology.

	. up to a subsequence. Thus, by the continuity of L we
	have that L(γ(t), u i (t), e t ♯η) → L(γ(t), u ∞ (t), e t ♯η) a.e. and by Remark 3.1
	the Lagrangian L is bounded from below. Therefore, by Fatou's Lemma we
	obtain that u Corollary 3.10. Assume (L1) -(L3). Then, the set-valued map
	ϕ : R d , | • | ⇒ Γ T , • ∞
	x → Γ * η (x)
	has closed graph.
	, there exists a constant R(α, [m 0 ] α ) >
	0 such that if R ≥ R(α, [m 0 ] α ) then E(η) is non-empty. Moreover, E(η) is
	convex and compact.
	Proof. We, first, prove that given m 0 ∈ P α (R d ) for any η ∈ P m 0 (Γ T , R) the
	set E(η) is non empty for some constant R ≥ [m 0 ] α . Indeed, we have that
	by Corollary 3.10 and [21, Proposition 9.5] the set-valued map x ⇒ Γ * η (x)
	is measurable with closed values. Thus, by [26, Theorem A 5.2], there
	exists a measurable selection γx ∈ Γ * η (x), that is γx (t) = γ(t, x, u *

Lemma 3.12. Assume (L1) -(L3). Then

	36	CHAPTER 3. MILD AND WEAK SOLUTIONS OF MFG
		, for any R ≥ R(α, [m 0 ] α ), the
	set-valued map
		E : P m 0 (Γ T , R), d 1 ⇒ P m 0 (Γ T , R), d 1
		η → E(η)
	has closed graph.
	Proof. The proof of this Lemma is a straightforward application of [18,
	Lemma 3.6].

Theorem 3.13 (Existence of Mean Field Games equilibria). Assume (L1) -(L3). Let

	R ≥ R(α, [m 0 ] α ), where R(α, [m 0 ] α ) is defined as in
	Lemma 3.11. Then, the set-valued map E has a fixed point.
	Proof. By the above lemmas the assumptions of Kakutani-Fan-Glicksberg's
	fixed point theorem (see, for instance, [4, Corollary 17.55]) are satisfied and
	therefore, there exists a fixed point of the map E, that is η ∈ E(η) and η is
	a Mean Field Games equilibrium.
	At this point, for α > 1 fix m 0

  is the projection function. Then, there exists at least one solution ( λ, μ) ∈ R × C of the ergodic MFG problem.Moreover, if F satisfies the monotonicity assumption (4.6) and if ( λ1 , μ1 ) and ( λ2 , μ2 ) are two solutions of the ergodic MFG problem, then λ1 = λ2 .

Proposition 4.23. Assume that F satisfies (F1') and (F2').

  Note that by the definition of μT in (4.60) and by (4.58) and (4.59), we have

	90CHAPTER 4. ERGODIC BEHAVIOR OF ACCELERATION MODEL
	(4.61)
	m) μT (dx, dv, dw).

  Remark 5.21. Since u 0 is locally Lipschitz continuous w.r.t. time and Lipschitz continuous w.r.t. space, we have that Proposition 5.20 holds for a.e. (t, x)

  .31) Remark 6.12. Recalling that r R is the uniform degree of nonholonomy of the distribution {f i } i=1,...,m associated with the compact B R , Corollary 2.7 and (6.31) yield

  .38) 3. For any x ∈ R d , any t ≥ 0 and any φ ∈ S we have that T t φ(x) ≥ 0. Moreover, for any (t, x) ∈ [0, ∞)×R d the infimum in (6.36) is attained. 138CHAPTER 6. ASYMPTOTIC BEHAVIOR OF SUB-RIEMANNIAN SYSTEMS 6. T t is continuous on S w.r.t. the topology induced by the uniform convergence on compact subsets. Remark 6.18. We recall that, according to [2, Theorem 3.31], a set K is compact in (R d , d SR ) if and only if K is compact in R d w.r.t. the Euclidean distance. Proof. We begin by proving (1). To do so, we consider two cases: first, we take (t, x) ∈ [D(|x|), ∞) × R d and, then, (t, x) ∈ [0, D(|x|)) × R d . Recall that D(•) is defined in (6.7) and satisfies δ * (x) ≤ D(|x|). Define the function N φ : R d → R as

	4. For any φ ∈ S and any c ∈ R we have that T t (φ + c) = T t φ + c for all
	t ≥ 0.
	5. T t is a semigroup on S, i.e., T t : S → S and for any s, t ≥ 0 and
	φ ∈ S
	T 0 φ = φ, T

s (T t φ) = T s+t φ.

  {T t χ} t≥0 is equibounded on B R ; (ii) {T t χ} t≥1 is equicontinuous on B R .

	Proposition 6.19. Assume (F0), and (L0) -(L2'). Then, for any R ≥ 0
	we have that
	(i)

  Thus, the level set is compact by Prokhorov's Theorem and [7, Theorem 5.2.2]. Moreover, from Lemma 7.7 we have that F(ψ, µ) is lower-semicontinuous w.r.t. µ in P 2 R (R d × R m ).

	Therefore, applying the
	Minimax Theorem ([64, Theorem A.1]) we obtain
	inf
	µ∈P

  t], ∀ n ≥ n {γ n } n∈N is uniformly bounded in W 1,2 (0, t; R d ) for any t > 0. Then, by the Ascoli-Arzela Theorem, up to extracting a further subsequence, {γ n } n∈N uniformly converges to a curve γ on every compact subset of [0, +∞). Now, we claim that (γ, ū) satisfies (7.1). Indeed, for any t ≥ 0 we have that γ n (t) = x + Thus, by the locally uniform convergence of γ n it follows that f i (γ n (t)) → f i (γ(t)), locally uniformly, for any t ≥ 0, as n → +∞ for any i = 1, . . . , m. Therefore, taking v ∈ R d we deduce that v, γ n (t) = v, x + Remark 7.16. Arguing as in the proof of Lemma 7.15, one can prove the following. Given h ∈ R, {t n } n∈N and (γ n , u n ) ∈ Γ x→y -tn,0 such that t n → +∞ and lim

	7.3. AUBRY SET					161
	Hence, m		t
				u n i (s)f i (γ n (s)) ds.
		i=1	0
		m	t
	i=1 i (s) f n→+∞ 0 u n 0 -tn
	and
	tn	| γn (s)| 2 ds ≤	tn	c 2 f 1 + |γ n (s)|	2 |u n (s)| 2 ds
	0			0
	≤ c 2

f (1 + Q R )2ℓ 1 (h + 1), ∀ s ∈ [0, t], ∀ n ≥ n. i (γ n (s)), v ds, ∀ t ≥ 0. As n → +∞ we get v, γ(t) = v, x + m i=1 t 0 ūi (s) f i (γ(s)), v ds, ∀ t ≥ 0. Since v ∈ R d is arbitrary the conclusion follows.

L(γ un (s), u n (s)) ds -α(L) t n = h.

  Then, since u 0 ∞ ≤ 1 it follows that )(ℓ 1 + ℓ 2 ) + (t k -δ(x 0 ))L(x * , 0).

	164					CHAPTER 7. ON THE AUBRY SET
	t k					t k
	0	L(γ k (s), u k (s)) ds ≤ ≤ δ(x 0 (7.34) 0 L( γ k (s), u k (s)) ds
	Thus, combining (7.33) and (7.34) and dividing by t k we get
		inf y∈K c L	L(y, 0) <	1 t k	δ(x 0 )(ℓ 1 + ℓ 2 ) + 1 -	δ(x 0 ) t k	L(x * , 0)
	Moreover, by (L3) we deduce that
		L(x * , 0) + δ L <	1 t k	δ(x 0 )(ℓ 1 + ℓ 2 ) + 1 -	δ(x 0 ) t k	L(x
			s),		s ∈ [0, δ(x 0 )]
			0,

s ∈ (δ(x 0 ), t k ]. * , 0)

  A is a closed subset of R d .Proof. Let {x n } n∈N be a sequence in A such that lim n→∞ x n = x ∈ R d . Then, we have to show that x ∈ A.By definition we have that there exist sequence {t kn } n∈N and{(γ kn , u kn )} n∈N ∈ kn (s), u kn (s)) ds -α(L) t kn ≤ 1 n .Then, by Lemma 7.15 there exists (γ, ū) such that u kn weakly converges to ū and γ kn uniformly converges to γ, on every compact subset of [0, ∞), respectively. Let us define d n = d SR (x n , x) and the control Γ xn→x t kn ,t kn +d kn are optimal for (2.5), on their respective intervals. Hence, we have that ( γ n , u n ) ∈ Γ x→x -d k ,t kn +d kn and, thus, we getA ∞ (x, x) ≤ lim inf n→∞ [A t kn +2dn (x, x) -α(L)(t kn + 2d n )]The fact that A is compact follows from Proposition 7.20 and Proposition 7.21. Moreover, (L3) and Corollary 7.10 ensure that A is nonempty since x

	Γ xn→xn 0,t kn	such that
				t kn
				0 L(γ u n (s)     	u n 1 (s), u kn (s), u n 2 (s),	s ∈ [-d n , 0] s ∈ (0, t kn ]
	≤ lim inf n→∞	-dn 0	L(γ n 1 (s), u n 1 (s)) ds +	t kn

7.4. HORIZONTAL REGULARITY OF CRITICAL SOLUTIONS 165 Proposition 7.21. Assume (F0) -(F2), (L0) -(L3) and (S). s ∈ (t kn , t kn + d kn ]

where

(γ n 1 , u n 1 ) ∈ Γ x→xn -dn,0 and (γ n 2 , u n 2 ) ∈ 0 L(γ kn (s), u kn (s)) ds -α(L) t kn + t kn +dn t kn L(γ n 2 (s), u n 2 (s)) ds -2 α(L) d n ≤ lim n→∞ d n (ℓ 1 + ℓ 2 ) + 1 n = 0.

The proof is thus complete since, by definition, A ∞ (x, x) ≥ 0 for any x ∈ R d . Theorem 7.22 (Compactness of the Aubry set). Assume (F0) -(F2), (L0) -(L3) and (S). Then, A is a nonempty compact set. Proof. * ∈ A.

  7.4. HORIZONTAL REGULARITY OF CRITICAL SOLUTIONS169(iii) χ is horizontally differentiable at x ∈ A.Proof. We start by proving(7.36). Since x ∈ A. We have that A ∞ (x, x) = 0 . So there exist {t n } n∈N and (γ + n , u + n ) ∈ Γ x→x 0,tn such that t n → +∞ and lim Then, by Lemma 7.15 there exists (γ + x , u + x ) such that u + n weakly converges to u + x and γ + n uniformly converges to γ + x , on every compact subset of [0, ∞), respectively. Fix t ∈ [0, +∞), fix n large enough such that d n:= d SR (γ + x (t), γ + n (t)) ≤ 1 and t + 1 < t n . Let (γ 0 , u 0 ) ∈ Γ a solution of (2.5) and let u n ∈ L 2 (t, t n + d n ) be such that u n (s) = u 0 (s), s ∈ [t, t + d n ] u + n (s), s ∈ (t + d n , t n ].Then, by (7.39), the uniform convergence of γ n and the fact that d n ↓ 0 we deduce that Similar arguments show that there exists(γ - x , u - x ) ∈ Γ →x -∞,0 such that (7.37) holds. Indeed, it is enough to consider {t n } n∈N and (γ - n , u - n ) ∈ Γ x→x -tn,0 such that t n → +∞ and lim

		170		CHAPTER 7. ON THE AUBRY SET
		Moreover, we also have that
				t
		A ∞ (x, γ + x (t)) = A ∞ (x, γ + x (t)) -A ∞ (x, x) ≤	L(γ + n (s), u + n (s)) ds -α(L) t
				0
		tn x (t)) + A ∞ (γ + and A ∞ (x, γ + x (t), x) ≥ 0. Therefore, we obtain
			n→+∞ t L(γ + 0 x (s), u + L(γ + n (s), u + n (s)) ds -α(L) t n = 0. (7.38) x (s)) ds -α(L) t + A ∞ (γ + x (t), x) = 0.
			0
			γ + x (t)→ γ + n (t)
			t,t+dn	0
			n→+∞	-tn	L(γ -
		tn-t
	≤ lim inf n→+∞	0	L( γ tn
			L(γ + n (s), u +
			t+dn
			tn	t+dn
			L(γ + n (s), u + n (s)) ds -α(L) t n -	L(γ + n (s), u + n (s)) ds + α(L) t
			0	0

be Then, recalling that u 0 ∞ ≤ 1 we obtain

tn t L( γ n (s), u n (s)) ds = t+dn t L(γ 0 (s), u 0 (s)) ds + tn t+dn L(γ n (s), u n (s)) ds ≤ (ℓ 1 + ℓ 2 )d n + tn t+dn L(γ n (s), u n (s)) ds.

Now, defining u n (s) = u n (s-t) and denoting by γ n the associated trajectory we get

A ∞ (γ + x (t), x) ≤ lim inf n→+∞ A tn-t -α(L)(t n -t) n (s), u n (s)) ds -α(L)(t n -t) ≤ lim inf n→+∞ t+dn t L(γ 0 (s), u 0 (s)) ds + tn t+dn L(γ n (s), u n (s)) ds -α(L)(t n -t) ≤ lim inf n→+∞ ℓ 1 + ℓ 2 d n + n (s)) ds -α(L)(t n -t) = lim inf n→+∞ ℓ 1 + ℓ 2 d n + t 0 L(γ + (s), u + x (s)) ds -α(L) t + A ∞ (γ + x (

t), x) ≤ 0. n (s), u - n (s)) ds -α(L) t n = 0. (7.39)

  it is enough to show that A ∞ (γ x (t), γ x (t)) ≤ 0, t ≥ 0(7.41) since it is always true that A ∞ (γ x (t), γ x (t)) ≥ 0 for any t ≥ 0. From (7.27) the following holdsA ∞ (γ x (t), γ x (t)) ≤ A ∞ (γ x (t), x) + A ∞ (x, γ x (t)), t ≥ 0. (7.42)Since γ x is calibrated for χ we deduce that A Proof. Let χ be a critical solution, let x ∈ A and let γ x be a calibrated curve for χ. Let u x be the control associated with γ x . Then, from the Maximum Principle and the inclusion of the dual arc into the superdifferential of the corresponding value function, e.g.[START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] Theorem 7.4.17], we have thatD F χ(γ x (t)), u x (t) = L(γ x (t), u x (t)) + L * (γ x (t), D F χ(γ x (t))for any t ≥ 0. Note that, D F χ(γ x (t)) is well-defined by Proposition 7.26. Hence, by the properties of the Legendre Transform we obtain u x (t) = D p L * (γ x (t), D F χ(γ

	(7.44)
	Hence, combining (7.43) and (7.44) with (7.42) we get (7.41) which we recall
	that it implies (7.40).

∞ (x, γ x (t)) = t 0 L(γ x (s), u x (s)) ds -α(L) t. (7.43) and A ∞ (γ x (t), x) = -t 0 L(γ x (s), u x (s)) ds + α(L) t.

Corollary 7.29. Assume (F0) -(F2), (L0) -(L3) and (S).

Let χ be a critical solution, let x ∈ A and let γ x be calibrated for χ. Then, γ x satisfies the state equation with control

u x (t) = D p L * (γ x (t), D F χ(γ x (t))), t ≥ 0. Moreover, D F χ(γ x (t)) = D u L(γ x (t), u x (t)), t ≥ 0. x (t))), t ≥ 0

and D F χ(γ x (t)) = D u L(γ x (t), u x (t)), t ≥ 0. □

(u1, . . . , um) ⋆ denotes the transpose of (u1, . . . , um)

where the intersection is taken over all the critical solutions χ. Note that such a set is nonempty since (x * , 0) ∈ A.

MOTS CLÉS

Jeux á champs moyen, Contrôle optimal, Equations de Hamilton-Jacobi RÉSUMÉ Les travaux de cette thése concernent l'analyse de systémes de jeu á champ moyen (MFG) du premier ordre avec contrôle de l'accélération et l'étude du comportement en temps moyen long de systémes de contrôle de type sous-riemannien. Plus précisément, dans la premiére partie nous commençons par étudier le caractére bien posé du systéme MFG associé á un probléme de commande á équation linéaire en espace et en état de commande. En particulier, nous prouvons l'existence et l'unicité des solutions généralisées et nous étudions également leur régularité. Ensuite, nous nous concentrons sur le systéme MFG avec contrôle de l'accélération, un cas particulier de celui décrit ci-dessus, et nous étudions le comportement en temps moyen long des solutions en montrant la convergence vers une constante ergodique. Ici, comme pour l'analyse précédente, le principal probléme est le manque de convexité et de coercivité stricte du Hamiltonien par rapport á la variable de quantité de mouvement. Cela conduit par exemple á la non-existence de solutions de viscosité continue aux équations ergodiques de Hamilton-Jacobi et, par conséquent, ce permet pas de définir le systéme MFG ergodique au sens classique. Nous concluons cette premiére partie en établissant un lien entre le systéme MFG avec contrôle de l'accélération et le systéme MFG classique. Pour ce faire, nous étudions le probléme de perturbation singuliére pour le systéme d'accélération MFG, c'est-á-dire que nous analysons le comportement des solutions aux systémes de jeu á champ moyen dont le coût d'accélération devient nul. Encore une fois, nous résolvons le probléme en utilisant des techniques de calcul des variations en raison du probléme résultant du manque de convexité et de coercivité strictes du Hamiltonien par rapport á la variable de quantité de mouvement. Dans la deuxiéme partie, nous nous concentrons sur les systémes de contrôle affine sans dérive (de type sousriemannien). A la différence du cas de l'accélération, nous montrons qu'il existe une constante critique et que l'équation ergodique de Hamilton-Jacobi associée á une telle constante qui posséde des solutions de viscosité continues. Pour cela nous faisons appel á la géométrie sous-riemannienne sur l'espace d'état. Toujours en utilisant les propriétés de cette géométrie, nous définissons le semi-groupe de Lax-Oleinink et nous prouvons l'existence d'un point fixe de ce semi-groupe. Nous concluons cette partie, et donc cette thése, en étendant la célébre théorie d'Aubry-Mather au cas du systéme de contrle sous-riemannien. Nous montrons d'abord une formule de représentation variationnelle de la constante critique et, á partir de celle-ci, nous définissons l'ensemble de Mather et l'ensemble d'Aubry. En utilisant une approche dynamique, nous étudions les propriétés analytiques et topologiques de tels ensembles comme, par exemple, la différentiabilité horizontale de la solution critique en tout point se trouvant dans l'un des deux ensembles. Enfin, nous appliquons ces résultats pour étudier le caractére bien posé du systéme MFG ergodique associé á de tels systémes de contrôle.

ABSTRACT

The work in this thesis concerns the analysis of first-order mean field game (MFG) systems with control of acceleration and the study of the long time-average behavior of control systems of sub-Riemannian type. More precisely, in the first part we begin by studying the well-posedness of the MFG system associated with a control problem with linear state equation. In particular, via a relaxed approach, we prove the existence and the uniqueness of mild solutions and we also study their regularity. Then, we focus on the MFG system with control of the acceleration, a particular case of the one above, and we investigate the long time-average behavior of solutions showing the convergence to the critical constant. Here, as for the previous analysis, the main issues are the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable. Indeed, for instance, when studying the asymptotic behavior of the control system this lead us to a non existence result of continuous viscosity solutions to the ergodic Hamilton-Jacobi equation. Consequently, it does not allowed us to the define the ergodic MFG system as one would expect. We conclude this first part establishing a connection between the MFG system with control of acceleration and the classical one. To do so, we study the singular perturbation problem for MFG system of acceleration, that is, we analyze the behavior of solutions to the system when the acceleration cost goes to zero. Again, we solve the problem by using variation techniques due to the problems arising from the lack of strict convexity and coercivity of the Hamiltonian with respect to the momentum variable.

In the second part, we concentrate the attention to drift-less affine control systems (sub-Riemannian type). Differently from the case of acceleration, we prove that there exists a critical constant and the ergodic Hamilton-Jacobi equation associated with such a constant has continuous viscosity solutions. This is possible appealing to the properties of the sub-Riemannian geometry on the state space. Still using the properties of this geometry we finally define the Lax-Oleinink semigroup and we prove the existence of a fixed point of such semigroup. We conclude this part, and thus this thesis, extending the celebrated Aubry-Mather Theory to the case of sub-Riemannian control system. We first show a variational representation formula for the critical constant and from this we define the Aubry set. By using a dynamical approach we study the analytical and topological properties of such sets as, for instance, horizontal differentiability of the critical solution at any points lying in such a set.

KEYWORDS

Mean field games, optimal control, Hamilton-Jacobi equations