François Delarue

Catherine Rainer

Daniela Tonon D'

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Je tiens tout d'abord à remercier mon directeur de thèse, Pierre Cardaliaguet, pour ses conseils, sa disponibilité et sa patience, tous autant inestimables, au cours de ces trois dernières années.

Je suis honoré qu'Yves Achdou et Dylan Possamaï aient accepté de rapporter cette thèse et je les remercie chaleureusement. Je suis aussi reconnaissant à

Chapter 1 Introduction

Mean-field stochastic control problems are stochastic control problems where the state dynamic as well as the criteria to optimize depend, in a non trivial way, on the statistical distribution of the system. They arise from two directions. On the one hand they are a convenient extension to the classical theory of stochastic control. They allow for cost criteria which are non-linear functionals of the probability distribution of the system. On the other hand, they arise as "mean-field" limit for control problems of large number of interacting agents. The theory was largely developed in the last ten years, in connection with the sibling theory of mean-field games. Mean-field games were introduced independently by Lasry and Lions in [START_REF] Lasry | Jeux à champ moyen. II -Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games. I -The stationary case[END_REF][START_REF] Lasry | Mean field games[END_REF] and Huang, Caines, Malhamé in [START_REF] Caines | Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF] and [START_REF] Caines | Large-Population Cost-Coupled LQG Problems With Nonuniform Agents : Individual-Mass Behavior and Decentralized epsilon-Nash Equilibria[END_REF]. It aims at describing stochastic differential games when the number of players tends to infinity. We refer to the textbooks [START_REF] Bensoussan | Mean Field Games and Mean Field Type Control Theory[END_REF][START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[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[END_REF], as well as the lecture notes [START_REF] Achdou | Mean field games and applications: Numerical aspects[END_REF][START_REF] Cardaliaguet | Notes on Mean Field Games (from P.-L. Lions' lectures at College de France)[END_REF][START_REF] Cardaliaguet | An introduction to mean field game theory[END_REF][START_REF] Delarue | From the master equation to mean field game limit theory: Large deviations and concentration of measure[END_REF][START_REF] Santambrogio | Lecture notes on variational mean field games[END_REF] and the video lectures [START_REF] Lions | Théorie des jeux à champs moyen[END_REF] for a full account of the theory.

From a theoretical point of view, the main questions in mean-field control theory fall into two categories. On the one hand, one wants to solve the mean-field stochastic control problem. This means finding the optimal value that can be achieved and, if possible, determining an optimal control. On the other hand, one would like to properly understand the connection between the mean-field problem and control problems for large number of interacting agents.

The goal of this introduction is threefold. First, we want to introduce the main questions of the theory and some available results from the literature. We do not aim for exhaustiveness and this is rather an opportunity to introduce basic concepts that will appear throughout the manuscript. Secondly, we want to address specifically mean-field control problems with constraints in law which are the main subject of this dissertation. Finally we present the contributions of this thesis.

Mean-field stochastic control

In their simplest version, stochastic mean-field control problems take the following form. The dynamic of the system is described by a stochastic differential equation of McKean-Vlasov type CHAPTER 1. INTRODUCTION dX t " bpX t , LpX t q, α t qdt `?2dB t .

(1.1)

The state of the system X t is a random variable taking values in R d . We denote by LpX t q the probability distribution of X t , sometimes referred to as its law. In the above equation, pα t q 0ďtďT is the control process, taking values in the control set A. Moreover the stochastic differential equation is driven by a standard Brownian motion pB t q 0ďtďT . The goal is to choose the control in order to minimize over the time interval r0, T s a cost in expectation form

Jpαq " E

"ż T 0 LpX t , α t qdt `ż T 0 FpLpX t qqdt `GpLpX T qq  . (1.2)
In the above expression, L is a running cost which depends on the state of the system and the control. The running cost F and the terminal cost G are of "mean-field" type. This means that they are functions of the probability distribution LpX t q of the process X t . They are defined over PpR d q, the infinite dimensional space of Borel probability measures over R d . This problem exceeds the scope of the standard theory of stochastic control (see [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF]) because of this dependence of b, F and G on LpX t q. Consequently, knowing the state of the system X t at time t is not sufficient to act optimally. One needs as well to know the full probability distribution of X t . This leads to an intrinsically infinite dimensional problem and requires analysis as well as differential calculus tools over PpR d q. For the same reasons, the dynamic programming approach requires new techniques to be adapted to the mean-field setting. The resulting Hamilton-Jacobi-Bellman (HJB) equation has to be stated over PpR d q and the standard techniques to study HJB equations break-down in infinite dimension.

The study of mean-field control problems is partly motivated because they arise as limit of control problems for large numbers of interacting agents. The pre-limit problem involves a huge number of systems and, presumably the "mean-field" limit can be easier to handle. The N -agent problem can be described as follows. Consider N ě 1 agents -or players, or particles-denoted X 1,N t , . . . , X N,N t evolving according to the stochastic differential equations dX i,N t " bpX i,N t , p µ N,x t , α i,N t qdt `?2dB i,N t ,

where p µ N,x t :" 1 N

N ÿ i"1 δ X i,N t
is the empirical distribution of the agents. The Brownian motions are independent and they represent the private noises to which each agent is subject. We assume that a central planner chooses the controls pα i,N t q 0ďtďT for each agent, in order to minimize the cost J N pppα i,N t q 0ďtďT q 1ďiďN q :" E

« ż T 0 1 N N ÿ i"1 LpX i,N t , α i,N t qdt `ż T 0 Fpp µ N,x t qdt `Gpp µ N,x T q ff . (1.4)
To understand, at least formally, the connection between the N -agent problem and the mean-field problem we can look for controls of the form α i,N t " αpt, X i,N t , p µ N,x t q, for some function α : r0, T s ˆRd ˆPpR d q Ñ A. This means that we expect that the optimal strategy for each player is a (common) function of its state X i,N t , time t P r0, T s and the distribution of the population p µ N,x t at time t. Provided α is regular enough with respect to its arguments and the agents are initialized from independent and identically distributed (iid) initial positions, propagation of chaos dictates that the players become increasingly independent as their number goes to infinity, see [START_REF] Gärtner | On the McKean-Vlasov Limit for Interacting Diffusions[END_REF][START_REF] Oelschlager | A Martingale Approach to the Law of Large Numbers for Weakly Interacting Stochastic Processes[END_REF][START_REF] Sznitman | Topics in propagation of chaos[END_REF]. Moreover the dynamic of a typical player is given by " dX t " bpX t , LpX t q, αpt, X t , µptqqdt `?2dB t µptq " LpX t q (1.5)

and the (stochastic) empirical distribution p µ N,x t of the players converges to the (deterministic) measure µptq. This means that a typical player does not interact anymore with the rest of the population but rather with its own statistical distribution. At the limit, the cost functional takes the form (1.2). We have reduced the complexity of the problem, from N to one player. The price to pay is that the dynamic of the system and the cost to minimize now depend on LpX t q. The rigorous connection between Problem (1.13) and Problem (1.10) is unfortunately not obvious. Indeed it is not clear that passing to the mean-field limit in the system and optimizing over the controls are commutative operations.

In the rest of this introduction we present the main approaches developed in the literature in order to study the mean-field problem and its connection with the N -agent problem.

Optimal control of SDEs of McKean-Vlasov type and the Pontryagin maximum principle

Depending on the objective to achieve, it is convenient to formulate the mean-field control problem alternatively as an optimal control problem for stochastic differential equations, as an optimal control problem for a parabolic equation, the Fokker-Planck equation and finally, as an optimization problem over a set of probability measures satisfying martingale constraints.

The first and most natural formulation of the control problem goes as follows. We consider a filtered probability space pΩ, F, F, Pq endowed with a d-dimensional Brownian motion B " pB t q 0ďtďT . An admissible control is a measurable process taking values in a control space A and satisfying the integrability condition E

"ż T 0 |α t | 2 dt  ă `8.
If we assume that b satisfies suitable Lipschitz continuity conditions, for any initial position X 0 P L 2 pΩ, F 0 , Pq, and any admissible pα t q 0ďtďT , there is a unique solution to the McKean-Vlasov stochastic differential equation X t " X 0 `ż t 0 bpX t , LpX t q, α t qdt `?2B t , @t P r0, T s.

(1.6)

We seek to minimize the cost Jppα t q 0ďtďT q :" E "ż T 0 LpX t , α t qdt `ż T 0 FpLpX t qqdt `GpX T q  , over all the admissible controls. This problem can be solved, under specific regularity and convexity assumptions, using the Pontryagin maximum principle. To this end, we introduce, for each admissible control pα t q 0ďtďT and associated process pX t q 0ďtďT , an adjoint state pY t , Z t q 0ďtďT as the solution of the backward stochastic differential equation,

"
dY t " ´rB x bpX t , LpX t q, α t qY t `Bx LpX t , α t q `Dm FpLpX t q, X t qs dt `Zt dB t , Y T " D m GpLpX T q, X T q, (1.7)

where "D m " denotes a suitable derivative with respect to the measure variable. See Section 1.3 of this introduction for the exact definition. Under specific regularity and growth assumptions on the data b, L, F and G which guarantee among other things, for each admissible control pα t q 0ďtďT and associated process pX t q 0ďtďT , the well-posedness of (1.7), the Pontryagin maximum principle can be stated as follows.

Theorem 1.1. Assume that the control set A is convex and a Þ Ñ bpx, µ, aq.y `Lpx, aq is convex over A for all px, µq P R d ˆPpR d q. Then we have the following necessary condition for optimality: if pα t q 0ďtďT is an optimal control and pX t q 0ďtďT is the associated optimally controlled process, then, for all a P A, bpX t , LpX t q, α t q.Y t `LpX t , α t q ď bpX t , LpX t q, aq.Y t `LpX t , aq, dt b P-almost surely, where pX t , Y t , Z t q 0ďtďT is solution to the forward-backward system of stochastic differential equations $ & % dX t " bpX t , LpX t q, α t qdt `?2dB t , dY t " ´rB x bpX t , LpX t q, α t qY t `Bx LpX t , α t q `Dm FpLpX t q, X t qs dt `Zt dB t , Y T " D m GpLpX T q, X T q.

(1.8)

Conversely, let pα t q 0ďtďT be an admissible control with associated process pX t q 0ďtďT and adjoint process pY t , Z t q 0ďtďT . We further assume that G is (displacement) convex and that px, µ, aq Þ Ñ bpx, µ, aq.Y t `Lpx, aq `Fpµq is, dt b P almost-everywhere, convex. If bpX t , LpX t q, α t q.Y t `LpX t , α t q " inf aPA tbpX t , LpX t q, aq.Y t `LpX t , aqu , dt b P almost-everywhere, then pα t q 0ďtďT is an optimal control. This theorem is an extension of the standard stochastic Pontryagin maximum principle which can be found in [START_REF] Yong | Stochastic Controls -Hamiltonian systems and HJB equations[END_REF]. It was proved by Carmona and Delarue (in a more general setting) in [START_REF] Carmona | Forward-Backward Stochastic Differential Equations and Controlled McKean-Vlasov Dynamics[END_REF] where the authors prove as well the well-posedness of the system of forwardbackward stochastic differential equations. We refer to the first book of Carmona and Delarue [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] for a detailed presentation of this result, including a discussion about the notion of convexity required for this theorem to hold.

The Pontryagin maximum principle is, by definition, limited to situations where strong optimal solutions are expected to exist. In the theory of stochastic control, this is more the exception than the rule. Therefore we need to introduce weak formulations to the optimal control problem. This is usually done by introducing measure-valued controls and controlled martingales.

Compactness methods and the martingale problem

The control problem in its weak formulation is described as follows. We denote by C d :" Cpr0, T s, R d q the path space and V the control space consisting of measures q over r0, T s ˆA with the Lebesgue measure as first marginal and such that ż T 0 ż A |a| 2 dqpt, aq ă `8. A control q P V can be identified as a flow pq t q tPr0,T s of probability measures over A by disintegration with respect to its first marginal. We let pX, Λq be the identity processes over pC d ˆVq. That is pX t , Λ t qpx, qq " pxptq, q t q for any px, qq P C d ˆV and all t P r0, T s. We look for probability measures m over C d ˆV such that X 0 is distributed according to µ 0 under m, ϕpX t q ´ż t 0 ż A LϕpX s , m x s , aqdΛ s paqds is a martingale under m for all smooth compactly supported ϕ : R d Ñ R, where we use the notations Lϕpt, x, µ, aq " bpx, µ, aq.Dϕpxq `∆ϕpxq for the generator of the diffusion and m x s :" X s #m for the push-forward of m by X s . We denote by R the set of such measures and we look for m P R which minimizes the cost function

Γpmq :" E m "ż T 0 ż R d LpX t , aqdΛ t paqdt  `ż T 0 Fpm x t qdt `Gpm x T q.
It is important to notice that a "strong" control pα t q 0ďtďT defined on some probability space pΩ, F, P, Fq with associated controlled process pX t q 0ďtďT solution to the McKean-Vlasov SDE (1.6) induces a relaxed control pX, dtδ αt pdaqqq#P P PpC d ˆVq. We denote by R s the set of such strong controls. Under appropriate assumptions on b to ensure the well-posedness of (1.6) for constant controls and appropriate assumptions on L, F, G and µ 0 to ensure the lower semi-continuity and coercivity of Γ, Lacker proved -in a more general and difficult setting where the dynamics have controlled, possibly degenerate volatility-the following result which can be found in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF].

Theorem 1.2. Optimal relaxed controls exist. Moreover, the infimum over the relaxed controls is the same as the infimum over the strong controls inf mPR Γpmq " inf mPRs Γpmq.

Under additional convexity assumptions, one can prove the existence of Markovian controls, that is controls m P R such that mpΛ t " δ αpt,Xtq q, for a.e. t P r0, T sq " 1 for some measurable function α : r0, T s ˆRd Ñ A. Assume that, for each px, µq P R d ˆPpR d q the set Kpx, µq :" tpbpx, µ, aq, zq : a P A, z ď Lpx, aq `Fpµqu Ă R d ˆR is convex. Under this additional assumption, the following result is proved in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF].

Theorem 1.3. For all m P R, there exists a Markovian control r m P R satisfying r m x t " m x t for all t P r0, T s, as well as Γp r mq ď Γpmq. In particular there exists an optimal Markovian control.

To prove the existence of optimal solutions, one usually relies on Aldou's criteria to prove compactness of solutions to stochastic differential equations with appropriate uniform time regularity. Uniform time regularity for candidates m P R such that Γpmq ď C for some C ą 0 follows from the martingale constraint and energy estimates obtained thanks to the coercivity of Γ. This is achieved, for instance in Proposition 3.5 of [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF]. The approximation of relaxed controls by strong controls is particularly technical when the volatility coefficient depends on the control, a situation that we do not consider here.

The deep connection between solutions to stochastic differential equations and solutions to martingale problems is demonstrated in the seminal work of Stroock and Varadhan [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF]. It was broadly used in the context of stochastic control, see [START_REF] Harold | Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems[END_REF], and notably for applications in large deviations theory [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Dupuis | A weak convergence approach to the theory of large deviations[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF][START_REF] Fischer | Continuous time mean-variance portfolio optimization through the mean field approach[END_REF]. In the theory of stochastic control, two milestones are the papers [START_REF] Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF] and more recently [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF]. In the first one, the authors investigate the existence of optimal Markovian control under very broad assumptions, using the martingale problem approach. In [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF], Lacker generalizes these results to mean-field systems and rigorously connects the mean-field problem and problems for large number of interacting particles. More recently these results were further extended to controlled McKean-Vlasov SDEs with a common noise in [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF].

Optimal control of the Fokker-Planck equation and the meanfield game system of partial differential equations

Mean-field stochastic control problems are control problems for stochastic processes. However it is convenient -especially for the type of constraints presented in Section 1.2 of this introduction-to directly formulate them as control problems for the law of this processes. If α t has the form α t " αpt, X t q then the law µptq :" LpX t q of X t solves, by Itô's lemma, the equation " B t µ `divpbpx, µptq, αpt, xqqµq ´∆µ " 0, in p0, T q ˆRd , µp0q " LpX 0 q.

(1.9)

This is a non-linear parabolic partial differential equation (pde) named the Fokker-Planck equation. We refer to [START_REF] Bogachev | Fokker-Planck-Kolmogorov Equations[END_REF] for its general theory, and to [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF][START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] for its in connections with stochastic differential equations. The cost functional (1.2) can be rewritten solely as a function of µ and α and we are led to consider the new problem inf pα,µq

ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0 Fpµptqqdt `GpµpT qq, (1.10)
where the minimum is taken over couples pα, µq satisfying the Fokker-Planck equation (1.9).

We are now facing a deterministic control problem for a (non-linear) parabolic pde.

In the special case where we control directly the drift of the diffusion, that is bpx, µ, αq " α, Problem (1.10) has been widely studied in the literature in the context of potential mean-field games. We denote by Hpx, pq :" sup qPR d t´p.q ´Lpx, qqu the Hamiltonian of the system and assume that L is convex and coercive with respect to the control variable.

Theorem 1.4. Under these conditions, optimal solutions pα, µq exist and satisfy α " ´Bp Hpx, Duq, for some solution pu, µq to the mean-field game system of partial differential equations

$ ' ' ' & ' ' ' %
´Bt u `Hpx, Duq ´∆u " δF δm pµptq, xq in p0, T q ˆRd , B t µ ´divpD p Hpx, Duqµq ´∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 , upT, xq " δG δm pµpT q, xq in R d .

(

The first equation is a backward Hamilton-Jacobi-Bellman equation satisfied by an adjoint state u and from which we derive the optimal control α. The second equation is a forward Fokker-Planck equation satisfied by the optimal solution. The existence of solutions and the derivation of the optimality conditions can first be found in the seminal work [START_REF] Lasry | Mean field games[END_REF], see also [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] and Chapters 2 and 3 of this dissertation. Usually the necessary conditions are obtained through convex duality techniques, using generally the Fenchel-Rockafellar theorem as in [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF][START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF] or the Von-Neumann theorem as in [START_REF] Orrieri | A variational approach to the mean field planning problem[END_REF]. The form of the optimality conditions as well as the techniques to derive them are reminiscent of the the Benamou-Brenier formulation of optimal transport, see [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF].

This system of partial differential equations is known as the mean-field game system. In mean-field game theory, u is the value function of an infinitesimal player. The other unknown µ represents the density of the players at equilibrium.

The mean-field game system was introduced by Lasry and Lions in [START_REF] Lasry | Mean field games[END_REF]. Since then, its analysis has generated numerous works and we refer to [START_REF] Bensoussan | Mean Field Games and Mean Field Type Control Theory[END_REF][START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[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[END_REF][START_REF] Diogo | Regularity theory for mean-field game systems[END_REF][START_REF] Lasry | Jeux à champ moyen. II -Horizon fini et contrôle optimal[END_REF][START_REF] Lasry | Mean field games. I -The stationary case[END_REF][START_REF] Lasry | Mean field games[END_REF] for existence and uniqueness results in various contexts. System (1.11) has a particular form among general mean-field game systems. Indeed, the source term and the terminal condition for the HJB equation are derivatives of the costs F and G. We refer to these games as potential mean-field games and they exhibit some particular structural conditions. They allow for instance for the manipulation of weak solutions to the pde system, as in [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF][START_REF] Cardaliaguet | First order mean field games with density constraints: Pressure equals price[END_REF]. They also permit to discuss the stability of solutions as in [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF]. The study of their long time behavior was also pursued in [START_REF] Cardaliaguet | Weak KAM theory for potential MFG[END_REF][START_REF] Masoero | On the long time convergence of potential MFG[END_REF]. System (1.11) was also analyzed for its connections with mean-field control problems in the early works [START_REF] Achdou | On the system of partial differential equations arising in mean field type control[END_REF][START_REF] Achdou | Mean Field Type Control with Congestion[END_REF][START_REF] Achdou | Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method[END_REF]. 1.1.4 Connection with control problems for large number of interacting agents.

In this section we explain how Problem (1.10) arises as limit problem for large number of interacting agents. To this end, consider N ě 1 agents (or players, or particles) pX 1,N t , . . . , X N,N t q, evolving according to the stochastic differential equations

$ ' & ' % dX i,N t " bpX i,N
t , α i,N t , p µ N,x t qdt `?2dB i,N t ,

p µ N,x t " 1 N N ÿ i"1 δ X i,N t , (1.12)
starting from independent initial positions X 1,N 0 , . . . , X N,N 0 , identically distributed according to µ 0 P PpR d q and driven by independent Brownian motions. A central planner chooses the controls pα i,N t q 0ďtďT for each agent, in order to minimize the cost

E « ż T 0 1 N N ÿ i"1
LpX i,N t , α i,N t qdt `ż T 0 Fpp µ N,x t qdt `Gpp µ N,x T q ff (1. [START_REF] Benamou | An entropy minimization approach to second-order variational mean-field games[END_REF] where pX i,N t q 0ďtďT satisfies the dynamic (1.12) for all 1 ď i ď N . We distinguish two main approaches to the convergence problem. On the one hand, we can argue by compactness methods. This is successfully achieved in various contexts in [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF][START_REF] Fischer | Continuous time mean-variance portfolio optimization through the mean field approach[END_REF][START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF]. One first introduces suitable weak formulations of the control problems. We already presented the martingale problem formulation of the mean-field problem. For the N -particle system we proceed similarly. We denote by pX i,N , Λ i,N q the canonical process on pC d ˆVq N and define the empirical measures

p µ N :" 1 N N ÿ i"1 δ pX i,N ,Λ i,N q , p µ N,x t :" 1 N N ÿ i"1 δ X i,N t .
We define R N as the set of probability P N P PppC d ˆVq N q under which pX i,N 0 q i"1,...,N are iid with law µ 0 and

ϕpX 1,N t , . . . , X N,N t q ´N ÿ i"1 ż t 0 ż R d L N
i ϕpX 1,N s , . . . , X N,N s , aqdΛ i,N s paqds is a martingale under P N , for all smooth, compactly supported ϕ with L N i ϕpx 1 , . . . , x N , µ, aq :" D x i ϕpx 1 , . . . , x N q.bpx i , µ, aq `∆x i ϕpx 1 , . . . , x N q, for all px 1 , . . . , x N , µ, aq P pR d q N ˆPpR d q ˆA. The N -state problem in its weak formulation is therefore to minimize over P N P R N the cost functional

E P N « ż T 0 ˜żR d 1 N N ÿ i"1
LpX i,N t , aqdΛ i,N t paq `Fpp µ N t,x q ¸dt `Gpp µ N,x T q ff where E P N is the expectation under P N . In the setting of Theorem (1.2), Lacker proved in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] the following result, where PpPpR d qq is endowed with a suitable topology.

Theorem 1.5. For each N ě 1, let P N P R N be an N -optimal control for the N -particle problem, for some sequence N Ñ 0. Then P N #p µ N is precompact in PpPpC d ˆVqq and every limit point is supported on the set R ˚of optimal solutions to the mean-field problem.

Conversely, if P P PpPpC d ˆVqq is supported on R ˚, then there exists a sequence N Ñ 0 and a sequence of relaxed N -optimal controls P N P R N such that P N #p µ N Ñ P .

There are two key steps to prove the above result. The first one is to prove the compactness of the sequence pP N #p µ N q N PN and to identify weak-limit points. This is achieved thanks to uniform time regularity inherited from the martingale constraint and energy estimates derived from the coercivity of Γ. Then we can pass to the limit in the martingale problems. The second step is to show that any relaxed control for the mean-field problem can be approximated by sequences of relaxed controls for the N -particle problems. This means that, for any m P R, one can find, for all N P N, P N P R N such that P N #p µ N Ñ δ m in PpPpC d ˆVqq and E P N " Γpp µ N q ‰ Ñ Γpmq. This is achieved thanks to coupling techniques from the theory of propagation of chaos.

The other way to prove the convergence and obtain rates of convergence along the way, is to rely on the regularity of the value function for the mean-field problem. We discuss this point in the special case where bpx, µ, aq " a and A " R d . The value function U : r0, T s ˆPpR d q Ñ R is then defined, for all pt 0 , µ 0 q P r0, T s ˆPpR d q by Upt 0 , µ 0 q :" inf pµ,αq

ż T t 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T t 0 Fpµptqqdt `GpµpT qq
where the infimum is taken over the couples pµ, αq in Cprt 0 , T s, PpR d qq ˆL2 dtbµptq prt 0 , T s Rd , R d q satisfying " B t µ `divpαµq ´∆µ " 0 in pt 0 , T q ˆRd , µpt 0 q " µ 0 .

(1.14)

If U is smooth enough, we can prove that it satisfies the dynamic programming equation

$ ' ' & ' ' % ´Bt U `żR d Hpx, D m U, mqdmpxq ´żR d div x D m Upm, xqdmpxq
" Fpmq in p0, T q ˆPpR d q, UpT, mq " Gpmq in PpR d q.

(1.15)

Now, let us define V N pt 0 , x N q for t P r0, T s and x N " px 1,N , . . . x N,N q P pR d q N as the value function for the N -agent problem when the players start at t 0 with X i,N t 0 " x i,N for all 1 ď i ď N . We are faced with a standard stochastic control problem and we can show under appropriate regularity assumptions on the data, that V N is a strong solution to the HJB equation pt, x N qq ´N ÿ i"1

∆ x i,N V N pt, x N q " Fp 1 N N ÿ i"1
δ x i,N q `EN pt, x N q in p0, T q ˆpR d q N , V N pT, x N q " Gp 1 N

N ÿ i"1 δ x i,N q in pR d q N , (1.17)
with

E N pt, x N q :" ´1 N 2 N ÿ j"1
TrpD 2 mm Upt,

1 N N ÿ i"1
δ x i,N , x j,N , x j,N q.

See Section 1.3 for the definition of the second order derivative D 2 mm . If D 2 mm U is bounded, we have that |E N | ď 1 N and by a standard comparison principle for parabolic equations, we can deduce the estimate

|Upt 0 , 1 N N ÿ i"1 δ x i,N q ´VN pt 0 , x N q| ď C N
for some C ą 0 independent from N ě 1, t 0 P r0, T s and x N P pR d q N . Moreover, still assuming that U is a smooth function, it provides us, for each initial positions pt, µq, with a Lipschitz continuous optimal feedback control α ˚pt, µ, xq :" ´Bp Hpx, D m Upt, µ, xqq for the mean-field problem. We can test this control for the population of particles and deduce proper propagation of chaos results for the optimal trajectories of the N -particle problem, see [START_REF] Germain | Rate of convergence for particle approximation of PDEs in Wasserstein space[END_REF]. Therefore the difficulty comes down to proving the desired regularity for the value function U over r0, T s ˆPpR d q. This is not merely a technical challenge or a by-product of the regularity of the data and this cannot be achieved without structural conditions which guarantee, at least, the uniqueness of solutions to the control problem for each initial position. In [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] Chapter 3.7 such regularity is proved (in a compact setting) under suitable convexity assumptions on the costs F and G, by analyzing the mean-field game system (1.11) and suitable linearizations of this system around the initial condition µ 0 . A similar approach is also carried on in [START_REF] Carmona | Forward-Backward Stochastic Differential Equations and Controlled McKean-Vlasov Dynamics[END_REF].

Finally we mention other approaches to the convergence problem. For problems with finite state space or for problems where all the players face the same noise, one can use techniques from the theory of viscosity solutions to HJB equations to prove the convergence of the value functions. This respectively done in [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] and [START_REF] Cecchin | Finite state N-agent and mean field control problems[END_REF]. For the deterministic setting (namely when problem (1.10) is considered without diffusion), we refer to [START_REF] Cavagnari | Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence[END_REF][START_REF] Fornasier | Mean-field optimal control as Gamma-limit of finite agent controls[END_REF][START_REF] Fornasier | Mean-Field Optimal Control[END_REF][START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF].

Mean-field stochastic control with constraints in law

Chapters 2, 3 and 4 of this manuscript are dedicated to the optimization problem (1.10) when the solution µ is constrained to satisfy, either the terminal constraint ΨpµpT qq ď 0, (1.18) in Chapter 2, or, in Chapters 3 and 4, the state constraint Ψpµptqq ď 0, @t P r0, T s,

where Ψ : P 2 pR d q Ñ R satisfies similar regularity conditions as F and G.

In this section we give some motivations to study these constrained problems.

Constraint in law.

Constraints in law arise naturally in applications in economy and finance, as a way to control the risk associated with a given strategy. The most striking examples are the Mean-Variance portfolio selection problem, see [START_REF] Markowitz | Porfolio Selection[END_REF], and models with value at risk constraints as in [START_REF] Krokhmal | Portfolio optimization with conditional value-at-risk objective and constraints[END_REF]. Probability constraints are also a convenient way to relax almost-sure constraints. In the classical super-hedging problem, the controlled process must match almost surely a given random variable at the terminal time. This might be too hard to replicate. To tackle this issue, Föllmer and Leukert introduced in [START_REF] Föllmer | Quantile hedging[END_REF] the method of quantile hedging which can be seen as a stochastic control problem with constraints in law.

Mean-field constraint. For mean-field control problems, the constraint can also arise as a limit "mean-field" constraint. We refer for instance to the works of Seguret [START_REF] Seguret | Optimal control of a first order Fokker-Planck equation with reaction term and density constraints[END_REF] in connection with the problem of smart charging of electric vehicles. Mean-field constraint also arise when the density of a population of agents is constrained to remain below a threshold.

This problem leads to local constraints (the constraint is a function of the density of the law of state process) and it was addressed in [START_REF] Cardaliaguet | First order mean field games with density constraints: Pressure equals price[END_REF][START_REF] Di | Uniqueness issues for evolution equations with density constraints[END_REF][START_REF] Mészáros | A variational approach to second order mean field games with density constraints : The stationary case[END_REF][START_REF] Mészáros | On The Variational Formulation Of Some Stationary Second-Order Mean Field Games Systems[END_REF][START_REF] Santambrogio | Advection-Diffusion Equations With Density Constraints[END_REF].

Large deviations for interacting diffusion processes. Finally we explain an application to large deviations theory. Consider N interacting (non-controlled) particles X i,N t subject to the dynamics

" dX i,N t " bpX i,N t , p µ N,x t qdt `dB i,N t for t P r0, T s, X i,N 0 " x i,N 0 P R d , (1.20)
where pB 1,N t q, . . . , pB N,N t q are N independent Brownian motions, p

µ N,x t " 1 N ř N i"1 δ X i,N t
is the empirical distribution of the particles and x 1,N 0 , . . . , x N,N

0 are deterministic initial positions in R d such that 1 N ř N i"1 δ x i,N 0 converges to some µ 0 in PpR d q.
We introduce the rate function I : Cpr0, T s, PpR d qq Ñ R defined by

Ipµq :" inf pαq ż T 0 ż R d 1 2 |αpt, xq| 2 dµptqpxqdt (1.21)
where the infimum is taken over the α P L 2 µptqbdt pr0, T sˆR d , R d q such that the time marginals pµptqq 0ďtďT of µ satisfy the Fokker-Planck equation

" B t µ `divprbpx, µptqq `αpt, xqs µq ´1 2 ∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 . (1.22)
In their seminal work [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF], Dawson and Gärtner proved that the family of probability measures pp µ N,x q N ě1 satisfies a large deviation principle with rate function I ( in [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF] the rate function is not exactly (1.21) but the two formulations are equivalent, see for instance [START_REF] Benamou | An entropy minimization approach to second-order variational mean-field games[END_REF]). This means that, for suitable subsets K Ă Cpr0, T s, PpR d qq, it holds lim

N Ñ`8 1 N log P " p µ N P K ‰ " ´inf µPK Ipµq.
Therefore, if one is interested in the asymptotic of the probabilities P " Ψpp µ N,x T q ď 0 ı as N tends to infinity, one is led to minimize (1.21) over couples pµ, αq P Cpr0, T s, PpR d qq L2

mptqbdt pr0, T s ˆRd , R d q satisfying (1.22) in the sense of distributions as well as the terminal constraint ΨpµpT qq ď 0. If one is interested in the asymptotic of the probabilities P " Ψpp µ N,x t q ď 0, @t P r0, T s ı , one is led to solve the same problem with, this time, the state constraint Ψpµptqq ď 0, @t P r0, T s.

We refer to the works [START_REF] Budhiraja | A variational representation for positive functionals of infinite dimensional Brownian motion[END_REF][START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF] for the deep connections between mean-field control and the theory of large deviations for weakly interacting diffusions and how the weak convergence methods introduced in Section 1.1.2 can be used to prove the large deviations principle.

The space of probability measures

In this section we recall some basic facts about the space of probability measures PpR d q, that will be used throughout the manuscript.

Metric aspect. In this dissertation, we mostly focus on subsets P p pR d q of PpR d q, for p ě 1. The space P p pR d q is the set of Borel probability measures µ over R d such that ż

R d |x| p dµpxq ă `8.
It is endowed with the p-th Wasserstein distance d p from optimal transport, defined, for µ, ν P P p pR d q by

d p p pµ, νq " inf γPΓpµ,νq ż R d ż R d |x ´y| p dπpx, yq, (1.23)
where Γ is the set of transport plans between µ and ν, that is the set of Borel probability measures γ over R d ˆRd satisfying, for all Borel subset A of R d , γpA ˆRd q " µpAq and γpR d ˆAq " νpAq. With this metric, P p pR d q is a complete separable metric space. Moreover, for a given a sequence pµ n q P pP p pR d qq N , lim nÑ`8 d p pµ n , µq " 0 ðñ " µ n narrowly converges to µ, pµ n q has uniformly integrable p-moments.

(1.24)

We recall that pµ n q narrowly converges to µ if

lim nÑ`8 ż R d f pxqdµ n pxq " ż R d f pxqdµpxq,
for every continuous and bounded function f : R d Ñ R. We refer to the books [START_REF] Ambrosio | Gradient Flows in Metric Spaces and in the Space of Probability Measures[END_REF] and [START_REF] Villani | Topics in Optimal Transportation[END_REF] for the problem of optimal transport and the properties of the Wassertein distance.

Differential calculus. We are going to solve optimization problems defined over the space P 2 pR d q and, to this end, we need a suitable notion of differentiability with respect to probability measures. We say that a map F : P 2 pR d q Ñ R is C 1 over P 2 pR d q if there is a jointly continuous map δF δm : P 2 pR d q ˆRd Ñ R satisfying the integrability condition

sup µPK 1 1 `|x| 2 ˇˇˇδ F δm pµ, xq ˇˇˇă `8, for all bounded subset K Ă P 2 pR d q,
and such that, for all µ, ν P P 2 pR d q, one has the following:

ż 1 0 ż R d δF δm pp1 ´hqµ `hν, xqdpν ´µqpxqdh " Fpνq ´Fpµq.
The derivative is defined up to an additive constant and we always assume that, for all

µ P P 2 pR d q, ż R d δF δm pµ, xqdµpxq " 0.
In the terminology of [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] it means that F admits a linear functional derivative. When δF δm is differentiable with respect to the space variable, we define the intrinsic derivative

D m Fpµ, xq :" D x δF δm pm, xq.
If, for every x P R d , the map µ Þ Ñ δF δm pµ, xq is C 1 we say that F is C 2 and we denote by δ 2 F δm 2 its derivative, defined, for all µ P P 2 pR d q and all x, y P R d by

δ 2 F δm 2 pµ, x, yq " δ δm ˆδF δm pµ, xq ˙pyq. Finally, if F is C 2 and if δ 2 F δm 2 pµ, x, yq is twice differentiable in px, yq, we let D 2 mm Fpµ, x, yq :" D 2 x,y δ 2 F δm 2 pµ, x, yq.
Equivalently D 2 mm Fpµ, x, yq " D m pD m Fp., xqq pµ, yq. We refer to the books [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] and [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] for details on the notion(s) of derivative over the Wasserstein space. The reader, not familiar with differential calculus in the space of probability measures, can assume that functionals like F : P 2 pR d q Ñ R are linear with respect to the measure variable. This means that there is a measurable function f : R d Ñ R with at most quadratic growth, such that, for all µ P P 2 pR d q,

Fpµq " ż R d f pxqdµpxq.
In this case F admits a linear derivative, given for all µ P P 2 pR d q, by δF δm pµ, xq " f pxq ´żR d f pyqdµpyq.

If f is a differentiable function, then F admits an intrinsic derivative D m F given, for all µ P P 2 pR d q by D m Fpµ, xq " D x f pxq.

Organization of the manuscript and summary of the main results

This dissertation contains four chapters, in addition to this introduction. In Chapter 2, which constitutes the first section, we investigate a stochastic control problem with terminal constraint in law. We prove the existence of optimal solutions and characterize them with a second-order mean-field game system of partial differential equations associated with an exclusion condition.

The second section of this manuscript is dedicated to optimal control problems where the state is a flow of probability measures constrained to stay, at all time, in some region of the Wasserstein space of probability measures. In Chapter 3 we prove the existence of optimal solutions and exhibit some optimality conditions, once again in the form of a mean-field game system of pde's. The state constraint gives rise to delicate questions of time regularity for the optimal solutions. We address this questions and propose a geometric condition on the constraint under which optimal controls are Lipschitz continuous in time. Chapter 4 is an extension of Chapter 3. We show how mean-field control problems with constraints in law arise as limit of control problems for interacting agents with symmetric, almost-sure constraints. We also discuss some connections of this result with large deviations principles for (uncontrolled) weakly interacting particle systems. Although the results of Chapter 4 rely on the main results of Chapter 3, the former can be read independently.

Finally, in Chapter 5 we investigate the convergence problem in mean-field control without constraint. In this joint work with Pierre Cardaliaguet, Joe Jackson and Panagiotis Souganidis, we find an algebraic rate of convergence for the convergence of the value functions of the N -particle problems toward the value function of the mean-field problem, in a setting where uniqueness of optimal controls for the mean-field problem is not expected and therefore the value function is not expected to be differentiable.

Chapter 2: Stochastic control with terminal constraint in law

This chapter is devoted to a stochastic control problem with terminal constraint in law. This problem was originally motivated by applications in economy and finance, where it is natural to impose terminal constraints on the statistical distribution of the controlled state, see [START_REF] Föllmer | Quantile hedging[END_REF][START_REF] Guo | Portfolio optimization with a prescribed terminal wealth distribution[END_REF][START_REF] Guo | Calibration of local-stochastic volatility models by optimal transport[END_REF][START_REF] Krokhmal | Portfolio optimization with conditional value-at-risk objective and constraints[END_REF][START_REF] Markowitz | Porfolio Selection[END_REF]. From a control theoretic perspective, optimality conditions were investigated in [START_REF] Frankowska | Stochastic optimal control problems with control and initial-final states constraints[END_REF][START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF] for linear constraints and in the paper of Pfeiffer [START_REF] Pfeiffer | Optimality conditions in variational form for non-linear constrained stochastic control problems[END_REF] for a problem with non-linear terminal constraint 1 . A dynamic programming principle for stochastic control problems with linear constraints can also be found in [START_REF] Chow | On Dynamic Programming Principle for Stochastic Control Under Expectation Constraints[END_REF]. Our contribution is to prove the existence of strong optimal Markovian controls for a problem with non-linear costs and constraint, and characterize these controls through a (second-order, fully non-linear) mean-field game system of pde's. Along the way we put forward a qualification assumption adapted to the infinite dimensional setting, as well as new weak formulation of the problem as an optimal control problem for the Fokker-Planck equation. We also exhibit some regularity and growth conditions on the Hamiltonian of the system to ensure simultaneously the (approximate) controllability of the Fokker-Planck equation and the regularity of the backward HJB equation which appear in the system of optimality conditions. The problem takes the following form, where A is an appropriate class of controls,

inf αtPA E "ż T 0 pf 1 pt, X t , α t q `f2 pt, LpX t qqqdt `gpLpX T qq



under the constraint ΨpLpX T qq ď 0 for the diffusion 1 A word of caution on the terminology: throughout this dissertation, linearity and convexity are usually understood with respect to the measure variable. For instance a functional

F : m Þ Ñ ż R d
f pxqdmpxq is linear and therefore convex but it does not bear any meaning on the linearity or convexity of f : R d Ñ R.

dX t " bpt, X t , α t qdt `?2σpt, X t , α t qdB t , with the initial condition given by LpX 0 q " m 0 P PpR d q. The constraint Ψ satisfies, among other regularity assumptions, the qualification condition # Dm f P PpR d q, Ψpm f q ă 0, δΨ δm pm, .q ‰ 0, whenever Ψpmq " 0.

(1.25)

We define the Hamiltonian of the system Hpt, x, p, M q :" sup aPA ´bpt, x, aq.p ´σt σpt, x, aq.M ´f1 pt, x, aq

( .
Under specific assumptions on f 1 , f 2 , g, b and σ which guarantee the regularity and the controllability of the system we prove, in Theorem (2.2), that optimal feedback controls α : r0, T s ˆRd Ñ A exist and satisfy upt, xqqq ij mq " 0 in p0, T q ˆRd upT, xq " λ δΨ δm pmpT q, xq `δg δm pmpT q, xq in R d , mp0q " m 0 λΨpmpT qq " 0, ΨpmpT qq ď 0, λ ě 0, Our strategy consists first in introducing a suitable relaxed problem (RP) and then showing that optimal solutions for the relaxed problem yield optimal solutions for the original problem. The relaxed problem has the following form, inf pm,ω,W qPK

J RP pm, ω, W q, (1.27)
where K is the set of triples pm, ω, W q P Cpr0, T s, PpR d qq ˆMpr0, T s ˆRd , R d q ˆMpr0, T s Rd , S d pRqq such that ω and W are absolutely continuous with respect to mptq b dt,

B t m `divω ´ÿ i,j B 2 ij W ij " 0 (1.28)
holds in the sense of distributions, mp0q " m 0 and ΨpmpT qq ď 0. Above S d pRq is the space of symmetric square matrices of size d and we denoted by Mpr0, T s ˆRd , R d q (respectively by Mpr0, T s ˆRd , S d pRq ) the space of finite R d -valued (respectively S d pRq-valued) Radon measures over r0, T s ˆRd . The cost J RP is defined on K by

J RP pm, ω, W q :" ż T 0 ż R d L ˆt, x, dω dt b dm pt, xq, dW dt b dm pt, xq ˙dmptqpxqdt `ż T 0 f 2 pt, mptqqdt `gpmpT qq, with
Lpt, x, q, N q :" sup pp,M qPR d ˆSd pRq t´p.q ´M.N ´Hpt, x, p, M qu " H ˚pt, x, ´q, ´N q.

By standard compactness methods, using the controllability of the system, we prove the existence of relaxed solutions p r m, r ω, Ă W q in Lemma 2.3. In Proposition 2.3 we proceed to show that solutions p r m, r ω, Ă W q to the relaxed problem are also minimizers of the linearized functional To solve the dual problem, we use the controllability of the system to find an a priori upper bound on λ in Lemma 2.5 and we show that, for any λ ě 0, there is a unique solution to the HJB equation in C 1,2 b pr0, T s ˆRd q in Theorem 2.1. This last step is, by far, the most technical since H is not Lipschitz continuous with respect to the gradient variable, and therefore the standard regularity results on fully non-linear, uniformly parabolic HJB equations do not apply. Using the weak Bernstein method we prove that viscosity solutions to the HJB equation are Lipschitz continuous which is, by standard arguments, enough to conclude.

J l RP pm, ω, W q :" ż T 0 ż R d L ˆt, x, dω dt b dm pt, xq, dW dt b dm pt, xq ˙dmptqpxqdt `ż T 0 ż R d δf 2 δm pt,
Once we have found optimal solutions p r λ, r φq for the dual problem, we can express optimal controls to the relaxed problem in terms of p r λ, r φq. Arguing by verification, we show that these optimal controls are also optimal for the original problem and deduce the optimality conditions.

Chapter 3: Optimal control of the Fokker-Planck equation under state constraint: optimality conditions

This chapter is devoted to an optimal control problem similar to the one of Chapter 2 but, this time, the constraint is imposed throughout the whole time horizon. For deterministic control problems in finite dimension, state constraints have been widely studied and we refer to the survey of Frankowska [START_REF] Frankowska | Optimal control under state constraints[END_REF] and the lecture notes of Bonnans [START_REF] Bonnans | Optimal Control Problems with State Constraints[END_REF] for an overview. As for unconstrained problems, the results usually take two forms. The dynamic programming principle leads to the characterization of the value function as the unique "constrained" viscosity solution to the corresponding HJB equation. This result was proved in the seminal work of Soner [START_REF] Soner | Optimal Control With State-Space Constraint I[END_REF]. The other strategy is to show optimality conditions, usually thanks to the Pontryagin maximum principle. The effect of the constraint is then captured by the presence, in the system, of an additional Lagrange multiplier. This approach is convenient to address the delicate question of time regularity of the optimal controls in the presence of state constraints. This is achieved in [START_REF] Galbraith | Regularity of optimal controls for state constrained problems[END_REF][START_REF] Hager | Lipschitz Continuity for Constrained Processes[END_REF]. First order control problems with state constraints in the Wasserstein space were investigated in [START_REF] Bonnet | A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems[END_REF][START_REF] Bonnet | Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces[END_REF]. First order mean-field game systems with state constraints were also studied in a series of papers [START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF][START_REF] Cannarsa | Mean field games with state constraints: from mild to pointwise solutions of the PDE system[END_REF] and the structure of the present paper is closely inspired from [START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF] (although our techniques are completely different). Finally let us mention the reference [START_REF] Frankowska | Necessary optimality conditions for local minimizers of stochastic optimal control problems with state constraints[END_REF] where the authors prove first and second order optimality conditions for stochastic control problems with linear state constraints. In this work the authors put forward a stochastic inward pointing condition to ensure the normality of the optimality conditions. This condition is the finite dimensional analog of our qualification condition 1.42, introduced below.

In this chapter, we study an optimal control problem for the Fokker-Planck equation where the state is constrained to stay in some region of the Wasserstein space of probability measures. We prove the existence of optimal feedback controls and characterize them with a second order mean-field game system of pdes associated with an exclusion condition. Our main contributions are to give the system of optimality conditions when the constraint and the cost are non-linear functionals of the measure variable and to prove the Lipschitz continuity of optimal controls under an appropriate geometric condition on the constraint.

More precisely the problem takes the following form. We seek to minimize a cost Jpα, mq :"

ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `GpmpT qq
over pairs pα, mq with m P Cpr0, T s, P 2 pR d qq and α P L 2 mptqbdt `r0, T s ˆRd , R d ˘satisfying in the sense of distributions the Fokker-Planck equation:

B t m `divpαmq ´∆m " 0 (1.33)
with the initial condition mp0q " m 0 P P 2 pR d q. The flow of probability measures m is constrained to satisfy the inequality Ψpmptqq ď 0, @t P r0, T s for some function Ψ :

P 2 pR d q Ñ R.
We assume, in addition to specific regularity assumptions, that the constraint Ψ satisfies

" Ψpm 0 q ă 0, Ψ is convex. (1.34)
This is a stronger requirement than the qualification condition (1.25) of Chapter 2. Our first main contribution in this chapter is to derive the system of optimality conditions for the problem with state constraint. Assuming as well some coercivity and regularity conditions for Hpx, pq :" sup

qPR d
t´p.q ´Lpx, pqu and some regularity conditions for the mean-field costs F and G, we prove, in Theorem 3.4, that optimal solutions pα, mq exist and satisfy α " ´Bp Hp., Duq for some solution pu, m, ν, ηq of the system of optimality conditions ηΨpmpT qq " 0, (1.37) where ν P M `pr0, T sq and η P R `are Lagrange multipliers associated to the constraint. We discuss in details the well-posedness of the backward HJB equation. It is a priori non standard since the multiplier ν is a measure. As a consequence u and the optimal control α are discontinuous in time. Yet we show that u is still smooth in the space variable. To prove this result we need to introduce, for small parameters , δ ą 0 solutions to the penalized problems inf pm,αq

J ,δ pα, mq (1.38)
where the infimum runs over all pm, αq satisfying (1.33) (but not necessarily the state constraint) and J ,δ is defined by

J ,δ pα, mq :" ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `1 ż T 0 Ψ `pmptqqdt `GpmpT qq `1 δ Ψ `pmpT qq,
where we used the notation Ψ `pmq " maxpΨpmq, 0q (1.39) where, λ P L 8 pr0, T sq and

β P R `satisfy λptq $ & % " 0 if Ψpmptqq ă 0 P r0, 1s if Ψpmptqq " 0 " 1 if Ψpmptqq ą 0 (1.40) β $ & % " 0 if ΨpmpT qq ă 0 P r0, 1s if ΨpmpT qq " 0 " 1 if ΨpmpT qq ą 0.
(1.41)

To pass to the limit, as and δ go to 0, we first use condition (1.34) to build, in Lemma 3.4, admissible candidates which remain uniformly inside the constraint at all times and deduce, in Lemma 3.5 some bounds on λ and β of the form

ż T 0 λptq dt `β δ ă C,
for some C ą 0 independent of and δ. By parabolic regularity, we show, in Theorem 3.6 that this is enough to bound uniformly u and its space derivatives in (1.39) and then we can pass to the limit as and δ go to 0 to obtain the optimality conditions for the constrained problem.

Our second main contribution in this chapter (although presented first in the manuscript) is to provide an additional geometric assumption on the constraint under which the Lagrange multiplier ν is in fact a bounded function over r0, T s. This condition reads as follows

ż R d |D m Ψpm, xq| 2 dmpxq ‰ 0, whenever Ψpmq " 0. (1.42) Notice that ż R d
|D m Ψpm, xq| 2 dmpxq is nothing but the squared norm of the Wasserstein gradient of Ψ in L 2 pdmq at the point m. To prove this additional regularity, we apply a method already used in [START_REF] Cannarsa | Regularity Properties of Attainable Sets Under State Constraints[END_REF] and [START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF] for control problems in finite dimension. We look at the second order derivative of t Þ Ñ Ψpm ,δ ptqq when m ,δ is a solution to the penalized problem. We show, in Proposition 3.5 that it can be written, whenever Ψpm ,δ ptqq ‰ 0 as

d 2 dt 2 Ψpm ,δ ptqq " λptq ż R d D m Ψpm ,δ ptq, xq.D 2 pp Hpx, Du ,δ pt, xqqD m Ψpm ,δ ptq, xqdm ,δ ptqpxq `Op1q
where Op1q is bounded independently from and δ and can be expressed in terms of m ,δ as well as u ,δ and its space derivatives. Using conditions (1.40) and (1.42) as well as the convexity of the Hamiltonian with respect to the momentum variable, we prove that d 2 dt 2 Ψpm ,δ ptqq cannot be negative when is small enough. As a consequence, maxima of t Þ Ñ Ψpm ,δ ptqq must occur when Ψpm ,δ ptqq ď 0 which means that optimal solutions to the penalized problem remain inside the constraint when the penalization is strong enough. As a direct byproduct, solutions to the constrained problem are also solutions to the penalized problem with a strong enough penalization and therefore enjoy the same regularity. In particular, we prove, in Theorem 3.3 that ν belongs to L 8 pr0, T sq and that optimal controls are Lipschitz continuous in time.

Chapter 4:

Optimal control of the Fokker-Planck equation under state constraint: the mean-field limit

In this chapter we apply the results of Chapter 3, in particular the existence of bounded, Lipschitz continuous optimal controls for the mean-field problem, in order to study the meanfield limit for problems with constraints. We show that the mean-field problem arises as limit of control problems for finitely many interacting agents subject to symmetric, almostsure constraints. The pre-limit and limit problems are very different in nature. For the N -particle system, the constraint has to be satisfied almost-surely and therefore optimal controls blow-up near the boundary to compensate the effect of the non-degenerate noise. At the limit, the problem is deterministic, the constraint is of mean-field type and optimal controls are bounded and Lipschitz continuous. The mean-field limit is now rather well understood with the different approaches introduced in Section 1.1.4, however it seems that our result is the first of his kind for second order problems with constraints. This being said, making logarithmic transforms reminiscent of [START_REF] Fleming | Exit probabilities and optimal stochastic control[END_REF][START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] in the special case of a purely quadratic Hamiltonian, we see that our result has well-known counterparts in the theory of large deviations for weakly interacting (uncontrolled) particles, see [START_REF] Budhiraja | A variational representation for positive functionals of infinite dimensional Brownian motion[END_REF][START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF][START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF][START_REF] Fischer | On the form of the large deviation rate function for the empirical measures of weakly interacting systems[END_REF]. More precisely, the goal of this chapter is to investigate the connection between the optimization problem:

inf pα,µq ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0 Fpµptqqdt `GpµpT qq (mfP) subject to $ ' ' &
and a control problem for a large number N of interacting particles:

inf pα i,N t q 1ďiďN E P γ N « ż T 0 1 N N ÿ i"1 LpX i,N t , α i,N t qdt `ż T 0 Fpp µ N,x t qdt `Gpp µ N,x T q ff (NP) subject to $ & % dX i,N
t " α i,N t dt `?2dB i,N t , pX 1,N 0 , . . . , X N,N 0 q " µ bN 0 under P, Ψpp µ N,x t q ď 0 for all t P r0, T s P γ N ´almost-surely.

In the latter problem, pB i,N q 1ďiďN are N independent standard Brownian motions supported on a probability space pΩ, F, Pq. We denoted by

p µ N,x t :" 1 N N ÿ i"1 δ X i,N
t the empirical measures and finally P γ N :" P " ¨|Ψpp µ N,x 0 q ď ´γN ı is the conditional probability with respect to the event

! Ψpp µ N,x 0 q ď ´γN )
for some suitable rate γ N ą 0 such that γ N Ñ 0 as N Ñ `8. The conditioning being necessary to ensure that the particles start from inside the constraint. If we denote by Upµ 0 q, the value of Problem (mfP) and by U N pµ 0 q, the value of Problem (NP) our main result, Theorem 4.1, is to prove the convergence of U N pµ 0 q toward Upµ 0 q, when N Ñ `8.

We proceed in two steps. First we prove that lim sup N Ñ`8 U N pµ 0 q ď Upµ 0 q in Theorem 4.2. To this end, we need to find a way to transform an admissible control for the mean-field problem into an admissible control for the N -particle problem. The difficulty is that, if the particles each follow a bounded control, admissible for the mean-field problem, then the empirical measure of the system will almost-surely leave the constraint. To overcome this issue, we build explicit feedback controls which allow the particles to stay strictly inside a little ball of pR d q N for as long as needed without paying to big a cost. Therefore we can virtually stop the particle system when the empirical measures get close to the boundary of the constraint.

In Theorem 4.3, we prove that lim sup N Ñ`8 Upµ 0 q ď U N pµ 0 q relying on the compactness methods introduced in Section 1.1.2. This boils down to finding weak limit points of sequences of nearly optimal weak solutions to the N -particle problem. Once we know that U N pµ 0 q is bounded independently from N , this follows from the line of arguments of [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] for problems without constraint.

In Section 4.4 we discuss connections of these results with the problem of large deviations for weakly interacting particle systems.

Chapter 5: A rate of convergence for the optimal control of McKean-Vlasov dynamics

Chapter 5 is a joint work with Pierre Cardaliaguet, Joe Jackson and Panagiotis Souganidis. The goal is to investigate quantitatively the validity of the mean-field approximation, in situations where uniqueness is not expected for the mean-field problem and, therefore, the value function is a priori not differentiable with respect to the measure argument. The motivation is to fill the gap between the two extreme regimes described in Section 1.1.4 of this introduction. We recall that the convergence of the value functions is well understood under mild regularity and coercivity assumptions since the works of Lacker [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF], without common noise and Djete, Possamaï and Tan [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] for problems with a common noise. On the other hand, in the special case where the diffusion is non-degenerate and the cost functionals are convex with respect to the measure variable, we can prove that the value function for the mean-field problem is a smooth function over the space of probability measures, see [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]. As a consequence we can prove propagation of chaos for the optimal trajectories and quantitative rates of convergence for the value functions. Let us mention that several other papers have studied the question of the mean field limit of optimal control problems, for example [START_REF] Cavagnari | Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence[END_REF] and [START_REF] Fornasier | Mean-field optimal control as Gamma-limit of finite agent controls[END_REF] investigate the problem without noise by Γ´convergence techniques. The recent contribution [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] studies the mean field limit without idiosyncratic but with common noise using partial differential equations (PDE for short) techniques.

The main contribution of this chapter is to provide an algebraic rate of convergence for the value functions in a setting where the value function of the limit problem is not expected to be smooth.

We use the notations of Section 1.1.4 in this section to describe our results. Our main theorem asserts that there exists β P p0, 1s (depending on the dimension only) and C ą 0 (depending on the data of the problem) such that, for all t P r0, T s and all x N " px 1,N , . . . , x N,N q P ˆpR d q N it holds

|Upt 0 , 1 N N ÿ i"1 δ x i,N q ´VN pt 0 , x N q| ď C N β p1 `M 1{2 2 p 1 N N ÿ i"1 δ x i,N qq (1.43)
where

M 2 p 1 N N ÿ i"1 δ x i,N q " 1 N N ÿ i"1 px i,N q 2 .
The first step is to prove, in Lemma 5.1 and Lemma 5.3, uniform in N , L 8 , Lipschitz and semi-concavity estimates for V N . This is possible under suitable regularity assumptions on the data, since V N solves, by dynamic programming, the uniformly parabolic backward HJB equation

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ´Bt V N pt, x N q `1 N N ÿ i"1 Hpx i,N , N D x i,N V N pt, x N qq ´N ÿ i"1 ∆ x i,N V N pt, x N q " Fp 1 N N ÿ i"1 δ x i,N q in p0, T q ˆpR d q N , V N pT, x N q " Gp 1 N N ÿ i"1 δ x i,N q in pR d q N .
(1.44)

Then we proceed, in Proposition 5.3 to show the inequality

V N pt 0 , x N q ď Upt 0 , 1 N N ÿ i"1 δ x i,N q `C N β p1 `M 1{2 2 p 1 N N ÿ i"1 δ x i,N qq. CHAPTER 1. INTRODUCTION
This not too difficult because we can check that the function p V N pt, µq :"

ż pR d q N
V N pt, x 1,N , . . . , x N,N qdµ bN px 1,N , . . . , x N,N q is a smooth subsolution to the dynamic programming equation of the mean-field problem, up to a small error that we can quantify using the results of [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF]. Indeed, it holds that

$ ' ' ' ' & ' ' ' ' % ´Bt p V N pt, µq ´żR d divpD m p V N pt, µ, xqqdµpxq `żR d Hpy, D m p V N pt, µ, xqqdµpxq ď p F N pµq in p0, T q ˆPpR d q, p V N pT, µq " p G N pµq in PpR d q,
where p F N pµq "

ż pR d q N Fp 1 N N ÿ i"1 δ x i,N qdµ bN px N q and p G N pmq " ż pR d q N Gp 1 N N ÿ i"1 δ x i,N qdµ bN px N q.
The results of [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] allow us to estimate the differences between p V N and V N , between p F N and F and between p G N and G thanks to the regularity of V N , F and G. The reverse inequality is more intricate. Indeed it is not clear how to transform an optimal control for V N , which depends on the position of each particles (two particles which are very close can go into opposite directions) into an admissible control for the mean-field problem. To overcome this issue, we break the particles into subgroups, so that, in each subgroup, the particles have approximately the same drift. This is possible because we proved a -priori that the controls for each particle are bounded independently from N . Now we can prove, using a concentration inequality, that the empirical measure of each sub-group converges toward the solution of a Fokker-Planck equation when the number of particles gets larger. We can interpolate between these solutions arising from each subgroup, to find a candidate for the mean-field problem which is quantitatively close, for a small time at least, to the empirical measure of the whole system of particles. We combine this construction in the proof of Proposition 5.4 with our Lipschitz and semi-concavity estimates as well as a convenient doubling of variables argument inspired by techniques from the theory of viscosity solutions for Hamilton-Jacobi equations.

Our strategy relies mostly on the regularity of the value function for the N -particle system and we only use the Lipschitz continuity of the value function and dynamic programming principle for the mean-field problem. Therefore, we can extend our result to problems with a common noise, as long as the private noises remain non-degenerate. This is possible thanks to the recent results of [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF][START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF][START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] on mean-field control problems in the presence of a common noise. In this more general setting, we define

V N pt 0 , x N 0 q :" inf αPA N E « ż T t 0 p 1 N N ÿ k"1 LpX k t , α k t q `Fpp µ N,x t qdt `Gpp µ N,x T q ff ,
where A N is an appropriate class of controls, x N 0 " px 1,N 0 , . . . , x N,N 0 q is the initial position of the particles and

X k t " x k,N 0 `ż t t 0
α k s ds `?2pB k t ´Bk t 0 q `?2a 0 pB 0 t ´B0 t 0 q t P rt 0 , T s.

The pB k q kě0 are independent d-dimensional Brownian motions defined on a fixed filtered probability space and a 0 ě 0 is the intensity of the common noise. We also define the value function for the mean-field control problem with common noise Upt 0 , µ 0 q :" inf

α Er ż T t 0 `LpX t , α t q `FpLpX t |F B 0 t qq ˘`GpLpX T |F B 0 T qqs,
where the infimum is taken, once again, over an appropriate set of admissible controls, F B 0 " pF B 0 t q 0ďtďT denotes the filtration generated by B 0 , LpX t |F B 0 t q is the law of X t conditioned upon F B 0 t , and

X t " X t 0 `ż t t 0
α s pX s qds `?2pB t ´Bt 0 q `?2a 0 pB 0 t ´B0 t 0 q, with B another Brownian motion, X t 0 a random initial condition with law µ 0 and B 0 , B and X t 0 mutually independent. Under appropriate assumptions on the data L, F and G we prove in Section 5.4 that the estimate (1.43) still holds in the presence of a common noise.

Part I Stochastic Control With Terminal Constraint in Law Introduction

This paper is devoted to the study of stochastic optimal control problems with constraints on the law LpX T q of the controlled process at the terminal time. Our problem takes the following form :

inf αtPA E "ż T 0 pf 1 pt, X t , α t q `f2 pt, LpX t qqqdt `gpLpX T qq  under the constraint ΨpLpX T qq ď 0 for the diffusion: dX t " bpt, X t , α t qdt `?2σpt, X t , α t qdB t with the initial condition given by LpX 0 q " m 0 for some m 0 in P 2 pR d q, the space of probability measures over R d with finite second order moment. Here, f 1 : r0, T s ˆRd ˆA Ñ R and f 2 : r0, T s ˆP2 pR d q Ñ R are the instantaneous costs, g : P 2 pR d q Ñ R is the terminal cost, Ψ : P 2 pR d q Ñ R is the final constraint, b : r0, T sˆR d ˆA Ñ R d and σ : r0, T sˆR d ˆA Ñ S d pRq are respectively the drift and the volatility of the controlled process X and α and is the control process valued in the control space A. We look in particular for optimal Markov policies, that is control processes pα t q which are optimal among all admissible controls and for which there exists some measurable function α : r0, T s ˆRd Ñ A such that, for all t P r0, T s, α t " αpt, X t q.

We are going to show that optimal Markov policies are related to the solutions of the following system of partial differential equations, where the unknown pλ, φ, mq belongs to R `ˆC 1,2 b pr0, T s ˆRd q ˆCpr0, T s, P hpxqmpdxq for some function h : R d Ñ R and for all m P P 2 pR d q, we say that the constraint is linear. When the costs f 2 and g are linear as well we recover the problem of stochastic optimal control under expectation constraint (as in [START_REF] Bouchard | Stochastic Target Problems with Controlled Loss[END_REF], [START_REF] Chow | On Dynamic Programming Principle for Stochastic Control Under Expectation Constraints[END_REF], [START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF]). Such problems arise in economy and finance when an agent tries to minimize a cost (maximize a utility function) under constraints on the probability distribution of the final output. These types of constraints can take into account the risk given by the dispersion of the cost. There has recently been a surge of interest for this kind of problems. For instance [START_REF] Guo | Calibration of local-stochastic volatility models by optimal transport[END_REF] and [START_REF] Guo | Portfolio optimization with a prescribed terminal wealth distribution[END_REF] use similar formulations to study respectively the problem of calibration of local-stochastic volatility models and the problem of portfolio allocation with prescribed terminal wealth distribution. Probability constraints of the form P rhpX T q ď 0s ď 1 ´ also fall into our analysis since they can be written as functions of the law LpX T q of X T . In state constrained problems, the constraint is directly imposed on the process X T and must be satisfied almost-surely. Such constraints might be too stringent or even impossible to satisfy and probability constraints might allow to find controls with a better reward and a controlled probability of failure/success.

Stochastic control problems with terminal constraints have been extensively studied in the literature. Optimal control problems under stochastic target constraints have been studied in Bouchard, Elie and Imbert [START_REF] Bouchard | Optimal control under stochastic target constraints[END_REF] using the geometric dynamic programming principle proposed in Soner and Touzi [START_REF] Soner | Dynamic programming for stochastic target problems and geometric flows[END_REF] . In Föllmer and Leukert [START_REF] Föllmer | Quantile hedging[END_REF], the authors introduce the notion of quantile hedging to relax almost-sure constraints into probability constraints. In Yong and Zhou [START_REF] Yong | Stochastic Controls -Hamiltonian systems and HJB equations[END_REF] Chapter 3, necessary optimality conditions are proved in the form of a system of forward/backward stochastic differential equations. More recently the problem with constraints on the law of the process has been studied in Pfeiffer [START_REF] Pfeiffer | Optimality conditions in variational form for non-linear constrained stochastic control problems[END_REF] and in Pfeiffer, Tan and Zhou [START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF]. In these works, the authors prove that the problem can be reduced to a "standard" problem (without terminal constraint) by adding a term involving λ ˚hin the case where the constraint has the form E rhpX T qs ď 0-to the final cost for some optimal Lagrange multiplier λ ˚. A dual problem over the Lagrange multipliers associated to the constraints is exhibited using abstract duality results. In Pfeiffer, Tan and Zhou [START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF], the authors provide necessary and sufficient optimality conditions for problems with multiple equality and inequality expectation constraints with much less restrictions on the data than we do and in a path dependent framework. However [START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF] needs to assume some controllability condition (Assumption 3.1.ii) and works with a compact control set. In our framework, the corresponding controllability condition would be to assume a priori that there exist some control α such that EphpX α T qq ă 0. In our analysis, we are able to prove such controllability condition when H satisfies suitable assumptions.

The novelty of the present work is to provide a framework in which both controllability and existence of strong regular solutions for the Stochastic Control problem can be proved. We also believe that our necessary conditions for optimality can lead to efficient numerical methods using techniques already developed for similar kind of coupled PDE systems as in Achdou and Capuzzo Dolcetta [START_REF] Achdou | Mean field games: Numerical methods[END_REF]. We are also able to handle costs of mean-field type.

Our strategy is to study a relaxed problem which is an optimal control problem for the Fokker-Planck equation and then rely on the regularity of the data to show that optimal controls for the relaxed problem yield optimal controls for the original problem. The relaxed problem is the following :

inf pm,ω,W q ż T 0 ż R d Lpt, x, dω dt b dm pt, xq, dW dt b dm pt, xqqdmptqpxqdt `ż T 0 f 2 pt, mptqqdt `gpmpT qq,
where Lpt, x, q, N q :" sup pp,M qPR d ˆSd pRq t´p.q ´M.N ´Hpt, x, p, M qu " H ˚pt, x, ´q, ´N q and the infimum is taken over the triples pm, ω, W q P Cpr0, T s, P 1 pR d qqˆMpr0, T sˆR d , R d qM pr0, T sˆR d , S d pRqq for which ω and W are absolutely continuous with respect to mptqbdt and pm, ω, W q satisfy in the sense of distribution the Fokker-Planck equation:

B t m `divω ´ÿ i,j B 2 ij W ij " 0
together with the initial condition mp0q " m 0 and the terminal constraint ΨpmpT qq ď 0.

Notice that here and in the following, we denote by S d pRq the space of symmetric matrices of size d, endowed with the inner product M.N :" TrpM N q and by Mpr0, T s ˆRd , R d q (respectively by Mpr0, T s ˆRd , S d pRqq) the space of R d -valued (respectively S d pRq-valued) Borel measures on r0, T s ˆRd with finite total variation.

In order to study the relaxed problem, we rely on duality techniques that originated in the theory of Optimal Transport (see [START_REF] Svetlozar | Mass Transportation Problems -Volume 1: Theory[END_REF], [START_REF] Villani | Topics in Optimal Transportation[END_REF], [START_REF] Villani | Optimal Transport Old and New[END_REF] and [START_REF] Benamou | A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem[END_REF]) and were further developed in the theory of Mean Field Games. Indeed, when the game has a potential structure -see for instance Lasry, Lions [START_REF] Lasry | Mean field games[END_REF], Cardaliaguet, Graber, Porretta and Tonon [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF], Briani and Cardaliaguet [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] and Orrieri, Porretta and Savaré [START_REF] Orrieri | A variational approach to the mean field planning problem[END_REF] -the system of partial differential equations which describes the distribution of the players and the value function of a typical infinitesimal player can be obtained as optimality conditions for an optimal control problem for the Fokker-Planck equation. In this framework, the necessary conditions are obtained through convex duality techniques, using generally the Fenchel-Rockafellar theorem as in [START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF], [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] or the Von-Neumann theorem as in [START_REF] Orrieri | A variational approach to the mean field planning problem[END_REF]. We follow this path and -when the final constraint as well as the costs f 2 and g are linear -we are able to exhibit a dual problem, which is an optimal control problem for the HJB equation involving the Lagrange multiplier λ P R `associated to the terminal constraint. It takes the following form : The necessary conditions for optimality then follow from the lack of duality gap between the relaxed and the dual problems. We can then address more general constraints Ψ : P 2 pR d q Ñ R and costs f 2 : r0, T s ˆP2 pR d q Ñ R, g : P 2 pR d q by "linearizing" the costs and the constraint around solutions of the relaxed problem.

Using convex duality techniques to solve optimal control problems for diffusion processes is of course not new. It can be traced back at least to Fleming and Vermes [START_REF] Fleming | Convex duality approach to the optimal control of diffusions[END_REF], where the philosophy is very close to ours. In Tan and Touzi, [START_REF] Tan | Optimal transportation under controlled stochastic dynamics[END_REF] the authors extend the usual Monge-Kantorovitch optimal transportation problem to a stochastic framework. The mass is transported along a continuous semimartingale and the initial and terminal distributions are prescribed. Studying optimal control problems for the Fokker-Planck equation in order to understand the stochastic control problem is less common and it seems adapted to problems where the constraints only act on the law of the process. We refer to the works of Blaquière [START_REF] Blaquière | Controllability of a Fokker-Planck Equation, The Schrödinger System, and a Related Stochastic Optimal Control[END_REF] and more recently, Mikami [START_REF] Mikami | Two End Points Marginal Problem by Stochastic Optimal Transportation[END_REF] and Mikami and Thieullen [START_REF] Mikami | Duality theorem for the stochastic optimal control problem[END_REF] where similar approaches are developed in connection with the so-called Schrödinger problem. This approach has been followed recently by Guo, Loeper and Wang [START_REF] Guo | Calibration of local-stochastic volatility models by optimal transport[END_REF] and Guo, Loeper, Langrené and Ning [START_REF] Guo | Portfolio optimization with a prescribed terminal wealth distribution[END_REF] for problems with various expectation constraints. In both papers, the authors show that their original problem is in duality with a problem of optimal control of sub-solutions of an HJB equation. This dual problem is solved numerically. Our relaxation is in the spirit of classical works in convex analysis (see [START_REF] Ekeland | Convex Analysis and Variational Problems[END_REF]) but usually probabilists prefer to study another relaxation of the initial problem through the martingale problem (see Stroock and Varadhan [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF]), as in El Karoui, Jeanblanc-Picqué and Nguyen [START_REF] Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF] or Lacker [START_REF] Lacker | Mean field games via controlled martingale problems: Existence of Markovian equilibria[END_REF]. These different ways to relax the initial problem are, of course, connected and the correspondences between the diffusion processes, the martingale problem and the Fokker-Planck equation are now well established starting from the seminal work of [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] and more recently Figalli [START_REF] Figalli | Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients[END_REF] and Trevisan [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF].

Under very general assumptions, as in [START_REF] Fleming | Convex duality approach to the optimal control of diffusions[END_REF], one is usually able to see that the original problem is in duality with a problem of optimal control of the HJB equation. However, existence of solutions for this dual problem is much harder to come by and requires particular structural conditions. Essentially, the dual problem has a solution if the Hamilton-Jacobi-Bellman equation admits a regular solution. This is of course rather difficult to obtain. Regularity results for the Hamilton-Jacobi-Bellman equation where the control appears in the volatility as in Fleming and Soner [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] Chapter IV.4, usually rely upon three things : the regularity of the coefficients of the diffusion and of the costs functionals, the compactness of the control set and finally the uniform parabolicity of the equation. The last point means that there must be some Λ ´ą 0 such that the volatility coefficient satisfies (uniformly in the time/state/control variables) σ t σ ě Λ ´Id .

In studying terminal constraints, compact control sets are not satisfactory since we would not be able to show, in full generality, that the constraint can indeed be reached with a finite cost. Part of the challenge of the paper is to find a framework in which the process is sufficiently "controllable" but the HJB equation is still solvable. For that we need to impose restrictions on the coefficients.

In particular, we require some growth assumptions on the Hamiltonians and its derivatives. This allows us to use the weak Bernstein method as in Ishii and Lions [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF], Barles [START_REF] Barles | A Weak Bernstein Method for Fully Nonlinear Elliptic Equations[END_REF], Lions and Souganidis [START_REF] Lions | Homogenization of degenerate secondorder PDE in periodic and almost periodic environments and applications[END_REF] and Armstrong and Cardaliaguet [START_REF] Armstrong | Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions[END_REF] (among others) to prove that the viscosity solution of the HJB equation is Lipschitz in time and space.

As it is well-known, controllability for such systems is related to the coercivity of the Hamiltonian H in the momentum variable. As we will show, imposing a strictly super-linear polynomial growth (in p) for Hpt, x, p, 0q :" sup aPA ´bpt, x, aq.p ´f1 pt, x, aq allows to show that the agent can take (with a relaxed control) any instantaneous drift without paying too big a cost.

The rest of the paper is organized as follows : in Section 2.1 we present our assumptions and the precise statement of the problem. We also give our main results there. In Section 2.2 we introduce and study the problem of optimal control of the Fokker-Planck equation. Our main results, Theorems 2.2 and 2.3 are then proved in Section 2.3. Finally we give in Section 2.4 a detailed study of the Hamilton-Jacobi-Bellman equation which is crucial to our analysis.

Main Results

In this section we first present our notations and our standing assumptions. Then we briefly discuss some properties of the Lagrangian L and finally we state our main results.

Notations and Functional Spaces

The d-dimensional euclidean space is denoted by R d and the space of real matrices of size d by M d pRq. The space of symmetric matrices of size d ˆd is denoted by S d pRq. The subset of S d pRq consisting of positive symmetric matrices is denoted by S d pRq and S `d pRq is the subset of S d pRq consisting of definite-positive symmetric matrices. Recall that S `d pRq is endowed with a smooth (analytic) square root : ? . : S `d pRq Ñ S `d pRq (see for instance [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF] Lemma 5.2.1). Sometimes we will use S p pAq to denote the set of eigenvalues of a square matrix A. The euclidean space R d is endowed with its canonical scalar product : x.y :" ř d i"1 x i y i and the associated norm |x| 2 :"

ř d i"1 x 2 i .
The space M d pRq is endowed with its canonical scalar product : M.N :" Trp t M N q and the associated norm |M | 2 :" Trp t M M q, where TrpM q is the trace of M and t M is the transpose of M . Sometimes we will use the operator norm on M d pRq : ~M ~:" sup xPR d

|M x|

|x| . For two real numbers r 1 and r 2 , r 1 ^r2 is the minimum of r 1 and r 2 and r 1 _ r 2 is the maximum of r 1 and r 2 . If η is a σ-finite positive measure on a measurable space pΩ, Fq, µ is a σ-finite vector measure on pΩ, Fq and µ is absolutely continuous with respect to η we write dµ dη P L 1 pηq for the Radon-Nikodym derivative of µ with respect to η. If E is a locally compact, complete, separable metric space and l ě 1 is an integer, C 0 pE, R l q is the space of R l -valued continuous functions on X, vanishing at infinity. It is endowed with the topology of uniform convergence. Its topological dual `C0 pE, R l q ˘can be identified thanks to Riesz theorem as the space MpE, R l q of R l -valued Borel measures with finite total variation on E, normed by total variation. We will often consider the weak-* topology on MpE, R l q. When l " 1 we simply note C 0 pEq and MpEq. M `pEq Ă MpEq is the cone of finite non-negative measures. The set of Borel probability measures over E is denoted by PpEq. If r ě 1, P r pEq is the set of Borel probability measures over E with finite moment of order r. It is endowed with the topology given by the Wasserstein distance d r of order r. If X is a random variable taking values into pR d , BpR d qq, its law is denoted by LpXq P PpR d q. We say that U :

P 1 pR d q Ñ R if C 1 if there is a bounded continuous function δU δm : P 1 pR d q ˆRd Ñ R such that, for any m 1 , m 2 P P 1 pR d q, U pm 1 q ´U pm 2 q " ż 1 0 ż R d δU δm pp1 ´tqm 2 `tm 1 , xqpm 1 ´m2 qpdxqdt.
This derivative is defined up to an additive constant and we use the standard normalization convention : [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] for details on the notion(s) of derivatives in the space of measures. We consider a finite, fixed horizon T ą 0. The set of continuous functions from r0, T s to PpR d q and from r0, T s to P r pR d q for r ě 1 are respectively denoted by Cpr0, T s, PpR d qq and by Cpr0, T s, P r pR d qq. The space of measurable functions defined on r0, T s ˆRd with values into the measurable space Y is denoted by L 0 pr0, T s ˆRd , Y q. If u : r0, T s ˆRd Ñ R is sufficiently smooth, Du : r0, T sˆR d Ñ R d and D 2 u : r0, T sˆR d Ñ S d pRq denote respectively the differential and the Hessian of u with respect to the space variable x. The space of continuous functions u on r0, T s ˆRd for which B t u, Du and D 2 u exist and are continuous is denoted by C 1,2 pr0, T sˆR d q and C 1,2 b pr0, T sˆR d q is the subspace of C 1,2 pr0, T sˆR d q consisting of functions u for which u, B t u, Du and D 2 u are bounded. If n P N ˚and α P p0, 1q, C n`α b pR d q is the space of bounded continuous real functions on R d for which the first n-derivatives are continuous and bounded and the n-th derivative is α-Hölder continuous. We say that φ

ż R d δU δm pm, xqmpdxq " 0. See
: r0, T s ˆRd Ñ R d is in C n`α 2 ,n`α b pr0, T s ˆRd q if φ is

Assumptions

In all the following, A is a closed subset of an euclidean space, T ą 0 is a finite horizon and r 2 ě r 1 ą 1 are two parameters. The conjugate exponents of r 1 and r 2 are respectively denoted by r 1 and r 2 . The data are:

pb, σ, f 1 q : r0, T s ˆRd ˆA Ñ R d ˆSd `pRq ˆR, f 2 : r0, T s ˆP1 pR d q Ñ R, g : P 1 pR d q Ñ R, Ψ : P 1 pR d q Ñ R, m 0 P P r 1 pR d q.
We define the Hamiltonian of the system, for all pt, x, p, M q P r0, T s ˆRd ˆRd ˆSd pRq : Hpt, x, p, M q " sup aPA ´bpt, x, aq.p ´σpt, x, aq t σpt, x, aq.M ´f1 pt, x, aq ( 1. Assumptions on b, σ, f 1 , f 2 and g (a) For all R ą 0, b, σ and f 1 as well as the partial derivatives

B x b, B t b, B 2 xx b, B x σ, B t σ, B 2 xx σ, B x f 1 , B t f 1 , B 2 xx f 1
, are continuous and bounded on r0, T sˆR d ˆpA X Bp0, Rqq ; B x b, B x σ and B x f 1 are globally bounded.

(b) b has at most a linear growth and σ satisfies Λ ´Id ď σ t σpt, x, aq ď Λ `Id for some Λ `ě Λ ´ą 0 uniformly in pt, x, aq.

(c) f 1 is continuous and coercive with respect to a: there is δ ą 0 and C 1 , C 2 ą 0 such that, for all pt, x, aq, f (b) There is some α 1 , α 2 ą 0 and C H ą 0 such that, for all pt, x, pq P r0, T s ˆRd ˆRd ,

1 pt, x, aq ě C 1 |a| 1`δ ´C2 . (d) f 2 is
α 1 |p| r 1 ´CH ď Hpt, x, p, 0q ď α 2 |p| r 2 `CH .
(c) B t Hpt, x, p, M q is bounded over r0, T s ˆRd ˆRd ˆSd pRq.

(d) There is some positive constant C BpH and an exponent ν ě 1 such that, for all pt, x, p, M q P r0, T s ˆRd ˆRd ˆSd pRq

|B p Hpt, x, p, M q| ď C BpH p1 `|p| ν q.
(e) B x H is uniformly in pt, x, pq P r0, T s ˆRd ˆRd Lipschitz continuous in M .

(f) i. Either f 2 " 0 and the limit lim

|p|Ñ`8
|p| 2 ´Bx Hpt, x, p, 0q.p H 2 pt, x, p, 0q " 0 holds uniformly in pt, xq P r0, T s ˆRd ii. or f 2 ‰ 0 and there is some C BxH ą 0 such that |B x Hpt, x, p, 0q| ď C BxH p1 |p|q.

Assumptions on the constraint Ψ

(a) Ψ is continuous and admits a functional derivative such that x Ñ δΨ δm pm, xq belongs to C 3`α b pR d q with bounds uniform in m.

(b) There is at least one m P P 1 pR d q such that Ψpmq ă 0.

(c) For all m P P 1 pR d q such that Ψpmq " 0 there exists x 0 P R d such that δΨ δm pm, x 0 q ă 0.

Remark 2.1. Assumption 1 is sufficient to uniquely define the controlled process X α for any control α P A (see below for the definitions). If A were compact with f 2 " 0, we would be in the setting of [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] Chapter IV.4 and these assumptions would guarantee the existence of a smooth value function (in C 1,2 b pr0, T s ˆRd q). Remark 2.2. The upper bound in Assumption 2b is a coercivity assumption on the cost f 1 relatively to the drift b. Taking the definition of H, we see that it is equivalent to ask that, for all pt, x, aq P r0, T s ˆRd ˆA, f 1 pt, x, aq ě α 1 2 |bpt, x, aq| r 2 ´CH , for some α 1 2 ą 0. It will be a source of compactness throughout the paper. The lower bound in Assumption 2b is a "weak"-controllability condition and we will discuss it further in Lemma 2.2.

Remark 2.3. Using the Envelope theorem (see for instance [START_REF] Paul Milgrom | Envelope theorems for arbitrary choice sets[END_REF]) we see that H being C 1 -Assumption 2a-in the p, M -variables implies that, for any apt, x, p, M q P A such that Hpt, x, p, M q " ´bpt, x, apt, x, p, M qq.p ´σt σpt, x, apt, x, p, M qq ´f1 pt, x, apt, x, p, M qq we get B p Hpt, x, p, M q " ´bpt, x, apt, x, p, M qq and B M Hpt, x, p, M q " ´σt σpt, x, apt, x, p, M qq.

Consequently, drift and volatility must agree on potentially different optimal controls with common values ´Bp Hpt, x, p, M q and a ´BM Hpt, x, p, M q respectively. Notice that the growth conditions on the cost f 1 and the drift b ensure that for any pt, x, p, M q P r0, T s Rd ˆRd ˆSd pRq, there exists at least one such apt, x, p, M q in A. Remark 2.4. Using the envelope theorem and the uniform ellipticity condition in Assumption 1b we see that for all pt, x, p, M q, Λ ´Id ď ´BM Hpt, x, p, M q ď Λ `Id , a fact that we will repeatedly used throughout the paper. Remark 2.5. We use (the restrictive) Assumptions 2c, 2d, 2e, 2f in order to find Lipschitz estimates for the solution of the Hamilton-Jacobi-Bellman equation and to deduce that it is well-posed in C 1,2 b pr0, T s ˆRd q. Assumptions 2a is then sufficient to show that the solution is actually in C 3`α 2 ,3`α b pr0, T s ˆRd q. When Assumption 2b hold, Assumption 2(f)ii is stronger than Assumption 2(f)i but we use it to find Lipschitz estimates which are independent from the time regularity of the source term of the HJB equation. Remark 2.6. Assumption 3c is a tranversality condition. When Ψ is convex, this assumption is equivalent to the existence of some probability measure m P P 1 pR d q such that Ψpmq ă 0.

The following observations will be useful in order to translate the properties of the Hamiltonian H into properties of the Lagrangian L defined for all pt, x, q, N q P r0, T s ˆRd ˆRd Ŝd pRq by Lpt, x, q, M q :" sup pp,M qPR d ˆSd pRq t´p.q ´M.N ´Hpt, x, p, M qu .

Taking convex conjugates in 2b we see that this assumption can be reformulated in terms of L: for all pt, x, qq P r0, T s ˆRd ˆRd ,

α 1 2 |q| r 2 ´CH ď Lpt, x, q, 0q ď α 1 1 |q| r 1 `CH , (2.2)
where, for i " 1, 2,

α 1 i " α ´1 r i ´1 i p r i ´1 q r ´ri r i ´1 i and r i " r i r i ´1 is the conjugate exponent of r i .
Throughout the article, the following dual representation for L will be useful.

Lemma 2.1. Under Assumption 1 above, for all pt, x, q, N q P r0, T s ˆRd ˆRd ˆSd pRq, Lpt, x, q, N q ă `8 if and only if there is q A P P 1 pAq such that ż A bpt, x, aqdq A paq " q and ż A σ t σpt, x, aqdq A paq " N and in this case Lpt, x, q, N q " min

q A ż A f 1 pt, x, aqdq A paq,
where the minimum is taken over the q A P P 1 pAq such that ż A bpt, x, aqdq A paq " q and ż A σ t σpt, x, aqdq A paq " N .

44CHAPTER 2. STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW

Proof. It is elementary to show that for all pt, x, p, M q P r0, T s ˆRd ˆRd ˆSd pRq, Hpt, x, p, M q " sup q A PP 1 pAq "ż A p´bpt, x, aq.p ´σt σpt, x, aq ´f1 pt, x, aqqdq A paq * and therefore L reads as follows for all pt, x, q, N q P r0, T s ˆRd ˆRd ˆSd pRq, Lpt, x, q, N q " sup p,M " inf q A PP 1 pAq " ´p.q ´M.N `żA pf 1 pt, x, aq `bpt, x, aq.p `σt σpt, x, aq.M qdq A paq ** .

The result follows by exchanging the "sup" and the "inf". To this end we use Von Neumann Theorem 2.6 in the Appendix. The coercivity of f 1 as well as results of [START_REF] Ambrosio | Gradient Flows in Metric Spaces and in the Space of Probability Measures[END_REF] (Proposition 7.1.5) about the lower semicontinuity of functions defined on the space of probability measures allow to ensure that the use of the minmax theorem is licit.

From this dual representation we can see that the lower-bound on Hpt, x, p, 0q -or equivalently the upper-bound on Lpt, x, q, 0q-is a "weak"-controllability condition. It ensures that the agent can take any drift with a relaxed (i.e measure-valued) control without paying more than the r 1 -power of the drift :

Lemma 2.2. Fix pt, xq P r0, T s ˆRd . It holds that Hpt, x, p, 0q ě α 1 |p| r 1 ´CH for all p P R d
if and only if, for all q P R d there exists q A P P 1 pAq such that q " ż A bpt, x, aqdq A paq and

ż A f 1 pt, x, aqdq A paq ď α 1 1 |q| r 1 `CH .
For example, the growth condition on H is satisfied if ConvpImpbpt, x, .qq " R d for all pt, xq P r0, T s ˆRd and for all pt, x, aq P r0, T s ˆRd ˆA, α 1 2 |bpt, x, aq| r 2 ´CH ď f 1 pt, x, aq ď α 1 1 |bpt, x, aq| r 1 `CH .

Main Results

Throughout the article, we consider a fixed filtered probability space pΩ, F, F, Pq with F " pF t q tě0 satisfying the usual conditions and supporting an adapted, standard d-dimensional Brownian motion pB t q tě0 . We fix a F 0 -measurable random variable X 0 , independent of pB t q and such that X 0 belongs to L r 1 pPq. The control process α " pα t q tě0 is a progressively measurable process valued in A with finite L 2 pΩ ˆr0, T qq-norm. We denote by A the set of control processes. From the Cauchy-Lipschitz theorem, we know that for every α P A, there exists a unique F-adapted process X α satisfying :

dX t " bpt, X t , α t qdt `?2σpt, X t , α t qdB t
with the initial condition X α 0 " X 0 . A particular class of controls which is of interest is the one of Markovian controls (or Markov policies). A control process α is a Markovian control if there is a measurable function α : r0, T s ˆRd Ñ such that, for all t P r0, T s, α t " αpt, X α t q. We now introduce the cost functional

J SP : A Ñ R Y t`8u J SP pαq :" E "ż T 0 pf 1 ps, X α s , α s q `f2 ps, LpX α s qqq ds `gpLpX α T qq  .
The optimal control problem we are interested in is to minimize J SP pαq over α P A under the constraint ΨpLpX T qq ď 0.

If there exists a continuous function h : R d Ñ R such that, for all m P P 1 pR d q, Ψpmq " ż R d hpxqdmpxq then will say that the final constraint is linear. We define the set of admissible controls U ad U ad :" tα P A : ΨpLpX α T qq ď 0 and J SP pαq ă `8u . The problem in strong formulation is thus :

inf αPU ad J SP pαq.
(SP)

The fact that U ad is not empty is not trivial in itself but in our setting we will show that there are indeed admissible controls. Our results are the following : Theorem 2.2 (General Constraint). Under Assumptions 1, 2 and 3, there exist optimal Markov policies. Moreover, if pα t q P A is an optimal Markov policy, then there exists pλ, φ, mq P R `ˆC 1,2 b pr0, T s ˆRd qq ˆCpr0, T s, P 2 pR d qq such that, for mptq b dt-almost all pt, xq in r0, T s ˆRd

Theorem 2.1 (HJB equation). Take g 1 P C 3`α b pR d q and f 1 2 P C b pr0, T s, C 3`α b pR d qq such that t Ñ f 1 2 pt, xq P C α pr0, T sq for all x P R d with bounds uniform in x. Assume further that Assumptions 1 and 2 hold with 2(f )i in force if f 1 2 " 0 and 2(f )ii in force if f 1 2 ‰ 0.
Hpt, x, Dφpt, xq, D 2 φpt, xqq " ´bpt, x, αpt, xqq.Dφpt, xq ´σt σpt, x, αps, xqq.D 2 φpt, xq (2.3) ´f1 pt, x, αpt, xqq
and pλ, φ, mq satisfies the system of optimality conditions :

$ ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' % ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq " δf 2 δm pt, mptq, xq in r0, T s ˆRd B t m ´divpB p Hpt, x, Dφpt, xq, D 2 φpt, xqqmq `ÿ i,j B 2 ij ppB M Hpt, x, Dφpt, xq, D 2 φpt, xqqq ij mq " 0 in r0, T s ˆRd φpT, xq " λ δΨ δm pmpT q, xq `δg δm pmpT q, xq in R d , mp0q " m 0 , λΨpmpT qq " 0, ΨpmpT qq ď 0, λ ě 0. (OC)
Furthermore, mptq is actually the law of the optimally controlled process X α t and the value of the problem -denoted by V SP pX 0 q-is given by

V SP pX 0 q :" inf αPU ad J SP pαq " ż R d φp0, xqdm 0 pxq `ż T 0 f 2 pt, mptqqdt `gpmpT qq.
When the constraint and the costs f 2 and g are convex in the measure variable, we are able to show that the conditions are also sufficient : Theorem 2.3 (Convex constraint and convex costs). If Ψ, f 2 and g are convex in the measure argument and Assumptions 1, 2 and 3 hold, then the conditions of Theorem 2.2 are also sufficient conditions: if α P L 0 pr0, T s ˆRd , Aq satisfies 2.3 for some pφ, m, λq satisfying OC then the SDE dX t " bpt, X t , αpt, X t qqdt `?2σpt, X t , αpt, X t qqdB t starting from X 0 has unique strong solution X t , it holds that mptq " LpX t q and α t :" αpt, X t q is a Markovian solution to SP. Remark 2.7. Using standard parabolic PDE techniques and the regularity of φ, we can show that, provided m 0 admits a density in C 2`α b pR d q, mptq in Theorem 2.2 admits a density mpt, xq with respect to the Lebesgue measure such that m P C 2`α 2 ,2`α b pr0, T s ˆRd q. Remark 2.8. In Theorems 2.2 and 2.3, the stochastic basis pΩ, F, F, Pq and the Brownian motion pB t q introduced at the beginning of this section are a priori fixed. In the terminology of stochastic control it means that we deal with strong solutions to the stochastic control problem.

Remark 2.9. In the spirit of the Karush-Kuhn-Tucker theorem, multiple inequality constraints Ψ i pmpT qq ď 0 @i P 1, n can be considered provided they satisfy some qualification condition. We would say that the constraint is qualified at r m P P 1 pR d q provided there exists some m P P 1 pR d q such that ż

R d δΨ i δm p r m, xqdmpxq ă 0 for all i P 1, n such that Ψ i p r mq " 0. If n " 2 a sufficient condition would be δΨ i δm p r m, .q P L 2 pR d q for i " 1, 2 and ż R d δΨ 1 δm p r m, xq δΨ 2 δm p r m, xqdx ą 0.
For n ě 2 the condition would be satisfied everywhere if the constraints Ψ i are convex, satisfy Assumption 3 and if there is some m P P 1 pR d q such that Ψ i pmq ă 0 for all i P 1, n .

A relaxed

J RP pm, ω, W q, (RP)
where K is the set of triples pm, ω, W q P Cpr0, T s, P 1 pR d qq ˆMpr0, T s ˆRd , R d q ˆMpr0, T s Rd , S d pRqq such that ω and W are absolutely continuous with respect to mptq b dt,

B t m `divω ´ÿ i,j B 2 ij W ij " 0 (2.4)
holds in the sense of distributions, mp0q " m 0 and ΨpmpT qq ď 0. The cost J RP is defined on K by

J RP pm, ω, W q :" ż T 0 ż R d L ˆt, x, dω dt b dm pt, xq, dW dt b dm pt, xq ˙dmptqpxqdt `ż T 0 f 2 pt, mptqqdt `gpmpT qq.
Notice that the first term in the objective function J RP is convex in the variables pm, ω, W q and that the Fokker-Planck equation and the initial condition are linear in pm, ω, W q. Therefore the problem is linear/convex when the final constraint as well as the costs f 2 and g are convex. We say that pm, ω, W q in Cpr0, T s, P 1 pR d qq ˆMpr0, T s ˆRd , R d q ˆMpr0, T s ˆRd , S d pRqq satisfies the Fokker-Planck equation (FPE) 2.4 with initital condition mp0q " m 0 if and only if, for all ϕ P CpR d q with compact support and all φ P C 1,2 pp0, T q ˆRd q with compact support we have

ż T 0 ż R d B t φpt, xqdmptqpxqdt `ż T 0 ż R d Dφpt, xq.dωpt, xq `ż T 0 ż R d D 2 φpt, xq.dW pt, xq " 0 and the initial condition ż R d ϕpxqdmp0qpxq " ż R d ϕpxqdm 0 pxq.
Moreover, if ω and W are absolutely continuous with respect to mptq b dt the above relations hold if ϕ and φ are respectively taken in C b pR d q and C 1,2 b pp0, T q ˆRd q (see [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] Remark 2.3). In this case, we have for all φ P C 1,2 b pr0, T s ˆRd q and for all t 1 , t 2 P r0, T s ż

R d φpt 2 , xqdmpt 2 qpxq " ż R d φpt 1 , xqdmpt 1 qpxq `ż t 2 t 1 " B t φpt, xq `Dφpt, xq. dω dm b dt pt, xq `dW dm b dt pt, xq.D 2 φpt, xq  dmptqpxqdt.
Let us recall some known results about the link between solutions of the FPE and solutions to the SDE.

Proposition 2.1.

1. Suppose that m is a solution to the Fokker-Planck equation

" B t m `divpbpt, xqmq ´ři,j B 2 i,j ppσ t σpt, xqq ij mq " 0 mp0q " m 0 .
(2.5)

with coefficients b : r0, T s ˆRd Ñ R d , σ : r0, T s ˆRd Ñ M d pRq, Borel functions satisfying ż T 0 ż R d `|bpt, xq| `|σpt, xq| 2 ˘dmptqpxqdt ă `8.
Then there is a filtered probability space pΩ, F, F, Pq, an adapted Brownian motion pB t q tě0 and an adapted process pX t q 0ďtďT such that

LpX 0 q " m 0 , dX t " bpt, X t qdt `?2σpt, X t qdB t .
Moreover, for all t P r0, T s, LpX t q " mptq.

2. Conversely, suppose that pX s q sě0 is a strong solution of the stochastic differential equation " dX s " bps, X s qds `?2σps, X s qdB s X| t"0 " X 0 on some filtered probability space pΩ, F, F, Pq endowed with an adapted Brownian motion pB t q with b : r0, T s ˆRd Ñ R d and σ : r0, T s ˆRd Ñ M d pRq Borel-measurable functions such that

P "ż T 0 `|bps, X s q| `|σps, X s q| 2 ˘ds ă `8 " 1
and let mptq :" LpX t q " X t #P, then m satisfies the Fokker-Planck equation 2.5.

Proof. The second part follows from Itô's lemma and is standard. For the first part we need to combine the argument of [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF] and [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF]. From [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] Theorem 2.5 we know that this statement is equivalent to the existence of a solution to the so-called martingale problem and from [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF] Chapter 4, we know that existence of a solution to the martingale problem is equivalent to the existence of a weak solution to the SDE.

Let V RP pm 0 q be the value of the relaxed problem. The link with the usual compactification / convexification (see [START_REF] Karoui | Compactification methods in the control of degenerate diffusions: existence of an optimal control[END_REF] and [START_REF] Lacker | Mean field games via controlled martingale problems: Existence of Markovian equilibria[END_REF]) method in stochastic optimal control is the following : Proposition 2.2.

V RP pm 0 q " inf q A ,m t ż T 0 ż R d ż A f 1 pt, x, aqdq A pt, xqpaqdmptqpxqdt `ż T 0 f 2 pt, mptqqdt `gpmpT qqu,
where the infimum is taken over the couples pq A , mq P L 0 pr0, T sˆR d , P 1 pAqqˆCpr0, T s, P 1 pR d qq that satisfy in the sense of distributions the Fokker-Planck equation

B t m `divp ż A bpt, x, aqdq A pt, xqpaqmq ´ÿ i,j B 2 ij p ˆżA σ t σpt, x, aqdq A pt, xqpaq ˙ij mq " 0
together with the initial condition mp0q " m 0 and the terminal constraint ΨpmpT qq ď 0.

A RELAXED PROBLEM: OPTIMAL CONTROL OF THE FOKKER-PLANCK EQUATION49

Proof. The proof follows from the dual representation of L in Lemma 2.1 and a measurable selection argument as in [START_REF] Stroock | Multidimensional Diffusion Processes[END_REF]

Analysis of the Relaxed Problem

We will need the following facts :

Lemma 2.3. There exists pm, ω, W q P K such that J RP pm, ω, W q ă `8.

Proof. We have to check that we can indeed reach the final constraint with a finite cost.

By continuity of Ψ we can find x 0 , ..., x n P R d such that Ψp 1 n ř n i"1 δ x i q ă 0. Fix some δ ą 0. Let i be in 0, n . For all c ą 0 we can find q c P L 0 pr0, T s ˆRd , P 1 pAqq such that ż A bpt, x, aqdq c pt, xqpaq " cpx i ´xq and . We can use the result of Krylov in part 2.6 of [START_REF] Nicolai | Controlled Diffusion Processes[END_REF] (existence of weak solutions to stochastic differential equations with bounded measurable coefficients and uniformly nondegenerate volatility) and find a filtered probability space pΩ 1 , F 1 , F 1 , P 1 q satisfying the usual conditions, an adapted Brownian motion pB t q, a F 1 0 measurable random variable X 0 with law m 0 and a solution Y c t of the stochastic differential equation

ż A f 1 pt, x, aqdq c pt, xqpaq ď α 1 1 c r 1 |x i ´x|
dY c t " ce ct x i dt `?2r σ 1 c pt, Y c t qdB t starting from X 0 with r σ 1 c pt, yq " e ct r σ c pt, e ´ct yq. By Ito's lemma, X c t :" e ´ct Y c t solves the SDE dX c
t " cpx i ´Xc t qdt `?2r σ c pt, X c t qdB t starting from X 0 and we have, for all t P r0, T s

X c t " x i `pX 0 ´xi qe ´ct `?2e ´ct ż t 0 r σ c ps, X c s qe cs dB s .
Using the Burkholder-Davis-Gundy inequality and the upper bound on σ t σ we get

E 1 p|X c t ´xi | r 1 q ď 2 r 1 ´1e ´r1 ct E 1 p|X 0 ´xi | r 1 q `2 3r 1 ´2 2 e ´r1 ct E 1 ˆ| ż t 0 r σ c ps, X c s qe cs dB s | r 1 ď 2 r 1 ´1e ´r1 ct E 1 p|X 0 ´xi | r 1 q `2 3r 1 ´2 2 e ´r1 ct E 1 ˆpż t 0 Trpr σ t r σ c ps, X c s qe 2cs dsq r 1 2 ď 2 r 1 ´1e ´r1 ct E 1 p|X 0 ´xi | r 1 q `2 3r 1 ´2 2 pdΛ `q r 1 2 ˆe´2ct ´1 2c ˙r1 2 ,
where E 1 is the expectation under P 1 . In particular, taking t " T we see that, for c sufficiently large we have d r 1 pLpX c T q, δ x i q ď δ. Now, for such a c, we let m i ptq "

LpX c t q, ω i " cpx i ´xqm i , W i " r σ t c r σ c pt, xqm i .
Since f 2 and g are bounded functions, and thanks to the upper bound on f 1 we have that

J RP pm i , ω i , W i q ď C ˆ1 `ż T 0 E 1 p|X c t ´xi | r 1 qdt ˙ă `8.
Now we do the same for all i P 0, n and we let pm, ω, W q :" 1 n

n ÿ i"1 pm i , ω i , W i q.
The triple pm, ω, W q solves the Fokker-Planck equation starting from m 0 . Now by convexity of

pm, ω, W q Ñ ż T 0 ż R d L ˆt, x, dω dm b dt pt, xq, dW dm b dt pt, xq ˙dmptqpxqdt
and using the fact that f 2 and g are bounded we get that J RP pm, ω, W q ă `8. Finally

d r 1 ˜1 n n ÿ i"1 m i pT q, 1 n n ÿ i"1 δ x i ¸ď Cpnqδ
for some non negative constant Cpnq. For δ small enough we get that J RP pm, ω, W q ă `8 and ΨpmpT qq ă 0 which concludes the proof. Lemma 2.4.

1. Any point pm, ω, W q P K with J RP pm, ω, W q ă `8, satisfies the following estimate for some constant C r 2 depending only on r 2 : for any 0 ă s ď t ă T ,

d r 2 pmpsq, mptqq r 2 ď C r 2 pt´sq r 2 ´1 ż R d ż T 0 ˇˇˇd ω dt b dm pu, xq ˇˇˇr 2 dmpuqpxqdu`C r 2 Λ `pt´sq r 2 2 .
(2.6)

2. There exists some M ą 0 such that

sup tPr0,T s ż R d |x| r 2 dmptqpxq `|ω|pr0, T s ˆRd q `|W |pr0, T s ˆRd q ď M (2.7)
whenever J RP pm, ω, W q ď inf J RP `1.

Proof. First observe that, since J RP pm, ω, W q ă `8, by the dual formula for L of Lemma

d r 2 pmpsq, mptqq r 2 ď E " |X t ´Xs | r 2 ı ď 2 r 2 ´1E " | ż t s dω dt b dm pu, X u qdu| r 2  `2r 2 ´1E « | ż t s c 2 dW dt b dm pu, X u qdB u | r 2 ff ď p2pt ´sqq r 2 ´1E « ż t s ˇˇˇd ω dt b dm pu, X u q ˇˇˇr 2 du ff `2r 2 M r 2 E ¨"ż t s Trp dW dt b dm qpu, X u qdu  r 2 2 ' ď C r 2 pt ´sq r 2 ´1 ż R d ż T 0 ˇˇˇd ω dt b dm pu, xq ˇˇˇr 2 dmpuqpxqdu `Cr 2 Λ `pt ´sq r 2 2 ,
where we used Jensen inequality for the term involving ω and Burkholder-Davis-Gundy inequality for the other one.

For the second part of the lemma, let us take pm, ω, W q P K such that J RP pm, ω, W q ď inf J RP `1. From the growth assumptions on L, there exists M 1 ą 0 (which does not depend on the particular pm, ω, W q) such that ż

R d ż T 0 ˇˇˇd ω dt b dm pu, xq ˇˇˇr 2 dmpuqpxqdu ď M 1 .
Using the estimate proven in the first part of the lemma, we see that for all t, s P r0, T s, d r 2 pmpsq, mptqq ď M 1 1 for some M 1 1 ą 0 which, once again, does not depend on the particular choice of pm, ω, W q. This yields the uniform estimate on From this we can conclude with:

Theorem 2.4. J RP achieves its minimum at some point p r m, r ω, Ă W q in K.

Proof. This follows from the direct method of calculus of variations. Let pm n , ω n , W n q be a minimizing sequence such that, for all n P N, J RP pm n , ω n , W n q ď inf J RP `1. Using the Estimate 2.7 in Lemma 2.4 we can use Arzela-Ascoli theorem on the one hand and Banach-Alaoglu theorem on the other hand to extract a subsequence (still denoted pm n , ω n , W n q) converging to p r m, r ω, Ă W q P Cpr0, T s, P r 2 pR d qq ˆMpr0, T s ˆRd , R d q ˆMpr0, T s ˆRd , S d pRqq in Cpr0, T s, P δ pR d qqˆMpr0, T sˆR d , R d qˆMpr0, T sˆR d , S d pRqq for any δ P p1, r 2 q. It remains to show that p r m, r ω, Ă W q belongs to K and is indeed a minimum. The Fokker-Planck equation and the initial and final conditions are easily deduced from the weak-˚convergence of measures. To conclude we can use Theorem 2.34 of [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] to show that absolute continuity of ω n and W n with respect to m n ptq b dt is preserved when we take limits and that

J RP p r m, r ω, Ă W q ď lim inf n J RP pm n , ω n , W n q. So p r m, r ω, Ă W q is indeed a minimum of J RP in K.

Necessary Conditions for the Linear Case

In this section we suppose that Ψ is linear: there is a function h : R d Ñ R such that, for all m P P 1 pR d q, Ψpmq " ż R d hpxqmpdxq. We also suppose that h belongs to C 3`α b pR d q for some α P p0, 1q and that there exists x T P R d such that hpx T q ă 0. Under these assumptions, Ψ satisfies Assumption 3. We also suppose that f 2 and g are linear in m with where A " C 1,2 b pr0, T s ˆRd q and, for all pλ, φq P R `ˆA, φ belongs to HJ ´pλh `g1 q if and only if :

f 2 pt, mq " ż R d f 1 2 pt,
" ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq ď f 1 2 pt, xq in r0, T s ˆRd φpT, xq ď λhpxq `g1 pxq in R d (2.8)
The main theorem of this part is a duality result between RP and DP :

Theorem 2.5.

min pm,ω,W qPK J RP pm, ω, W q " sup pλ,φqPR `ˆA,φPHJ ´pλh`gq ż R d φp0, xqm 0 pdxq.

A RELAXED PROBLEM: OPTIMAL CONTROL OF THE FOKKER-PLANCK EQUATION53

To prove Theorem 2.5 the idea is to write the relaxed problem RP as a min/max problem and use the Von Neumann theorem to conclude. The statement of the Von-Neumann theorem is given in Appendix 2.5.1.

Proof of Theorem 2.5. Step 1: Further Relaxation

First we need to enlarge the space of test functions A to allow for functions with linear growth. More precisely, we define A 1 as the subset of C 1,2 pr0, T s ˆRd q consisting of functions φ such that

}pB t φq ´}8 `}φ `}8 `}Dφ} 8 `}D 2 φ} 8 `› › › › |φ| `|B t φ| 1 `|x| › › › › 8 ă `8,
where pB t φq ´" minpB t φ, 0q and φ `" maxpφ, 0q.

Owing to the estimates of Lemma 2.4 and using an approximation argument similar to [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] Remark 2.3 we see that any minimizer of the relaxed problem RP satisfies the Fokker-Planck equation against any function φ P A 1 . Now we define B to be the set of tuples pm, ω, W, nq in M `pr0, T s ˆRd q ˆMpr0, T s ˆRd , R d q ˆMpr0, T s ˆRd , S d pRqq ˆM`p R d q such that ω and W are absolutely continuous with respect to m. The cost J 1

RP is defined on B by

J 1 RP pm, ω, W, nq " ż T 0 ż R d " L ˆt, x, dω dm pt, xq, dW dm pt, xq ˙`f 1 2 pt, xq  dmpt, xq`ż R d g 1 pxqdnpxq.
If p r m, r ω, Ă W q is a solution of the relaxed problem we claim that

J RP p r m, r ω, Ă W q " J 1 RP p r m, r ω, Ă W , r mpT, dxqq " inf pm,ω,W,nq J 1 RP pm, ω, W, nq,
where the infimum is taken over the pm, ω, W, nq in B satisfying,

@φ P A 1 , ż T 0 ż R d pB t φm `Dφ.ω `D2 φ.W q `żR d φp0, xqdm 0 pxq ´żR d φpT, xqdnpxq " 0, (2.9) ż R d hpxqdnpxq ď 0. (2.10)
Indeed, since p r m, r ω, Ă W , r mpT qq belongs to B and satisfies the Fokker-Planck equation, it is clear that J 1 RP p r m, r ω, Ă W , r mpT qq ě inf pm,ω,W,nq J 1 RP pm, ω, W, nq. Now let us take pm, ω, W, nq P B satisfying (2.9) for every φ P A 1 and such that J 1 RP pm, ω, W, nq ă `8. Testing (2.9) for constants functions we get that npR d q " 1 and for any f P Cpr0, T sq, taking φptq " ż t 0 f psqds as a test function in (2.9) we get

ż T 0 ż R d f ptqdmpt, xq " ż T 0 f ptqdt.
This means that the time marginal of m is the Lebesgue measure over r0, T s. If m " dm t pxqdt is a disintegration of m with respect to its time marginal, we deduce that

ż T 0 f ptqm t pR d qdt " ż T 0 f ptqdt
for all f P Cpr0, T sq and therefore m t pR d q " 1, for dt-almost-all t P r0, T s. Now we can follow Lemma 2.6 and the discussion in the proof of Theorem 2.4 to deduce that m admits a continuous representative m 1 P Cpr0, T s, P r 2 pR d qq. We then get n " m 1 pT q from (2.9). Therefore pm 1 , ω, W q belongs to K, Therefore we deduce that

J 1 RP pm, ω, W, nq " J RP pm 1 , ω, W q ě J RP p r m,
V RP pm 0 q :" min pm,ω,W qPK J RP pm, ω, W q " inf pm,ω,W,nqPB sup pλ,φqPR `ˆA 1
Lppλ, φq, pm, ω, W, nqq, where L : R `ˆA 1 ˆB Ñ R is defined by: Lppλ, φq, pm, ω, W, nqq " Step 2: Analysis of the Lagrangian We immediately check that for all pλ, φq P R `ˆA 1 , pm, ω, W, nq Ñ Lppλ, φq, pm, ω, W, nqq is convex and for all pm, ω, W, nq P B, pλ, φq Ñ Lppλ, φq, pm, ω, W, nqq is concave. Now L can be rewritten as the sum of four terms, L " L 1 `L2 `L3 `żR d φp0, xqm 0 pdxq where, L 1 ppλ, φq, mq :" with f pλ,φq : r0, T s ˆRd ˆRd ˆSd pRq Ñ R defined by, f pλ,φq pt, x, q, N q " L pt, x, q, N q `Hpt, x, Dφpt, xq, D 2 φpt, xqq `q.Dφpt, xq `N.D 2 φpt, xq.

ż T 0 ż R d pL ˆt,
ż T 0 ż R d " B t φpt, xq ´Hpt, x, Dφpt, xq, D 2 φpt, xqq `f 1 2 pt, xq ‰ dmpt, xq, L 2 ppλ, φq, pm, ω, W qq :" ż T 0 ż R d f pλ,
Now suppose that pm k , ω k , W k , n k q kPN weakly-˚converges to some point pm, ω, W, nq and satisfies the uniform estimate maxt

ż T 0 ż R d p1 `|x|qdm k pt, xq, ż R d p1 `|x|qdn k pxq, ż T 0 ż R d ˇˇˇd ω k dm k pt, xq ˇˇˇr 2 dm k pt, xq, ż T 0 ż R d χ r0,Λ `s ˆdW k dm k pt, xq ˙dm k pt, xqu ď M (2.11)
for some M ą 0 and for all k P N. For L 1 and L

L 3 ppλ, φq, n k q ě sup φ 1 P C 0 pR d q, φ 1 ď λh `g1 ´φpT q lim inf kÑ`8 ż R d φ 1 pxqdn k pxq " sup φ 1 P C 0 pR d q, φ 1 ď λh `g1 ´φpT q ż R d φ 1 pxqdnpxq " ż R d
rλhpxq `g1 pxq ´φpT, xqs dnpxq " L 3 ppλ, φq, nq where we can argue by approximation, using Lebesgue dominated convergence and (2.11) to prove the second inequality.

For L 2 we need to proceed differently. Being f pλ,φq is nonnegative, lower-semicontinuous and for all pt, xq P r0, T s ˆRd , pq, N q Ñ f pλ,φq pt, x, q, N q being convex, we can follow [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] Theorem 2.34 and Example 2.36 to prove that ω and W are absolutely continuous with respect to m and L 2 ppλ, φq, pm, ω, W qq ď lim inf

kÑ`8 L 2 ppλ, φq, pm k , ω k , W k qq. Finally, we have that Lppλ, φq, pm, ω, W, nqq ď lim sup kÑ`8 Lppλ, φq, pm k , ω k , W k , n k qq.
(2.12)

Step

J RP pm, ω, W q " sup pλ,φqPR `ˆA 1 ,φPHJ ´pλh`g 1 q ż R d φp0, xqm 0 pdxq.
Notice that this duality is not surprising and holds under very general conditions (see for instance [START_REF] Fleming | Convex duality approach to the optimal control of diffusions[END_REF]). In particular the volatility σ can be degenerate. However the existence of solutions to the dual problem requires stronger assumptions. In particular we need strong solutions to the HJB equation and that is why we need Theorem 2.1.

Lemma 2.5. The dual problem has a finite value which is achieved at some point p r λ, r φq P R `ˆC 1,2 b pr0, T s ˆRd q such that :

# ´Bt r φ `Hpt, x, D r φpt, xq, D 2 r φpt, xqq " f 1 2 pt, xq in r0, T s ˆRd r φpT, xq " r λhpxq `g1 pxq in R d .
Proof. The finiteness follows from the fact that sup

pλ,φqPR `ˆA 1 ,φPHJ ´pλh`gq ż R d φp0, xqdm 0 pxq " min pm,ω,W qPK J RP pm, ω, W q ă `8.
Using the continuity of h and the fact that hpx T q ă 0 for some x T P R d , we can follow Lemma 2.3 to build pm, ω, W q P K such that J RP pm, ω, W q ă `8 and ż Now if we take pφ n , λ n q nPN a maximizing sequence, the above inequality shows that pλ n q nPN is bounded. Taking a subsequence we can suppose that pλ n q nPN converges to some r λ ě 0. By comparison, p r φ, r λq is a solution of the dual problem, where r φ P C 1,2 b pr0, T s ˆRd q is solution to

R
" ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq " f 1 2 pt, xq in r0, T s ˆRd φpT, xq " r λhpxq `g1 pxq in R d . (2.14)
The existence of r φ and the well-posedness of (2.14) in C 1,2 b pr0, T s ˆRd q is guaranteed by Theorem 2.1.

Remark 2.10. In the proof of the previous lemma, we showed as a by product that λ is bounded independently from φ, m. In particular using inequality (2.13) for a maximizing sequence and using the duality result of Theorem 2.5 we get that r λ satisfies r λ ď J RP pm, ω, W q ´VRP pm 0 q ´żR d hpxqdmpT qpxq

for any candidate pm, ω, W q such that ż R d hpxqdmpT qpxq ă 0.

Corollary 2.1. If p r m, r ω, Ă W q and p r λ, r φq are points where respectively the primal and the dual problems are achieved, then

r ω " ´Bp Hpt, x, D r φ, D 2 r φpt, xqq r mptq b dt, Ă W " ´BM Hpt, x, D r φ, D 2 r φpt,

Linearization

Let us fix p r m, r ω, Ă W q a solution of the relaxed problem. The linearized problem is to minimize Proof. Suppose that Ψp r mpT qq " 0. By Assumption (3c) there is some x 0 P R d such that δΨ δm p r mpT q, x 0 q ă 0 and we can proceed as in Lemma 2.3 (the constraint being then the linear

J l RP pm, ω, W q :" ż T 0 ż R d L ˆt, x, dω dt b dm pt, xq, dW dt b dm pt, xq ˙dmptqpxqdt `ż T 0 ż R d δf 2 δm pt,
one: r Ψpmq " ż R d δΨ δm p r mpT q, xqdmpxq) and find pm 1 , ω 1 , W 1 q such that $ ' ' ' ' ' ' & ' ' ' ' ' ' % m 1 p0q " m 0 B t m 1 `divpω 1 q ´ÿ i,j B 2 ij W 1 ij " 0 ż R d δΨ δm p r mpT q, xqdm 1 pT qpxq ă 0 J l RP pm 1 , ω 1 , W 1 q ă `8.
Now let pm, ω, W q be any candidate for the linearized problem (in particular pm, ω, W q satisfies the linearized constraint (2.17)). Let P p0, 1q and define pm , ω , W q :" p1 ´ qpm, ω, W q ` pm 1 , ω 1 , W 1 q (we perturb pm, ω, W q a little bit so that it satisfies strictly the linearized constraint). Let λ P p0, 1q and define pm λ , ω λ , W λ q :" p1 ´λqp r m, r ω, Ă W q `λpm , ω , W q.

We have that Now we let λ go to 0 and use once again the convexity of Γ to get

J l RP p r m, r ω, Ă W q " Γp r m, r ω, Ă W q ď Γpm , ω , W q `ż T 0 ż R d δf 2 δm pt, r mptq, xqdm ptqpxqdt `żR d δg δm p r mpT q, xqdm pT qpxq ď J l RP pm, ω, W q ` `Jl RP pm 1 , ω 1 , W 1 q ´Jl RP pm, ω, W q ˘,
where the first equality comes from the normalization condition in the definition of the linear derivative. We get the result letting Ñ 0. When Ψp r mpT qq ă 0 there is no need to perturb pm, ω, W q since (2.18) shows that Ψpm 0 λ pT qq ď 0 for small enough λ independently from the sign of ż R d δΨ δm p r mpT q, xqdmpT qpxq and we can take " 0 in the rest of the proof.

General Constraint

Proof of Theorem 2.2. Recall that, on the one hand we want to prove the existence of optimal Markovian controls for (SP) and on the other hand we want to prove that optimal controls, if Markovian, satisfy some necessary conditions. Let p r m, r ω, Ă W q be a solution of the relaxed problem. We can apply Proposition 2.3 and Corollary 2.1 to find some p r λ, r

φq in R `Ĉ 1,2
b pr0, T s ˆRd q such that p r m, r λ, r φq satisfies the system of optimality conditions (2.15) with We use the assumption that H is continuously differentiable in pp, M q. Indeed, in this case one has, thanks to the Envelope theorem (see [START_REF] Paul Milgrom | Envelope theorems for arbitrary choice sets[END_REF]), starting from X 0 . Therefore Lp r X t q " r mptq for all t P r0, T s and, in particular, ΨpLp r X T qq ď 0. This means that r α t :" r αpt, r X t q is admissible for the strong problem. Since H is C 1 we know that for all pt, x, p, M q P r0, T s ˆRd ˆRd ˆSd pRq, Hpt, x, p, M q " p.B p Hpt, x, p, M q `M.B M Hpt, x, p, M q ´Lpt, x, ´Bp Hpt, x, p, M q, ´BM Hpt, and thus J SP pr αq " J RP p r m, r ω, Ă W q " V SP pX 0 q from which it comes that V RP pm 0 q ě V SP pX 0 q. The reverse inequality being clear, we get V RP pm 0 q " V SP pX 0 q and r α is a solution to the strong problem. This shows in particular that optimal controls for the strong problem (SP) do exist. Now take α a Markovian solution to the strong problem. If X is the corresponding process, we take pm, ω, W q " pm, bpt, x, αqm, σ t σpt, x, αqmq. Then, pm, ω, W q is admissible for the relaxed problem and we have J RP pm, ω, W q ď J SP pαq " V SP pX 0 q. And thus, J RP pm, ω, W q " V RP pm 0 q. Finally, pm, ω, W q is optimal for the relaxed problem and we can apply Proposition 2.3 and Corollary 2.1 to conclude. Now if we use r φ in (OC) as a test function for the Fokker-Planck equation, recalling (2.20) as well as the normalization convention for the linear functional derivative we get that ż

f 1 2 pt,
# B p Hpt, x, D r φpt,
R d r φp0qdm 0 pxq " ż T 0 ż R d L ´t, x,

Convex Constraint and Convex Costs

Now we show that the conditions are also sufficient when Ψ, f 2 and g are convex functions in the measure variable. Notice that this case covers in particular the problem with expectation constraint and costs in expectation form when Ψ, f 2 and g are linear.

Proof of Theorem 2.3. Let p r λ, r φ, r mq be a solution to the system of optimality conditions (OC) and let r X t be the solution to

# d r X t " bpt, X t , r αpt, r X t qqdt `?2σpt, r X t , r αpt, r X t qqdB t r X 0 " X 0 .
for some measurable function r α : r0, T s ˆRd Ñ A such that, for any pt, xq P r0, T s ˆRd , pr0, T s ˆRd q the SDE admits a unique strong solution. We are going to show that r α t :" r αpt, r X t q is a solution to the optimal control problem. The law of r X t is r mptq and we deduce that ΨpLp r X T qq ď 0 and r α t is admissible. Now we show that r α t is indeed optimal among the admissible strategies. Let α t be an admissible control, X t the corresponding process and mptq :" LpX t q. Let also J 1 SP be defined on U ad as follows

Hpt
64CHAPTER 2. STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW J 1 SP pα t q :" E ˆż T 0 ˆf1 pt, X t , α t q `δf 2 δm pt, r mptq, X t q ˙dt `δg δm p r mptq, X T q `r λ δΨ δm p r mpT q, X T q ˙.
Using a classical verification argument and the fact that r φ solves the HJB equation, we get that J 1 SP pr α t q ď J 1 SP pα t q. Now by convexity of Ψ, f 2 and g we get

E "ż T 0 f 2 pt, r mptqqdt ´ż T 0 f 2 pt, mptqqdt `ż T 0 δf 2 δm pt, r mptq, X t qdt  ď 0, E " gp r mpT qq ´gpmpT qq `δg δm p r mpT q, X T q  ď 0 and r λE " δΨ δm p r mpT q, X T q  " r λ ˆΨp r mpT qq `E " δΨ δm p r mpT q, X T q ˙ď r λΨpmpT qq ď 0.
Therefore we get that J SP pr αq ď J SP pαq and r α is optimal for the strong problem.

The HJB Equation

The aim of this section is to show that the HJB equation We start by observing that it is enough to prove that (2.21) admits a bounded Lipschitz continuous viscosity solution. Indeed, if u is such a solution, it follows from the following line of arguments that u belongs to C 3`α 2 ,3`α b pr0, T s ˆRd q. If u is Lipschitz continuous in space we can use Theorem VII.3 in [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF] to deduce that u is semi-concave with a modulus of semi-concavity uniform in pt, xq. Now, using the uniform parabolicity of the equation, the fact that u is Lipschitz and semi-concave we can prove that u is also semi-convex (see [START_REF] Imbert | Convexity of solutions and C1,1 estimates for fully nonlinear elliptic equations[END_REF] Theorem 4 with the help of [START_REF] Alvarez | Convex viscosity solutions and state constraints[END_REF] ) and therefore Du is continuous and Lipschitz in space. At this point, using the uniform parabolicity of the equation and the Hölder regularity in time of f 1 2 , we can use the results of [START_REF] Wang | On the regularity theory of fully nonlinear parabolic equations: I[END_REF] and [START_REF] Wang | On the regularity theory of fully nonlinear parabolic equations: II[END_REF] (see also the last section of [START_REF] Bourgoing | C1,β regularity of viscosity solutions via a continuous-dependence result[END_REF]) to deduce that u belongs to C 1,2 b pr0, T s ˆRd q. Finally, by differentiating the equation we can use results on uniformly parabolic linear PDEs (Theorem IV.5.1 of [START_REF] Ladyženskaja | Linear and Quasi-linear Equations of Parabolic Type[END_REF]) to conclude that u belongs to

C 3`α 2 ,3`α b pr0, T s ˆRd q.
We first show the existence of bounded, Lipschitz continuous viscosity solutions to (2.21) when f 1 2 is also globally Lipschitz continuous (and not just Hölder continuous in time) and then we use an approximation argument.

Lemma 2.6. Suppose that Assumptions (1) and (2) hold. Take g 1 P C 2 b pR d q and suppose that f 1 2 P C b pr0, T s ˆRd q is globally Lipschitz continuous in pt, xq and C 1 in x. Then (2.21) admits a unique bounded, globally Lischitz continuous viscosity solution.

Proof. We proceed as follows. For K ě }Dg 1 } 8 we let u be the unique viscosity solution to

" maxt´B t u `Hpt, x, Du, D 2 uq ´f 1 2 pt, xq, |Du| ´2Ku " 0 in r0, T s ˆRd upT, xq " g 1 pxq in R d . (2.22)
The existence of u is guaranteed by Proposition 2.4 in Appendix 2.5.2. In particular, u is 2K-Lipschitz continuous in x and L-Lipschitz continuous in t for some L ą 0 independent from K. We are going to show that, for K large enough, u is K-Lipschitz continuous in x.

As a direct by-product, u will be the unique viscosity solution to (2.21) and it will satisfy the Lipschitz regularity given by Proposition 2.4. Now we proceed to show that u is K-Lipschitz continuous in x. Let K ě 2}Dg 1 } 8 such that Hpt, x, p, 0q ą L `}f 1 2 } 8 for all pt, x, pq P r0, T s ˆRd ˆRd such that |p| ě K ´1, where L is an upper bound for the time-Lipschitz constant of u. We are going to show that for all pt, x, yq P r0, T s ˆRd ˆRd , upt, xq ´upt, yq ď K|x ´y| when K is large enough. Suppose on the contrary that δ :" sup pt,x,yqPtr0,T sˆR d ˆRd tupt, xq ´upt, yq ´K|x ´y|u is positive. Let β be a small positive parameter and define φ β pt, x, yq :" upt, xq ´upt, yq ´K|x ´y| ´β|y| 2 ´β 1 t .

The function φ β reaches its maximum at some point pt, x, yq P p0, T s ˆRd ˆRd and there is β 0 ą 0 such that for 0 ă β ď β 0 φ β pt, x, yq ě δ 2 .

(2.23)

Suppose that β ď β 0 and t " T , then

δ 2 ď φ β pT, x, yq " upT, xq ´upT, yq ´K|x ´y| ´β|y| 2 ´β T ď p}Dg 1 } 8 ´Kq|x ´y|.
But this is impossible since K ą }Dg 1 } 8 and δ ą 0. Thus for all β ď β 0 , t ‰ T . From (2.23) we deduce that β|y| 2 ď 2}u} 8 and therefore β|y| Ñ 0 as β Ñ 0. We also deduce that From now on, we let ξ :" K x´y |x´y| and γ :" 3K |x´y| . We are going to show that the information

δ 2 ď upt,
$ ' ' ' ' & ' ' ' ' % |ν| ď L T |ξ| " K ˆX 0 0 ´pY `2βI d q ˙ď γ ˆId ´Id ´Id I d Ḣ `t, x, ξ, X ˘´f 1 2 pt, xq ď ν ď H `t, y, ξ ´2βy, Y ˘´f 1 2 pt, yq (2.25)
is inconsistent whenever K is sufficiently large. Let η :" ξ ´2βy and for any λ P r0, 1s, x λ :" p1 ´λqx `λy and ξ λ :" p1 ´λqξ `λη " ξ ´2λβy. From [START_REF] Armstrong | Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions[END_REF], Lemma A.2, there exists a C 1 map, λ Ñ Z λ from r0, 1s Ñ S d pRq such that

$ & % d dλ Z λ " γ ´1Z 2 λ , Z 0 " X @λ P r0, 1s, X ď Z λ ď Y `2βI d .
Let us define l : r0, 1s Ñ R by lpλq " Hpt, x λ , ξ λ , Z λ q ´f 1 2 pt, x λ q, so that lp0q " H `t, x, ξ, X ˘´f 1 2 pt, xq ď ν and (using Z 1 ď Y `2βI d and the boundness of B M H) lp1q " Hpt, y, η, Z 1 q ě Hpt, y, η, Y `2βI d q ´f 1 2 pt, yq ě Hpt, y, η, Y q ´Cβ ´f

Hpt, x λ , ξ λ , 0q ´f 1 2 pt, x λ q ´ν `Cλβ " Hpt, x λ , ξ λ , 0q ´Hpt, x λ , ξ λ , Z λ q ď ´BM Hpt, x λ , ξ λ , Z λ q.Z λ ď ? dΛ `|Z λ |,
where we used Cauchy-Schwarz inequality at the last step. Therefore we have

|Z λ | ě Hpt, x λ , ξ λ , 0q ´f 1 2 pt, x λ q ´ν `Cλβ ? dΛ `.
(2.28)

Now we use (2.27). Computing the derivative of l gives l 1 pλq " B x Hpt, x λ , ξ λ , Z λ q.py ´xq ´Bx f 1 2 pt, x λ q.py ´xq ´2βB p Hpt, x λ , ξ λ , Z λ q.y `γ´1 B M Hpt, x λ , ξ λ , Z λ q.Z 2 λ ě ´Cβ and, since ´BM H ě Λ ´Id and B x H satisfies Assumption (2e), we get

|Z λ | 2 ď 1 Λ
´"γC β `γB x Hpt, x λ , ξ λ , Z λ q.py ´xq ´γB x f 1 2 pt, x λ q.py ´xq ´2βγB p Hpt, x λ , ξ λ , Z λ q.y ‰ ď 1 Λ

´"γC β `γB x Hpt, x λ , ξ λ , 0q.py ´xq ´γB x f 1 2 pt, x λ q.py ´xq ´2βγB p Hpt, x λ , ξ λ , Z λ q.y ‰ `C Λ ´γpy ´xq.Z λ , and therefore, for another C ą 0, we find |Z λ | 2 ď CpγB x Hpt, x λ , ξ λ , 0q.py ´xq ´γB x f 1 2 pt, x λ q.py ´xq `γ2 |x ´y| 2 `γβ ´2βγB p Hpt, x λ , ξ λ , Z λ q.yq.

On the one hand, being |ξ| " K, γ " 3K |x´y| ď 6K 2 δ (thanks to (2.24) ), lim βÑ0 β|y| " 0 and recalling Assumption (2d) on B p H we find that |γβ| `|2βγB p Hpt, x λ , ξ λ , Z λ q.yq| ď 1 for β (depending on K) small enough. Finally, being B x f 1 2 bounded and γpx ´yq " 3ξ " 3ξ λ `6λβy we find that

|Z λ | 2 ď Cp1 `|ξ λ | 2 ´Bx Hpt, x λ , ξ λ , 0q.ξ λ q (2.29)
for some C ą 0, as soon as β is small enough. Combining (2.28) and (2.29), we get, for some new C ą 0 independent from pK, δ, β, λq,

Hpt, x λ , ξ λ , 0q 2 ď C `1 `ν2 `|ξ λ | 2 ´ξλ .B x Hpt, x λ , ξ λ , 0q ˘.
We get a contradiction letting β Ñ 0 as soon as K is big enough since |ξ| " K, Hpt, x, p, 0q ě α 1 |p| r 1 ´CH with r 1 ą 1 for all pt, x, pq and B x H satisfies Assumption (2f) (either (2(f)i) or (2(f)ii) is enough to conclude here).

To conclude the proof of Theorem 2.1 we need to show Lipschitz estimates which hold when f 1 2 is merely Hölder continuous in time. This requires more space regularity for f 1 2 and also Assumption (2(f)ii) instead of Assumption (2(f)i).

Proof of Theorem 2.1. When f 1 2 " 0, the previous lemma and the discussion at the beginning of this section are enough to conclude. In the general case, take a smooth kernel ρ with support in r´1, 1s and define for all n P N ˚, ρ n prq :" nρpnrq and f 1pnq 2 pt, xq :" ż 1 ´1 f 1 2 ps, xqρ n pt ´sqds, where we extended f 1 2 to r´1, T `1s ˆRd by f 1 2 pt, xq " f 1 2 p0, xq for t P r´1, 0s and f 1 2 pt, xq " f 1 2 pT, xq for t P rT, T `1s. We also define u n to be the viscosity solution to

" ´Bt u n pt, xq `Hpt, x, Du n pt, xq, D 2 u n pt, xqq " f 1pnq 2 pt, xq in r0, T s ˆRd upT, xq " g 1 pxq in R d .
Thanks to the previous lemma and the discussion at the beginning of this section we know that u n actually belongs to C 3`α 2 ,3`α b pr0, T s ˆRd q. Now we use Bernstein method. We define w n :" Fix pt 0 , x 0 q P r0, T s ˆRd and consider a weak solution m n P Cprt 0 , T s, P 2 pR d qq to the adjoint equation

$ ' ' ' & ' ' ' % B t m n ´divpB p Hpt, x, Du n pt, xq, D 2 u n pt, xqqm n q `d ÿ i,j"1 B 2
i,j ppB M Hpt, x, Du n pt, xq, D 2 u n pt, xqqq i,j mq " 0 in rt 0 , T s ˆRd m n pt 0 q " δ x 0 Integrating v n against m n gives, after integration by part and reorganizing the terms:

v n pt 0 , x 0 q " ż R d " HpT, x, Dg 1 pxq, D 2 g 1 pxqq ´f 1pnq 2 pT, xq ı dm n pT qpxq ´ż T t 0 ż R d B t Hpt, x, Du n pt, xq, D 2 u n pt, xqqdm n ptqpxqdt `ż T t 0 ż R d B t f 1pnq 2 pt, xqdm n ptqpxqdt
But, again by integration by part, we have

ż T t 0 ż R d B t f 1pnq 2 pt, xqdm n ptqpxqdt " ż R d f 1pnq 2 pT, xqdmpT qpxq ´f 1pnq 2 pt 0 , x 0 q `ż T t 0 ż R d B p Hpt, x, Du n pt, xq, D 2 u n pt, xqq.Df 1pnq 2 pt, xqdm n ptqpxqdt `ż T t 0 ż R d B M Hpt, x, Du n pt, xq, D 2 u n pt, xqq.D 2 f 1pnq 2 pt, xqdm n ptqpxqdt
and we can conclude, using the growth assumption on B p H, Assumption (2d), and the boundness of B M H and B t H, that }B t u n } 8 ď C for some C ą 0 depending only on

}Du n } 8 , }f 1 2 } 8 , }Df 1 2 } 8 , }D 2 f 1 2 } 8 , }Dg 1 } 8 , }D 2 g 1 } 8 but not on }B t f 1pnq 2 } 8 .
Combining the two above estimates, we can use the stability of viscosity solutions to show that u n converges locally uniformly to u, the unique viscosity solution to (2.21) and that u is therefore a globally Lipschitz function. Following the discussion at the beginning of this section this is enough to conclude that u belongs to C

Conclusion

In this paper we investigated a stochastic control problem with constraints on the probability distribution of the output. By reformulating the problem as a control problem for the PDE satisfied by the time marginals of the process we were able to prove the existence of solutions and characterize them. The optimal trajectories and associated controls are given by a coupled system of PDEs associated with an exclusion condition. We proved the sufficiency of these conditions under suitable convexity assumptions.

Appendix

Von-Neumann Theorem

Since it appears twice in our article and in particular in the proof of Theorem 2.5 we recall the statement of the Von-Neumann theorem we are using. The statement and proof can be found in the Appendix of [START_REF] Orrieri | A variational approach to the mean field planning problem[END_REF] and in a slightly different setting, in [START_REF] Simons | Minimax and Monoticity[END_REF].

Theorem 2.6. (Von Neumann) Let A and B be convex sets of some vector spaces and suppose that B is endowed with some Hausdorff topology. Let L be a function satisfying :

a Ñ Lpa, bq is concave in A for every b P B, b Ñ Lpa, bq is convex in B for every a P A.
Suppose also that there exists a ˚P A and C ˚ą sup aPA inf bPB Lpa, bq such that : Remark 2.11. The fact that the infimum in the "inf sup" problem is in fact a minimum is part of the theorem.

A comparison principle

In this section we prove comparison for viscosity solutions of the following equation maxt´B t u `Hpt, x, Du, D 2 uq ´f 1 2 pt, xq, |Du| ´Ku " 0 in p0, T q ˆRd (2.30)

where K ą 0.

Proposition 2.4. Suppose that Assumptions (1) and (2) hold and suppose that f 1 2 P C b pr0, T sR d q is Lipschitz continuous in x P R d uniformly in t P r0, T s. Since u and v are bounded, φ α,β achieves its maximum at some point pt, x, yq P p0, T s Rd ˆRd . Moreover, there exists 0 P p0, 1q such that, for |α| `|β| ď 0 , φ α,β pt, x, yq ě δ 2 . In particular, it holds that β|y| 2 ď }u} 8 `}v} 8 `δ 2 ": C 1 and therefore β|y| ď ? βC 1 . First we suppose that there exists 1 P p0, 0 q such that t ‰ T as soon as |α|`|β| ď 1 . We fix α P p0, 1 2 q and we let β P p0, 1 2 q. We can apply the maximum principle for semicontinuous functions from [START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF] and find that there exist ν P R, X, Y P S d pRq such that # pν, 2 α px ´yq, Xq P P Recalling that β|y| ď ? C 1 β and that 2 α |x ´y| ď K we get a contradiction letting first β Ñ 0 and then α Ñ 0.

Remember that we assumed the existence of 1 P p0, 0 q such that t ‰ T whenever |α| `|β| ď 1 . Now we suppose that for all 1 ą 0 there exist α, β with |α| `|β| ď 1 and t " T . Then we can construct a sequence pα n , β n q such that for all n P N ˚, |α n | `|β n | ď 1 n and t " T . Let ω be a modulus of continuity for x Ñ upT, xq. We have δ 2 ď φ αn,βn pT, x, yq ď upT, xq ´p1 `α2 n qvpT, yq ď upT, yq ´vpT, yq `ωp|x ´y|q ´α2 n vpT, yq ď ωp|x ´y|q ´α2 n vpT, yq and we get a contradiction letting n Ñ `8. Thus we get that, for all ą 0, u ď v on r0, T s ˆRd . Letting go to 0 gives, for all pt, xq P r0, T s ˆRd , upt, xq ď vpt, xq. (Existence of viscosity solutions) For the second point of the proposition, we observe that, being g 1 in C 2 b pR d q, g 1 pxq ´Cg 1 pT ´tq and g 1 pxq `Cg 1 pT ´tq are respectively viscosity sub-solution and super-solution to (2.30) with terminal condition upT, xq " g 1 pxq over R d , as soon as K ě 2}Dg 1 } 8 . Using Perron's method we get, for all K ě 2}Dg} 8 , the existence and uniqueness of a bounded, continuous viscosity solution u to " maxt´B t u `Hpt, x, Du, D

ď sup xPR d g 1 pxq ´upT ´s, xq `C1 T s ď C g 1 s `C1 T s.
Doing the same with v ´we get that |upt, xq ´upt ´s, xq| ď pC g 1 `C1 T qs for all s P r0, T s, all t P rs, T s and all x P R d . As a consequence, we find L ą 0, independent of K ą 0 such that, for all pt, xq, ps, yq P r0, T s ˆRd it holds |upt, xq ´upt, yq| ď L|t ´s| `K|x ´y|.

This concludes the proof of the proposition.

Part II

Optimal control of the Fokker-Planck equation with state constraints in the Wasserstein space Chapter 3

Optimality Conditions Introduction

This paper is devoted to the study of an optimal control problem of the Fokker-Planck equation under state constraints on the space of probability measures. The formulation of the problem is the following. We seek to minimize a cost Jpα, mq :"

ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `GpmpT qq
over pairs pα, mq with m P Cpr0, T s, P 2 pR d qq and α P L 2 mptqbdt `r0, T s ˆRd , R d ˘(the control) satisfying in the sense of distributions the Fokker-Planck equation:

B t m `divpαmq ´∆m " 0 (3.1)
with the initial condition mp0q " m 0 P P 2 pR d q. The flow of probability measures m is also constrained to satisfy the inequality Ψpmptqq ď 0, @t P r0, T s for some function Ψ : P 2 pR d q Ñ R satisfying additional conditions. Here P 2 pR d q is the set of probability measures over R d with finite second order moment. The functions L : R d ˆRd Ñ R and F : P 2 pR d q Ñ R are the running costs and g : P 2 pR d q Ñ R is the final cost.

Our main motivation comes from the theory of stochastic control. The corresponding problem is to minimize:

E "ż T 0 LpX t , α t qdt `ż T 0 FpLpX t qqdt `GpLpX T qq
 over solutions of the stochastic differential equation

dX t " α t dt `?2dB t ,
where the controller controls their drift α t starting from a random position X 0 such that LpX 0 q " m 0 and under the constraint that ΨpLpX t qq ď 0 for all t P r0, T s. In this context, 79 it is well-known that LpX t q solves Equation (3.1) in the sense of distributions and therefore the stochastic control problem reduces to a problem of optimal control of the Fokker-Planck equation (see Chapter 2 and the references therein). Stochastic optimal control problems with constraints on the probability distribution of the output have raised some interest in the past few years in connection with quantile hedging in [START_REF] Föllmer | Quantile hedging[END_REF], stochastic target problems with [START_REF] Bouchard | Optimal control under stochastic target constraints[END_REF][START_REF] Bouchard | Stochastic Target Problems with Controlled Loss[END_REF] and stochastic control problems with expectation constraints -see [START_REF] Chow | On Dynamic Programming Principle for Stochastic Control Under Expectation Constraints[END_REF][START_REF] Guo | Portfolio optimization with a prescribed terminal wealth distribution[END_REF][START_REF] Guo | Calibration of local-stochastic volatility models by optimal transport[END_REF][START_REF] Pfeiffer | Optimality conditions in variational form for non-linear constrained stochastic control problems[END_REF][START_REF] Pfeiffer | Duality and approximation of stochastic optimal control problems under expectation constraints[END_REF] -to name a few. Given the type of constraints we are studying, here it is convenient to state our problem directly as an optimal control problem in the Wasserstein space. Such problems have been studied recently but mostly for control problems for the continuity equation (namely without diffusion term). Different approaches have been considered. In [START_REF] Jimenez | Optimal control of multiagent systems in the Wasserstein space[END_REF][START_REF] Marigonda | Mayer control problem with probabilistic uncertainty on initial positions[END_REF] the authors use the dynamic programming approach and prove that the value function is the viscosity (in a sense adapted to the infinite dimensional setting) of an HJB equation. Whereas in [START_REF] Bonnet | A Pontryagin Maximum Principle in Wasserstein Spaces for Constrained Optimal Control Problems[END_REF][START_REF] Bonnet | Necessary Optimality Conditions for Optimal Control Problems in Wasserstein Spaces[END_REF][START_REF] Bonnet | The Pontryagin Maximum Principle in the Wasserstein Space[END_REF] the authors prove some adapted forms of the Pontryagin maximum principle. Notice that optimal control problems for the Fokker-Planck equation were previously considered in [START_REF] José | On a mean field optimal control problem[END_REF][START_REF] Fleig | Optimal Control of the Fokker-Planck Equation with Space-Dependent Controls[END_REF] but without constraint. Here we emphasize that the constraint is a smooth function defined on the Wasserstein space. In particular, our results do not cover the case of local constraints where the constraint acts on the density (when it exists) of m. This latter problem was addressed in [START_REF] Cardaliaguet | First order mean field games with density constraints: Pressure equals price[END_REF][START_REF] Di | Uniqueness issues for evolution equations with density constraints[END_REF][START_REF] Mészáros | A variational approach to second order mean field games with density constraints : The stationary case[END_REF][START_REF] Mészáros | On The Variational Formulation Of Some Stationary Second-Order Mean Field Games Systems[END_REF][START_REF] Santambrogio | Advection-Diffusion Equations With Density Constraints[END_REF].

Here we follow the path initiated in Chapter 2 of this thesis for a problem with terminal constraint and prove some optimality conditions in the form of a coupled system of partial differential equations associated with an exclusion condition. One of the equations is a Fokker-Planck equation satisfied by the solution of the problem. The other equation is a Hamilton-Jacobi-Bellman equation which is satisfied by an adjoint state, and from which we derive an optimal control. Besides these two equations, the exclusion condition reflects the effect of the constraint on the system. Our strategy is to proceed by penalization. We solve the penalized problem in a way that is closely related to Mean Field Game theory. Indeed, when the game has a potential structure -see for instance [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF][START_REF] Cardaliaguet | Second order mean field games with degenerate diffusion and local coupling[END_REF][START_REF] Lasry | Mean field games[END_REF][START_REF] Orrieri | A variational approach to the mean field planning problem[END_REF] -the system of partial differential equations which describes the value function of a typical infinitesimal player and the distribution of the players can be obtained as optimality conditions for an optimal control problem for the Fokker-Planck equation. With this optimality conditions at hand we proceed to show that solutions to the penalized problem -when the penalization term is large enough-stay inside the constraint at all times and are therefore solutions to the constrained problem. This second step is inspired by ideas in finite dimensional optimal control theory (see [START_REF] Frankowska | Optimal control under state constraints[END_REF]). In particular we follow a method used in [START_REF] Cannarsa | Regularity Properties of Attainable Sets Under State Constraints[END_REF][START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF]. The idea is to look at local maximum points of the function t Þ Ñ Ψpmptqq for some solution m of the penalized problem and prove that they cannot satisfy Ψpmptqq ą 0 when the penalization is strong enough. To this end we compute the second order derivative of t Þ Ñ Ψpmptqq thanks to the optimality conditions previously proved. An interplay between the convexity of the Hamiltonian of the system, a tranversality assumption on the constraint and various estimates on the solutions of the optimality conditions of the penalized problem allows us to conclude. As a by-product of this method we can show that the solutions of the constrained problem enjoy the same regularity as the solutions of the penalized problem. In particular optimal controls are proved to be Lipschitz continuous. This result might seem surprising since the presence of state constraints generally leads to optimal controls which behave badly in time (see [START_REF] Frankowska | Optimal control under state constraints[END_REF] and the references therein). However it is reminiscent of classical results in finite dimensional optimal control theory in the presence of suitable regularity, growth and convexity assumptions as in see [START_REF] Galbraith | Regularity of optimal controls for state constrained problems[END_REF][START_REF] Hager | Lipschitz Continuity for Constrained Processes[END_REF].

The rest of the paper is organized as follows. In Section 3.1 we introduce the notations and state some useful preliminary results on the Fokker-Planck equation and the HJB equation on the one hand, and on the differentiability of maps defined on the space of measures on the other hand. We also state a form of Itô's lemma for flows of probability measures. In Section 3.2 we state the standing assumptions and our main results. In Section 3.3 we obtain optimality conditions for the penalized problem. In Section 3.4 we prove our main theorem. In section 3.5 we extend our results to a more general setting. Finally, we postpone to Section 3.6 some technical results for the Hamilton-Jacobi equation satisfied by the adjoint state, that we use throughout the paper.

Notations For a map u defined on r0, T s ˆRd we will frequently use the notation uptq to denote the function x Þ Ñ upt, xq. Notice that uptq is therefore a function defined on R d . If a function u defined on r0, T sˆR d is sufficiently smooth, we denote by B t u the partial derivative with respect to t and by Du, ∆u :" divDu, D 2 u (if u is a scalar function) or Du, Ý Ñ ∆u :" Ý Ñ divDu if u is vector-valued, the derivatives with respect to x. The Wasserstein space of Borel probability measures over R d with finite moment of order r ě 1 is denoted by P r pR d q. It is endowed with the r-Wasserstein distance d r . The space of n-times differentiable bounded real functions over R d with continuous and bounded derivatives is denoted by C n b pR d q. Given m P Cpr0, T s, P 2 pR d qq we denote by L 2 mptqbdt pr0, T sˆR d , R d q the space of R d -valued, mptqbdtsquare-integrable functions over r0, T sˆR d . The space of finite Radon measures over r0, T s is denoted by Mpr0, T sq, the subset of non-negative measures by M `pr0, T sq and the set of R dvalued Borel measures over r0, T s ˆRd with finite total variation by Mpr0, T s ˆRd , R d q. The space of symmetric matrices of size d is denoted by S d pRq. We denote by C 1,2 b pr0, T sˆR d q the space of bounded functions with one bounded continuous derivative in time and two bounded continuous derivatives in space. Finally we denote by W 1,8 pr0, T s ˆRd q the subspace of L 8 pr0, T s ˆRd q consisting of functions which have one bounded distributional derivative in space and one bounded distributional derivative in time. For n ě 1 we denote by E n the subspace of C n pR d q consisting of functions u such that }u} n :" sup

xPR d |upxq| 1 `|x| `n ÿ k"1 sup xPR d ˇˇD k upxq ˇˇă `8.
Similarly we define E n`α for n ě 1 and α P p0, 1q to be the subset of E n consisting of functions u satisfying }u} n`α :" }u} n `sup x‰y |D n upxq ´Dn upyq| |x ´y| α ă `8.

For α P p0, 1q we say that u P Cpr0, T s ˆRd q belongs to the parabolic Hölder space C p1`αq{2,1`α pr0, T s ˆRd q if u is differentiable in x and }u}1`α is finite. Finally we will use the heat kernel P t associated to ´∆ defined, when it makes sense, by

P t f pxq :" ż R d 1 p4πtq d{2 e ´|x´y| 2 4t
f pyqdy.

Preliminaries

We start by introducing the main protagonists of this paper. The first one is the Fokker-Planck equation.

The Fokker-Planck equation. Given m P Cpr0, T s, P 2 pR d qq and α P L 2 mptqbdt `r0, T s ˆRd , R d ˘, we say that pm, αq satisfies the Fokker-Planck equation:

B t m `divpαmq ´∆m " 0 (3.2)
if for all ϕ P C 8 c pp0, T q ˆRd q we have ż T b pr0, T s ˆRd q and for all ϕ P C 1,2 b pr0, T s ˆRd q and all t 1 , t 2 P r0, T s it holds

ż R d ϕpt 2 , xqdmpt 2 qpxq " ż R d ϕpt 1 , xqdmpt 1 qpxq `ż t 2 t 1 ż R d rB t ϕpt, xq `Dϕpt, xq.αpt, xq `∆ϕpt, xqs dmptqpxqdt.
Throughout the paper, we will repeatedly use the following properties of solutions to the Fokker-Planck equation. The proofs are given in the appendix.

PRELIMINARIES

We also have the following compactness result.

Proposition 3.2. Assume that, for all k ě 1, pm k , ω k q solve the Fokker-Planck equation starting from m 0 and satisfies the uniform energy estimate

ż T 0 ż R d |α k pt, xq| 2 dm k ptqpxqdt ď C,
for some C ą 0 independent of k. Then, up to taking a sub-sequence, pm k , α k m k q converges in C 1´δ 2 pr0, T s, P 2´δ pR d qq ˆMpr0, T s ˆRd , R d q for any δ P p0, 1q toward some pm, ωq. The curve m belongs to C 1{2 pr0, T s, P 2 pR d qq, ω is absolutely continuous with respect to mptq b dt, it holds that

ż T 0 ż R d ˇˇˇd ω dmptq b dt pt, xq ˇˇˇ2 dmptqpxqdt ď lim inf kÑ`8 ż T 0 ż R d ˇˇˇd ω k dm k ptq b dt pt, xq ˇˇˇ2 dm k ptqpxqdt
and, finally, pm, dω dtbdm q solves the Fokker-Planck equation (3.2) starting from m 0 .

The HJB equation The second protagonist of this paper is the following Hamilton-Jacobi-Bellman equation. It involves the Hamiltonian H : R d ˆRd Ñ R d of the system. For the following definition to make sense and the next theorem to hold, H is assumed to satisfy Assumption (AH), introduced in the next section.

Definition 3.1. Let f P L 1 pr0, T s, E n q and g P E n`α for some n ě 2. We say that u P L 1 pr0, T s, E n q is a solution to Let us point out that a solution u P Cpr0, T s, E n q for n ě 3 is differentiable in time whenever f is continuous and, at these times, the HJB equation is satisfied in the usual sense.

" ´Bt u `Hpx, Duq ´∆u " f in r0, T s ˆRd , upT, xq " g in R d , (3.4
We introduce this notion to handle solutions which are smooth in x at each time but not necessarily regular in the time variable.

The following theorem is proved in Section 3.6.1.

Theorem 3.1. Take n ě 2. Assume that f belongs to L 1 pr0, T s, E n q, g belongs to E n`α and H satisfies Assumption (AH) then,

• The HJB equation (3.4) admits a unique solution u in Cpr0, T s, E n q in the sense of definition 3. • Assume that pf m , g m q belongs to L 1 pr0, T s, E n q ˆEn`α for all m ě 1 and that f m converges to f in L 1 pr0, T s, E n q and g m converges to g in E n`α . Let u m be the solution to (3.4) with data pf m , g m q, then u m converges to u in L 8 pr0, T s, E n q.

Differentiability on the Wasserstein space and Itô's formula for flows of probability measures. We say that a map U : P 2 pR d q Ñ R m is C 1 if there exists a jointly continuous map δU δm : P 2 pR d q ˆRd Ñ R m such that, for any bounded subset K Ă P 2 pR d q,

x Ñ δU δm pm, xq has at most quadratic growth in x uniformly in m P K and such that, for all m, m 1 P P 2 pR d q, U pm 1 q ´U pmq "

ż 1 0 ż R d δU δm pp1 ´hqm `hm 1 , xqdpm 1 ´mqpxqdh.
The function δU δm is defined up to an additive constant and we adopt the normalization convention

ż R d δU δm pm, xqdmpxq " 0.
In the terminology of [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] it means that U admits a linear functional derivative. When the

map x Þ Ñ δU δm pm, xq is differentiable we define the intrinsic derivative of U D m U pm, xq :" D x δU δm pm, xq.
The following form of Itô's lemma -formulated in terms of SDEs-is proved (under more general assumptions) in [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] Theorem 5.99. Proposition 3.3. Take m P Cpr0, T s, P 2 pR d qq and α P L 2 mptqbdt `r0, T s ˆRd , R d ˘such that pm, αq is a solution of the Fokker-Planck equation (3.2) and suppose that U :

P 2 pR d qˆR d Ñ R is C 1 with δU δm satisfying @m P P 2 pR d q, x Þ Ñ δU δm pm, xq P C 2 pR d q, with pm, xq Þ Ñ D m U pm, xq and pm, xq Þ Ñ D x D m U pm,
xq being bounded on P 2 pR d q ˆRd and jointly continuous. Then, for all t P r0, T s, it holds that

U pmptqq " U pmp0qq `ż t 0 ż R d D m U pmpsq, xq.αps, xqdmpsqpxqds `ż t 0 ż R d div x D m U pmpsq, xqdmpsqpxqds.
Notice that Proposition 5.48 of [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] ensures that U satisfies the assumptions of Theorem 5.99.

Main results and assumptions

where Jpα, mq :"

ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `GpmpT qq
is the total cost and the infimum runs over all pα, mq such that

$ ' ' & ' ' % m P Cpr0, T s, P 2 pR d qq, α P L 2 dtbmptq pr0, T s ˆRd , R d q, B t m `divpαmq ´∆m " 0 in p0, T q ˆRd , mp0q " m 0 , (3.5)
where the Fokker-Planck equation is understood in the sense of distributions. Here, the Lagrangian L is defined by Lpx, qq :" sup pPR d t´p.q ´Hpx, pqu and the data are the finite horizon T ą 0, the Hamiltonian H : R d ˆRd Ñ R, the mean-field costs F : P 2 pR d q Ñ R and G : P 2 pR d q Ñ R and the initial measure m 0 P P 2 pR d q. The above data are supposed to satisfy the following conditions for some fixed integer n ě 3.

For U " F, G, the map U : P 2 pR d q Ñ R d satisfies # U is a bounded from below, C 1 map and δU δm belongs to CpP 2 pR d q, E n`α q.

(Ureg)

For some positive constant C 0 ą 0 it holds " for all px, pq P R d ˆRd ,

C ´1 0 |p| 2 ´C0 ď Hpx, pq ď C 0 |p| 2 `C0 . (AH2) H belongs to C n pR d ˆRd q. ( AH3
where the unknown pu, mq belong to C 1,2 pp0, T q ˆRd q.

The purpose of the present work is to investigate the effect of a state constraint Ψpmptqq ď 0, @t P r0, T s, on the problem above. Here Ψ : P 2 pR d q Ñ R satisfies the regularity assumption (Ureg) and is convex for the linear structure of P 2 pR d q " For all m 1 , m 2 P P 2 pR d q and all λ P r0, 1s, Ψpp1 ´λqm 1 `λm 2 q ď p1 ´λqΨpm 1 q `λΨpm 2 q.

(APsiConv)

We also need to assume that the problem is initialized at a point m 0 in the interior of the constraint that is Ψpm 0 q ă 0.

(APsiInside)

In addition to the previous assumptions we will ask for second-order differentiability with respect to the measure variable for Ψ.

$ ' ' ' & ' ' ' % For all x P R d , m Þ Ñ δΨ δm pm, xq is C 1 with px, yq Þ Ñ δ 2 Ψ δm 2 pm, x, yq :" δ 2 Ψ δm 2 pm, xqpyq in C 2 pR d ˆRd
q for all m P P 2 pR d q and δ 2 Ψ δm 2 pm, x, yq and its derivatives being jointly continuous and bounded in P 2 pR d q ˆRd ˆRd .

(APsiC2) Notice that Assumption (APsiC2) implies in particular (see for instance [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] Remark 5.27) that the map pm, xq Þ Ñ D m Ψpm, xq is uniformly Lipschitz continuous over P 1 pR d q ˆRd . Finally we require the following geometric assumption on the constraint.

ż R d |D m Ψpm, xq| 2 dmpxq ‰ 0 whenever Ψpmq " 0. (APsiTrans)
The transversality assumption (APsiTrans) is not necessary to get the optimality conditions however it is the key assumption to obtain the time regularity of optimal controls.

An example of constraint Ψ : with x 0 P R d , δ ą 0 and κ ą 0. We can compute the derivatives, for all m P P 2 pR d q and all x P R d :

P 2 pR d q Ñ R
δΨ δm pm, xq " a |x ´x0 | 2 `δ2 ´δ ´żR d ´a|y ´x0 | 2 `δ2 ´δ¯d mpyq, D m Ψpm, xq " x ´x0 a |x ´x0 | 2 `δ2 , div x D m Ψpm, xq " d a |x ´x0 | 2 `δ2 ´|x ´x0 | 2 p|x ´x0 | 2 `δ2 q 3{2 , δ 2 Ψ δm 2 pm, x 1 , x 2 q " 0. In particular, it holds that ż R d |D m Ψpm, xq| 2 dmpxq " ż R d |x ´x0 | 2 |x ´x0 | 2 `δ2 dmpxq,
and it is easily seen that Ψ satisfies the tranversality condition APsiTrans.

We can finally state the main problem of interest in this paper:

inf pα,mq ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `GpmpT qq (P)
where the infimum runs over the pairs pm, αq satisfying (3.5) and the state constraint

@t P r0, T s, Ψpmptqq ď 0.
Over the course of the paper we will introduce several auxiliary problems. The main one is the following. For , δ ą 0 the penalized problem (P ,δ ) is inf pm,αq J ,δ pα, mq (P ,δ )

where the infimum runs over all pm, αq satisfying (3.5) (but not necessarily the state constraint) and J ,δ is defined by

J ,δ pα, mq :" ż T 0 ż R d Lpx, αpt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `1 ż T 0 Ψ `pmptqqdt `GpmpT qq `1 δ Ψ `pmpT qq " Jpα, mq `1 ż T 0 Ψ `pmptqqdt `1 δ Ψ `pmpT qq.
Here and in the following, Ψ `pmq " Ψpmq _ 0 " maxpΨpmq, 0q. Notice that Problem (P ,δ ) is very similar to Problem (uP) although we have to deal with the non-differentiability at 0 of the map r Þ Ñ maxpr, 0q. We now state our main results. The first one is not expected without Assumption (APsiTrans). Roughly speaking, it asserts that optimal solutions to the penalized problems P ,δ stay inside the constraint when the penalization is strong enough. Theorem 3.2. Take n ě 3. Assume that (AH) holds for H, (Ureg) holds for F and G. Assume further that Ψ satisfies Assumptions (Ureg), (APsiConv), (APsiInside), (APsiC2) and (APsiTrans). Then there exist 0 , δ 0 ą 0 depending on m 0 only through the value Ψpm 0 q such that, for all p , δq in p0, 0 q ˆp0, δ 0 q Problems (P ,δ ) and (P) have the same solutions.

As a consequence we find the following optimality conditions for the optimal control problem with constraint. Theorem 3.3. Under the same assumptions as Theorem 3.2, Problem (P) admits at least one solution and, for any solution pα, mq there exist u P Cpr0, T s, E n q, ν P L 8 pr0, T sq and η P R `such that α " ´Dp Hpx, Duq

where the Fokker-Planck equation is understood in the sense of distributions and u solves the HJB equation in the sense of Definition 3.1.

The Lagrange multipliers ν and η satisfy νptq "

" 0 if Ψpmptqq ă 0 νptq P R `if Ψpmptqq " 0, (3.9)
η " " 0 if ΨpmpT qq ă 0 η P R `if ΨpmpT qq " 0.

(3.10)

If we also assume that F and G are convex in the measure variable, then the above conditions are sufficient conditions: if pm, αq satisfies Ψpmptqq ď 0 for all t P r0, T s and if there exists pu, ν, ηq such that (3.7), (3.8), (3.9) and (3.10) hold then pα, mq is a solution to (P).

The strength of the above result relies on the regularity of the Lagrange multiplier ν associated to the constraint that for all t P r0, T s, Ψpmptqq ď 0. Indeed we would a priori expect ν to be a finite Radon measure over r0, T s but here we find that ν belongs to L 8 pr0, T sq. As a consequence -and as explained in Remark 3.3 below-optimal controls are Lipschitz continuous in time.

We complete this section with a few comments.

Remark 3.1. Arguing as in [START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF], in the proof of Theorem 3.1, we can use the expression of

d 2
dt 2 Ψpmptqq given by Proposition 3.5 to express νptq as a (non-local) feedback function of Duptq, D 2 uptq and mptq.

Remark 3.2. Computing the cost of an optimal control we see that the value of the problem denoted by Upm 0 q is given by

Upm 0 q " ż R d up0, xqdm 0 pxq `ż T 0 Fpmptqqdt `GpmpT qq
for any solution pm, ´Dp Hpx, Duqq of (P).

Remark 3.3. Differentiating the HJB equation with respect to x shows that Du actually belongs to W 1,8 pr0, T sˆR d , R d q and since Du is also continuous and D p H is locally Lipschitz we get that α is Lipschitz continuous. In particular the Stochastic Differential Equation X t " X 0 `ż t 0 αps, X s qds `?2B t where X 0 " m 0 , admits a unique strong solution and we can proceed as in Chapter 2 in the proof of Theorem 2.2 to find strong solutions to the stochastic analog of Problem (P) (as stated in the introduction). (which does not satisfy the growth conditions of Assumptions (Ureg) and (APsiC2)) for some κ ą 0. However this would significantly increase the technicality of the paper and we leave this case for future research. Among other difficulties we would have to solve the backward HJB equation in (3.8) when the source term has a quadratic growth in the space variable.

Remark 3.5. Our results could be naturally extended to multiple (possibly time dependent) equality or inequality constraints under suitable qualification conditions but we focus on this case of just one inequality constraint for the sake of clarity in an already technical paper.

Optimality conditions without Assumptions (APsiC2) and (APsiTrans).

When Assumptions (APsiC2) and (APsiTrans) are not satisfied we do not expect the conclusions of Theorem 3.2 to hold. However, we can pass to the limit as , δ go to 0 in the Penalized problem (P ,δ ) and find the optimality conditions for the constrained problem. Theorem 3.4. Assume that AH holds for H, Ureg holds for F and G. Assume further that Ψ satisfies Assumptions (Ureg), (APsiConv) and (APsiInside). Then the conclusions of Theorem (3.3) hold true with ν P M `pr0, T sq, and u P L 8 pr0, T s, E n q. The exclusion condition for ν now reads Ψpmptqq " 0, for ν-almost all t P r0, T s. Finally the optimal control α belongs to

BV loc pr0, T s ˆRd , R d q Ş L 8 pr0, T s, C n´1 b pR d , R d qq.
In this (slightly more) general case, we lose the time regularity of the optimal controls. This is due to the shocks that can occur when the optimal curve t Ñ mptq touches the constraint. Indeed, the set of times where the optimal control is not continuous, is contained into the support of the singular part of the Lagrange multiplier ν. However, the space regularity of the backward component u of the system and of the optimal control ´Dp Hpx, Duq remains.

The proof of Theorem 3.4 is the aim of section 3.5 where we discuss in particular the well-posedness of the HJB equation when the Lagrange multiplier ν belongs to M `pr0, T sq.

The penalized problem

In this section we analyze the penalized problem (P ,δ ). The main result is the following. Theorem 3.5. Problem (P ,δ ) admits at least one solution and, for any solution pα, mq of (P ,δ ) there exist u P Cpr0, T s, E n q, λ P L 8 pr0, T sq and β P r0, 1s such that

$ & % " 0 if Ψpmptqq ă 0 P r0, 1s if Ψpmptqq " 0 " 1 if Ψpmptqq ą 0, (3.12) β $ & % " 0 if ΨpmpT qq ă 0 P r0, 1s if ΨpmpT qq " 0 " 1 if ΨpmpT qq ą 0. (3.13)
The proof of Theorem 3.5 will be divided into three steps. First we are going to prove the existence of (relaxed) solutions to the problem. This is Lemma 3.1. In the second step, we will show that these relaxed solutions are actually solutions of a suitable linearized problem. This is Lemma 3.2. Finally, we will conclude the proof of Theorem 3.5 by computing the optimality conditions for this linearized problem. The three steps above are very similar to what is done in [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] Lemma 3.1 and in Chapter 2, Section 2.2 of this thesis. Here, however we have to deal with the lack of differentiability at 0 of the function r Þ Ñ maxp0, rq. We also proceed differently at the end of the proof of Theorem 3.5, where we argue by verification to avoid the unnecessary use of a min/max argument.

We start with the existence of relaxed solutions. A relaxed candidate is a pair pm, ωq such that $ ' ' & ' ' % m P Cpr0, T s, P 2 pR d qq, ω P Mpr0, T s ˆRd , R d q, B t m `divpωq ´∆m " 0 in p0, T q ˆRd , mp0q " m 0 , (3.14) where the Fokker-Planck equation is once again understood in the sense of distributions.

A relaxed solution is a minimizer over all the relaxed candidates of the following functional still denoted (with a slight abuse of notations) by J ,δ J ,δ pm, ωq :"

ż T 0 ż R d Lpx, dω dt b dmptq pt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `1 ż T 0 Ψ `pmptqqdt `GpmpT qq `1 δ Ψ `pmpT qq,
where we set J ,δ pm, ωq " `8 if ω is not absolutely continuous with respect to dt b mptq.

Lemma 3.1. Problem (P ,δ ) admits at least one relaxed solution.

The existence of relaxed solutions is standard (see [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] or Chapter 2 Section 2.2) but we give the proof in Appendix 3.7.1 for the sake of completeness and because we will use the same line of arguments at different points in our analysis.

Notice that it would not be more difficult to obtain weak solutions directly for the constrained problem. However, for the constrained problem, we don't know how to directly compute the optimality conditions and more importantly they would not give us the regularity of the Lagrange multipliers that we get thanks to our penalization procedure. Now we fix a solution p r m, r ωq of the penalized problem and we proceed to show that p r m, r ωq is solution to a suitable linearized problem for which it will be easier to compute the optimality conditions. In the proof of the following lemma we will use a smooth distancelike function. To this end we consider a family pϕ i q iPN of functions in C 2 b pR d q such that for m 1 , m 2 P P 2 pR d q we have

m 1 " m 2 ô @i P N ż R d ϕ i pxqdpm 1 ´m2 qpxq " 0,
and we define q : P 2 pR d q ˆP2 pR d q Ñ R by

qpm 1 , m 2 q :" `8 ÿ i"0 ˇˇˇż R d ϕ i dpm 1 ´m2 q ˇˇˇ2 2 i p1 `}ϕ i } 2 8 `}Dϕ i } 2 8 q
.

Notice that q satisfies " qpm 1 , m 2 q ě 0 @m 1 , m 2 P P 2 pR d q qpm 1 , m 2 q " 0 if and only if m 1 " m 2 .

(3.15)

It is straightforward to verify that q is C 1 with respect to both of its arguments and that

δq δm 1 pm 1 , m 2 qpxq " `8 ÿ i"0 2 ż R d ϕ i dpm 1 ´m2 q 2 i p1 `}ϕ i } 2 8 `}Dϕ i } 2 8 q pϕ i pxq ´żR d ϕ i dm 1 q.
In particular we have

$ ' & ' % ż R d δq δm 1 pm 1 , m 2 qpyqdm 1 pyq " 0 @m 1 , m 2 P P 2 pR d q, δq δm 1
pm 1 , m 1 qpxq " 0 @m 1 P P 2 pR d q and @x P R d .

( If p r m 1 , r ω 1 q is a solution of the above problem, then r m 1 " r m. This is a direct consequence of (3.15) and the fact that p r m, r ωq is a solution of the penalized problem. We use this function q (and not the Wasserstein distance for instance) because it is smooth and therefore we can differentiate it to get optimality conditions and also because δq δm p r m, r m, xq " 0 for all

x P R d (see (3.16)): therefore q will not appear in the optimality conditions for p r m, r ωq. Now, we introduce a suitable regularization of the function r Þ Ñ maxp0, rq. For all h ą 0, let γ h : R Ñ R `be functions satisfying

$ ' ' ' ' & ' ' ' ' % γ h P C 2 pRq, γ h ě 0, γ h prq " maxp0, rq in Rzr´h, hs, sup rPR |γ 1 h prq| ď 1, sup rPR |γ h prq ´maxp0, rq| Ñ 0 as h Ñ 0.
We consider the regularized, penalized cost functionals J ,δ,h pm, ωq :"

ż T 0 ż R d Lpx, dω dt b dmptq pt, xqqdmptqpxqdt `ż T 0 Fpmptqqdt `1 ż T 0 Ψ h pmptqqdt `GpmpT qq `1 δ Ψ h pmpT qq
where Ψ h is defined for all m P P 2 pR d q by Ψ h pmq " γ h pΨpmqq. Now we argue as in the proof of Lemma 3.1 (see Appendix 3.7.1) and find for all h P p0, 1q a solution pm h , ω h q of inf " J ,δ,h pm, ωq

`ż T 0 qpmptq, r mptqqdt  . (3.20)
Taking for granted that we can find a candidate pm, ωq such that Jpm, ωq ă `8 and Ψpmptqq ď 0 for all t P r0, T s (we explicitly construct such a candidate in Lemma 3.4 in Section 3.4.1 below) we find that J ,δ,h pm h , ω h q is bounded from above by Jpm, ωq independently of , δ and h. By coercivity of L we deduce that

ż T 0 ż R d ˇˇˇd ω h dt b dm h ptq pt, xq ˇˇˇ2 dm h ptqpxqdt ď C
for some C ą 0 independent of , δ and h. Following the proof of Lemma 3.1 in Appendix 3.7.1, we deduce that pm h , ω h q converges, up to a sub-sequence, in Cpr0, T s, P r pR d qq Mpr0, T sˆR d , R d q for some r P p1, 2q to an element pm 1 , ω 1 q of Cpr0, T s, P 2 pR d qqˆMpr0, T sR d , R d q satisfying (3.14) with ω 1 absolutely continuous with respect to m 1 . Let us prove that pm 1 , ω 1 q is a minimizer of (3.19) and therefore, by uniqueness -that is why we added the q-term in the cost functional-, m 1 " r m. We just need to show that

J ,δ pm 1 , ω 1 q `ż T 0 qpm 1 ptq, r mptqqdt ď J ,δ p r m, r ωq.
However, for any h P p0, 1q, using the minimality of pm h , ω h q for Problem (3.20) it holds,

J ,δ pm 1 , ω 1 q `ż T 0 qpm 1 ptq, r mptqqdt ´J ,δ p r m, r ωq " J ,δ,h pm h , ω h q `ż T 0 qpm h ptq, r mptqqdt ´J ,δ,h p r m, r ωq `J ,δ pm 1 , ω 1 q ´J ,δ,h pm h , ω h q `ż T 0 qpm 1 ptq, r mptqqdt ´ż T 0 qpm h ptq, r mptqqdt `J ,δ,h p r m, r ωq ´J ,δ p r m, r ωq ď J ,δ pm 1 , ω 1 q ´J ,δ,h pm h , ω h q `ż T 0 qpm 1 ptq, r mptqqdt ´ż T 0 qpm h ptq, r mptqqdt `J ,δ,h p r m, r ωq ´J ,δ p r m, r ωq.
Since ż T 0 qpm 1 ptq, r mptqqdt ´ż T 0 qpm h ptq, r mptqqdt and J ,δ,h p r m, r ωq ´J ,δ p r m, r ωq converge to 0 as h converges to 0, it is sufficient to prove that J ,δ pm 1 , ω 1 q ď lim inf hÑ0 J ,δ,h pm h , ω h q. For all h ą 0 we can rewrite

J ,δ,h pm h , ω h q " J ,δ pm h , ω h q`1 ż T 0 " Ψ h pm h ptqq ´Ψ`p m h ptqq ‰ dt`1 δ " Ψpm h pT qq ´Ψ`p m h pT qq ‰ but lim hÑ0 1 ż T 0 " Ψ h pm h ptqq ´Ψ`p m h ptqq ‰ dt `1 δ " Ψ h pm h pT qq ´Ψ`p m h pT qq ‰ " 0
and therefore lim inf hÑ0 J ,δ,h pm h , ω h q " lim inf hÑ0 J ,δ pm h , ω h q. Finally we can conclude by lower semi-continuity of J ,δ that lim inf hÑ0 J ,δ pm h , ω h q ď J ,δ pm 1 , ω 1 q. The lower semicontinuity of J ,δ can be proved following Theorem 2.34 of [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]. Now we argue as in Chapter 3.7 Section 2.3.1 to show that, for all h ą 0, pm h , ω h q is actually an infimum of the linearized problem inf J l ,δ,h pm, ωq

`ż T 0 ż R d δq δm 1 pm h ptq, r mptq, xqdmptqpxqdt (3.21)
where the infimum is still taken over relaxed candidates pm, ωq satisfying (3.14) with the linearized cost functional J l ,δ,h defined by

J l ,δ,h pω, mq " ż T 0 ż R d Lpx, dω dt b dmptq pt, xqqdmptqpxqdt `ż T 0 ż R d " 1 δΨ h δm pm h ptq, xq `δF δm pm h ptq, xq  dmptqpxqdt `żR d " 1 δ δΨ h δm pm h pT q, xq `δG δm pm h pT q, xq  dmpT qpxq,
with, once again J l ,δ,h pω, mq " `8 if ω is not absolutely continuous with respect to mptqbdt. Indeed, take a candidate pm, ωq with finite cost, take l P p0, 1q and define pm l , ω l q :" p1 ´lqpm h , ω h q `lpm, ωq. By minimality of pm h , ω h q we have, for all l P p0, 1q

1 l " J ,δ,h pm h , ω h q `ż T 0 qpm h ptq, r mptqqdt ´J ,δ,h pm l , ω l q ´ż T 0 qpm l ptq, r mptqqdt  ď 0.
The statement is proved letting l Ñ 0 in the expression above and using, on the one hand, the convexity of pm, ωq Þ Ñ

ż T 0 ż R d
Lpx, dω dt b dmptq pt, xqqdmptqpxq and, on the other hand, the differentiability of the mean-field costs. Now we are going to pass to the limit in the linearized problems when h Ñ 0. On the one hand, being the family of functions t Þ Ñ γ 1 h pΨpm h ptqq bounded in L 8 pr0, T sq, it converges -up to a sub-sequence-for the weak-˚topology σpL 8 , L 1 q of L 8 pr0, T sq to a function λ in L 8 pr0, T sq. It is easily seen that λ satisfies (3.17). On the other hand the

functions t Þ Ñ ż R d δΨ δm pm h ptq, xqdmptqpxq converge uniformly to t Þ Ñ ż R d δΨ δm p r mptq, xqdmptqpxq
as h goes to 0. Therefore we can conclude that, up to a sub-sequence,

ż T 0 ż R d δΨ h δm pm h ptq, xqdmptqpxqdt " ż T 0 γ 1 h pΨpm h ptqq ż R d δΨ δm pm h ptq, xqdmptqpxqdt Ñ ż T 0 λptq ż R d δΨ δm p r mptq, xqdmptqpxqdt
as h goes to 0. A similar statement holds for 1 δ ż R d δΨ h δm pm h pT q, xqdmpT qpxq and we can conclude that, up to a sub-sequence, J l ,δ,h pm, ωq converges to J l ,δ pm, ωq for any relaxed candidate pm, ωq, where J l ,δ is defined in the statement of the lemma for some λ, β satisfying the conditions (3.17) and (3.18). We deduce that p r m, ω 1 q is an infimum of J l ,δ . Notice that the term involving δq δm 1 in (3.21) disappeared since δq δm 1 p r mptq, r mptq, xq " 0 for all x P R d .

To conclude that p r m, r ωq is a solution to the linearized problem, it suffices to notice that, p r m, r ωq being a solution to the penalized problem it must hold that

ż T 0 ż R d Lpx, dr ω dt b d r mptq pt, xqqd r mptqpxqdt ď ż T 0 ż R d Lpx, dω 1 dt b d r mptq pt, xqqd r mptqpxqdt
(all the other terms in the J ,δ only involve r m) and therefore J l ,δ p r m, r ωq ď J l ,δ p r m, ω 1 q. This concludes the proof of the lemma.

Before we can prove Theorem 3.5 we need the following duality formula. Lemma 3.3. Assume that pm, αq P Cpr0, T s, P 2 pR d qq ˆL2

dtbdmptq pr0, T s ˆRd , R d q solves the Fokker-Planck equation (3.2) in the sense of distributions. Assume that u P Cpr0, T s, E n q is a solution to the HJB equation (3.4) in the sense of Definition 3.4 with inputs pf, gq P L 1 pr0, T s, E n q ˆEn`α . Then, for all t 1 , t 2 P r0, T s it holds

ż R d upt 2 , xqdmpt 2 qpxq " ż R d upt 1 , xqdmpt 2 qpxq ´ż t 2 t 1 ż R d f pt, xqdmptqpxqdt `ż t 2 t 1 ż R d rHpx, Dupt, xqq `αpt, xq.Dupt, xqs dmptqpxqdt. (3.22)
Proof. We take a sequence of functions f m P Cpr0, T s, E n q converging to f in L 1 pr0, T s, E n q and we let u m be the corresponding solutions to the HJB equation with data pf m , gq. Being f m in Cpr0, T s, E n q, it is straightforward from the definition of solution 3.4 that u m is differentiable in time, B t u m belongs to L 8 pr0, T s, E n´2 q and the HJB equation is satisfied in the strong sense. The curve mptq being bounded in P 2 pR d q, an approximation argument similar to [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] where we used the equation satisfied by u m at the last line. Now we can use the stability result of Theorem 3.1 to pass to the limit as m Ñ `8 and conclude the proof of the proposition.

Finally we can conclude the proof of Theorem 3.5.

Proof of Theorem 3. Here we used the equation satisfied by r u and the convention Combining the Fokker-Planck equation in (3.14) where r ω is replaced by ´Dp Hpx, Dr upt, xqq r mptqb dt with the HJB equation (3.23) and recalling that λ and β satisfy the conditions of Lemma 3.2 concludes the proof of the theorem.

From the penalized problems to the constrained one

The first goal of this section is to find estimates on the system of optimality conditions (3.11) which are independent from and δ. This is Section 3.4.1. Next we prove the regularity and find suitable expressions for the first two derivatives of the map t Þ Ñ Ψpmptqq when pm, αq is a solution to the penalized problem. This is Section 3.4.2. Finally we prove Theorems 3.2 and 3.3 in Section 3.4.3.

Uniform (in epsilon, delta) estimates

First we construct a candidate pm, αq which stays uniformly inside the constraint at all time with a finite cost.

Lemma 3.4. Provided Ψpm 0 q ă 0, we can build a trajectory pm, αq in Cpr0, T s, P 2 pR d qq L2

dtbmptq pr0, T s ˆRd , R d q such that Jpα, mq ă `8 and Ψpmptqq ď ´θ for all t in r0, T s, for some θ ą 0.

Proof. First we introduce a filtered probability space pΩ, F, F, Pq satisfying the usual conditions and supporting a standard, adapted Brownian motion pB t q. Take c ą 0 and consider a solution to the SDE dX t " ´cpX t ´X0 qdt `?2dB t where X 0 is F 0 -measurable, with law m 0 and independent from the Brownian motion. A simple application of Itô's lemma proves that X t can be rewritten as

X t " X 0 `?2 ż t 0 e ´cpt´sq dB s (3.25)
and therefore

E " |X t ´X0 | 2 ‰ " 2 ż t 0 e ´2cpt´sq ds " 1 c p1 ´e´2ct q.
Now let mptq be the law of X t . The above computation shows that

d 2 2 pmptq, m 0 q ď 1 c
for all t P r0, T s. With an abstract mimicking argument as in [START_REF] Brunick | Mimicking an Itô process by a solution of a stochastic differential equation[END_REF] we can find a measurable drift α : r0, T s ˆRd Ñ R d such that

B t m `divpαmq ´∆m " 0 and ż T 0 ż R d |αpt, xq| 2 dmptqpxqdt ď E "ż T 0 c 2 |X t ´X0 | 2 dt  ď cT.
However a direct computation, using Jensen's inequality, shows that it is enough to take, for all pt, xq P p0, T s ˆRd , B t m y ´cdivppx ´yqm y q ´∆m y " 0 m y p0q " δ y .

Notice that X 0 being independent from the Brownian motion, we easily deduce from (3.25) that mpt, xq ą 0 for all pt, xq P p0, T s ˆRd . Being Ψ Lipschitz continuous and Ψpm 0 q ă 0 we can choose c large enough so that Ψpmptqq ď Ψpm 0 q 2 for all t P r0, T s and this concludes the proof of the lemma.

Using this particular candidate and the convexity of the constraint we can obtain the following estimate which is crucial to find compactness in the problem.

Although the notations do not make it clear, from now on pm, u, λ, βq will generally denote a solution to the optimality conditions (3.11) for the penalized problem (P ,δ ) and therefore depend upon a particular p , δq. Lemma 3.5. There is a constant M " M pΨpm 0 qq ą 0 such that, for all , δ ą 0 and for all tuple pu, m, λ, βq satisfying the conditions of Theorem 3.5 it holds

1 ż T 0 λptqdt `β δ ď M.
Proof. By Lemma 3.4 we can build a solution of the Fokker-Planck equation pα, mq such that Jpα, mq ă `8 and, for all t P r0, T s, Ψpmptqq ď ´θ for some θ ą 0 independent of t.

Using the fact that pm, αq solves the Fokker-Planck equation, we can apply Lemma 3.3 to get and by definition of λ and β we have λptqΨpmptqq ě 0 for all t P r0, T s and βΨpmpT qq ě 0 and thus, if C ą 0 is an upper bound for the right-hand side of (3.26) we get

ż T 0 ż R d " αpt,
ż T 0 λptq dt `β δ ď C θ ,
which concludes the proof of the Lemma.

Remark 3.6. Notice that this estimate, together with the construction of Lemma (3.4) are the only steps which require the convexity of Ψ, Assumption (APsiConv) as well as the condition that Ψpm 0 q must be strictly negative, Assumption (APsiInside). We can combine this Lemma with Theorem 3.1 to find uniform in , δ estimates for the system of Optimality Conditions (3.11).

Proposition 3.4.

There is some C ą 0 such that, for any , δ ą 0 and any solution pm, u, λ, βq of (3.11) satisfying (3.12) and (3.13) it holds sup tPr0,T s }uptq} n ď C. At this stage, the above estimates would be sufficient to pass to the limit when and δ go to zero in the penalized problem (P ,δ ). We would find, at the limit, solutions of the constrained problem (P) and passing to the limit in the optimality conditions we would find that the solutions to the constrained problem satisfy similar conditions with λ replaced by a non-negative Radon measure ν P M `pr0, T sq. This would lead to a priori discontinuous (in time) optimal controls. However, we refrain from following such approach for now. Instead we are going to exhibit a special behavior of the optimal solutions of the penalized problem. Indeed we are going to show in the next section that solutions of the penalized problem stay inside the constraint when the penalization is strong enough. Consequently it is sufficient to take and δ small to get solutions to the constrained problem and optimal controls for the constrained problem are still continuous.

Second order analysis

The special behavior (described just above) of the solutions will be a simple consequence of the fact that we cannot have simultaneously Ψpmptqq ą 0 and d 2 dt 2 Ψpmptqq ď 0 (here m is a solution to (P ,δ )) when the penalization is strong enough. The purpose of this section is to prove the regularity and a suitable expansion of the map t Þ Ñ Ψpmptqq.

Recall that S d pRq is the space of symmetric matrices of size d. for some functional F : C b pR d , R d q ˆCb pR d , S d pRqq ˆCb pR d , R d q ˆP2 pR d q Ñ R independent of and δ and bounded in sets of the form A ˆP2 pR d q for bounded subsets

A of C b pR d , R d q Ĉb pR d , S d pRqq ˆCb pR d , R d q.
Proof. Since Ψ is supposed to satisfy Assumption (Ureg), we can use Proposition 3.3 and, for all t P r0, T s we get The parameter λ is constant (equal to 0 or 1) in a neighborhood pt 1 , t 2 q of t because of the exclusion condition (3.12) and u solves the HJB equation according to Definition 3.1 so we have that u belongs to C 1,2 ppt 1 , t 2 q ˆRd q. Moreover,

Ψpmptqq " Ψpm 0 q ´ż t 0 ż R d D m Ψpmpsq,
B
`żR d ∆ x div x D m Ψpx, mptqqdmptqpxq `żR d ż R d div x div y D 2 mm Ψpmptq, x, yqdmptqpxqdmptqpyq ´2 ż R d ż R d div y D 2 mm Ψpmptq, x, yq.D p Hpx, Dupt, xqqdmptqpxqdmptqpyq ´2 ż R d Ý Ñ ∆ x D m Ψpmptq,
´2 n ÿ i"1 ż R d B x i δΨ δm pmptq, xqD 2 xp B p i Hpx, Dupt, xqq.D 2 upt, xqdmptqpxq.
The formula above shows in particular that the terms in D∆u cancel out and thus F depends only on the derivatives of u up to order two.

Proof of the main theorems

Proposition 3.6. There is some 0 , δ 0 ą 0 such that any solution pm, αq of Problem (P ,δ ) for some p , δq P p0, 0 s ˆp0, δ 0 s stays inside the constraint at all time: @t P r0, T s, Ψpmptqq ď 0.

Proof. The proof follows closely the methodology of [START_REF] Cannarsa | C1;1-smoothness of constrained solutions in the calculus of variations with application to mean field games[END_REF] Lemma 3.7. Toward a contradiction we suppose that there exist a sequence p k , δ k q kPN P pp0, 1q ˆp0, 1qq N converging to p0, 0q, corresponding solutions pm k , ´Dp Hpx, Du k pt, xqqq kPN satisfying the conditions of Theorem 3.5 with corresponding multipliers pλ k , β k q and times pt k q kPN P p0, T s which are local maximum points of t Þ Ñ Ψpm k ptqq and such that Ψpm k pt k qq ą 0. The couples pm k , ω k q are uniformly bounded in C 1{2 pr0, T s, P 2 pR d qq ˆMpr0, T s ˆRd , R d q and we can assume that they converge in C 1{2´δ pr0, T s, P 2´δ pR d qq ˆMpr0, T s ˆRd , R d q, for some δ P p0, 1q, toward some solution p r m, r ωq to the constrained problem. In particular, Ψp r mptqq ď 0 for all t P r0, T s. We first notice that, thanks to Proposition 3.4, for large enough k, β k ă 1 and therefore Ψpm k pT qq ď 0 and t k ‰ T .

Using Proposition 3.5 yields that t Þ Ñ Ψpm k ptqq is C 2 in a neighborhood of t k and,

d 2 dt 2 Ψpm k ptqq| t"t k " 1 k ż R d D m Ψpm k pt k q, xq.D 2 pp Hpx, Du k pt k , xqqD m Ψpm k pt k q, xqdm k pt k qpxq `F pDu k pt k q, D 2 u k pt k q, D∆u k pt k q, m k pt k qq ě 1 µ k ż R d |D m Ψpm k pt k q, xq| 2 dm k pt k qpxq `F pDu k pt k q, D 2 u k pt k q, D∆u k pt k q, m k pt k qq,
where we used the convexity assumption (AH6) on the Hamiltonian H. On the one hand, using the estimates of Proposition 3.4 we have that F pDu k ptq, D 2 u k ptq, D∆u k ptq, m k ptqq is bounded independently from k. On the other hand, using the regularity assumption (APsiC2) and up to taking a subsequence we can assume that lim

kÑ`8 ż R d |D m Ψpm k pt k q, xq| 2 dm k pt k qpxq " ż R d |D m Ψp r mp r tq, xq| 2 d r mp r tqpxq
for some r t P r0, T s such that Ψp r mp r tqq " 0. This is where Assumption (APsiTrans) comes into play. Since Ψp r mp r tqq " 0, we have that

ż R d |D m Ψp r mp r tq, xq| 2 d r mp r tqpxq ą 0,
and we deduce that, d 2 dt 2 Ψpm k ptqq| t"t k ą 0 for k large enough. This leads to a contradiction since t k is assumed to be a local maximum point of t Ñ Ψpm k ptqq. Theorem 3.2 is a direct consequence of the above proposition.

Proof of Theorem 3.2. Denote by U ,δ the value of Problem (P ,δ ) and by U the value of the constrained problem (P). We assume that p , δq belongs to p0, 0 q ˆp0, δ 0 q with p 0 , δ 0 q the parameters from Proposition 3.6.

We have that U ,δ " U and the minimizers for problems (P ,δ ) and (P) coincide. Indeed, it is straightforward that U ,δ ď U. Now if pm 1 , α 1 q is a solution to Problem (P ,δ ), by Proposition 3.6, pm 1 , α 1 q is admissible for Problem (P). This means that U ,δ " J ,δ pm 1 , α 1 q " Jpm 1 , α 1 q ě U and, therefore U ,δ " U and pm 1 , α 1 q is a solution to (P).

Conversely, if pm 2 , α 2 q is a solution to (P) then J ,δ pm 2 , α 2 q " Jpm 2 , α 2 q " U " U ,δ and pm 2 , α 2 q is a solution to (P ,δ ).

Looking carefully at the proof of Proposition 3.6, using Theorem (3.1) with the estimates given by Proposition 3.4 and Lemma 3.4 we see that the threshold p 0 , δ 0 q depends on m 0 only through the value Ψpm 0 q. Now we are finally able to conclude the proof of Theorem 3.3.

Proof of Theorem 3.3. We use Theorem 3.2 and the optimality conditions for the penalized problem: If pm, αq is any solution to Problem (P), we can find p , δq P p0, 0 q ˆp0, δ 0 q, λ P L 8 pr0, T sq, β ě 0, u P Cpr0, T s, C n b pR d qq such that αpt, xq " ´Dp Hpx, Dupt, xqq for all pt, xq P r0, T sˆR d and pm, u, λ, βq satisfies the conditions of Theorem 3.5. Taking νptq :" λptq and η :" β δ concludes the proof of the first part of the theorem. Now, if we suppose that F and G are convex in the measure variable we can proceed as in Chapter 2 Section 2.3.3 and easily show that the conditions are sufficient.

The general case

The goal of this section is to prove Theorem 3.4 . We first need to extend the results of Theorem 3.1 to HJB equations with right hand-side of the form νψ 1 `ϕ1 where ν belongs to M `pr0, T sq and ψ 1 , ϕ 1 belong to Cpr0, T s, E n q.

3.5.1 The HJB equation Definition 3.2. Suppose that n ě 3. Let ψ 1 , ϕ 1 be in Cpr0, T s, E n q and ψ 2 be in E n`α . Let also ν be in M `pr0, T sq. We say that u P L 1 pr0, T s, E n q is a solution to

" ´Bt u `Hpx, Duq ´∆u " νψ 1 `ϕ1 , in r0, T s ˆRd upT, xq " ψ 2 , in R d , (3.27)
if, for almost all t P r0, T s, for all x P R d , upt, xq " P T ´tψ 2 pxq `ż T 0 1 pt,T s P s´t ψ 1 psqpxqdνpsq `ż T t P s´t ϕ 1 psqpxqds ´ż T t P s´t rHp., Dups, .qqs pxqds.

Theorem 3.6. Suppose that n ě 3. Let ψ 1 , ϕ 1 be in Cpr0, T s, E n q and ψ 2 be in E n . Let also ν be in M `pr0, T sq. Under these conditions, there is a unique solution u P L 8 pr0, T s, E n q to (3.27) in the sense of Definition 3.2. Moreover it satisfies

essup tPr0,T s }uptq} n ď Cp|ν|, sup tPr0,T s }ψ 1 ptq} n , sup tPr0,T s }ϕ 1 ptq} n , }ψ 2 } n q,
where |ν| is the total variation norm of ν.

In the following proposition, we abuse notations and we let u : r0, T s ˆRd Ñ R d be the unique element in its equivalence class of L 8 pr0, T s, E n q satisfying, for all pt, xq P r0, T s ˆRd upt, xq " P T ´tψ 2 pxq `ż T 0 1 pt,T s psqP s´t ψ 1 psqpxqdνpsq `ż T t P s´t ϕ 1 psqpxqds ´ż T t P s´t rHp., Dups, .qqs pxqds.

We have the following stability result.

Proposition 3.7. Assume that pν m q mě1 P L 8 pr0, T sq converges in M `pr0, T sq toward ν. Let u m P Cpr0, T s, E n q be the solution to the HJB equation (3.27) with data pν m , ψ 1 , ϕ 1 , ψ 2 q with ψ 1 , ϕ 1 P Cpr0, T s, E n q and ψ 2 P E n`α . Then, for all pt, xq P r0, T s ˆRd such that νpttuq " 0, it holds:

lim mÑ`8 u m pt, xq " upt, xq, lim mÑ`8
Du m pt, xq " Dupt, xq.

Once again, these technical results are postponed to Section 3.6.2.

Optimality conditions in the general case

We first prove a lemma similar to Lemma 3.2.

Lemma 3.6. Let p r m, r ωq be a solution to the constrained Problem (P). Then there exist ν P M `pr0, T sq and η P R `satisfying Ψp r mptqq " 0, ν ´ae (3.28) ηΨp r mpT qq " 0, (3.29) and such that p r m, r ωq minimizes J l pω, mq :"

ż T 0 ż R d Lpx, dω dt b dmptq pt, xqqdmptqpxqdt `ż T 0 ż R d δΨ δm p r mptq, xqdmptqpxqdνptq `ż T 0 ż R d δF δm p r mptq, xqdmptqpxqdt `żR d " η δΨ δm p r mpT q, xq `δG δm p r mpT q, xq  dmpT qpxq, (3.30)
over the pairs pm, ωq satisfying (3.14) and where we set, J l pm, ωq " `8 if ω is not absolutely continuous with respect to dt b mptq.

Proof. We take , δ ą 0 and pm ,δ , ω ,δ q solutions to the penalized problems P ,δ . As , δ Ñ 0, pm ,δ , ω ,δ q converges in Cpr0, T s, P 2´δ pR d qq ˆMpr0, T s ˆRd , ˆRd q for δ P p0, 1q to a solution to the constrained problem that we can assume, without loss of generality, to be p r m, r ωq. Now pm ,δ , ω ,δ q is also a solutions to the linearized problems of Lemma 3.2 for some λ ,δ , β ,δ P L 8 pr0, T sq ˆR`s atisfying the exclusion conditions

λ ,δ ptq $ & % " 0 if Ψpm ,δ ptqq ă 0 P r0, 1s if Ψpm ,δ ptqq " 0 " 1 if Ψpm ,δ ptqq ą 0, β ,δ $ & % " 0 if Ψpm ,δ pT qq ă 0 P r0, 1s if Ψpm ,δ pT qq " 0 " 1 if Ψpm ,δ pT qq ą 0.
Using the controllability lemma 3.4 and arguing as in Lemma 3.5 we can infer that λ ,δ is bounded in L 1 pr0, T sq independently from p , δq ą 0 and β ,δ δ is also bounded in R `. Let us take ν P M `pr0, T sq to be a limit point of λ ,δ and η a limit point of β ,δ δ as , δ Ñ 0. It is plain to check that Ψp r mptqq " 0 for ν-almost all t P r0, T s and ηΨp r mpT qq " 0. Now we can argue as in the proof of Lemma 3.2, passing to the limit in the linearized problems to conclude that p r m, r ωq is indeed a minimum of (3.30).

We now take u P L 8 pr0, T s, E n q to be the solution to

$ ' & ' % ´Bt u `Hpx,
Using an approximation argument and Proposition 3.7, we have the following duality relation: Proposition 3.8. Let u P L 8 pr0, T s, E n q be a solution to (3.31) satisfying (3.32) for all pt, xq P r0, T s ˆRd . Let also pm, αq P Cpr0, T s, P 2 pR d qq ˆL2

dtbdmptq pr0, T s ˆRd , R d q be a solution in the sense of distributions to " B t m `divpαmq ´∆m " 0, in p0, T q ˆRd , mp0q " m 0 .

Then the following duality formula holds for any t 1 P r0, T s such that νptt 1 uq " 0,

ż R d upt 1 , xqdmpt 1 qpxq " η ż R d δΨ δm p r mpT q, xqdmpT qpxq `żR d δG δm p r mpT q, xqdmpT qpxq ´ż T t 1 ż R d rHpx, Dupt, xqq `αpt, xq.Dupt, xqs dmptqpxqdt `ż T t 1 ż R d δΨ δm p r mptq, xqdmptqpxqdνptq `ż T t 1 ż R d δF δm p r mptq, xqdmptqpxqdt. (3.33)
We can conclude with the proof of Theorem 3.4.

Proof of Theorem 3.4. We proceed similarly to the proof of Theorem 3.5. Take p r m, r ωq a relaxed solution to the constrained problem P. Let also u P L 8 pr0, T s, E n q be the solution to (3.31) satisfying (3.32) with ν and η satisfying respectively (3.28) and (3.29).

Recall that the linearized cost J l is defined in Lemma 3.6. On the one hand, by definition of L, it holds that

J l p r m, r ωq " ż T 0 ż R d
Being Ψpm 0 q ă 0, it holds that νpt0uq " 0 because of the exclusion condition (3.28) and we can use the duality relation (3.33) with t 1 " 0 and α " dr ω dt b d r mptq to conclude that

J l p r m, r ωq ě ż R d up0, xqdm 0 pxq.
On the other hand, we can apply relation (3.33) to the candidate pm 1 , ´Dp Hpx, Dupt, xqqm 1 q where m 1 is solution to " B t m 1 ´divpD p Hpx, Dupt, xqqm 1 q ´∆m 1 " 0, in p0, T q ˆRd m 1 p0q " m 0 .

We get J l pm 1 , ´Dp Hpx, Dupt, xqqm

Technical Results about the HJB equation

We start with a (slightly unusual) version of Grönwall lemma. Lemma 3.7. Assume that l : r0, T s Ñ R `is a bounded measurable map which satisfies, for some C 1 , C 2 ą 0 and almost all t P r0, T s,

lptq ď C 1 `C2 ż T t lpsq ? s ´t ds. (3.35)
Then, for almost all t P r0, T s,

lptq ď C 1 p1 `C2 ? π ? T ´tqe C 2 2 πpT ´tq .
Proof. Arguing by induction, using (3.35) we find that, for all t P r0, T s and all n P N ˚, it holds

lptq ď C 1 ˜1 `n ÿ k"1 C k 2 I k ptq ¸`C n`1 2 ż T t ż T t 1 . . . ż T tn lpt n`1 q ? t 1 ´t . . . ? t n`1 ´tn dt 1 . . . dt n`1 ď C 1 ˜1 `n ÿ k"1 C k 2 I k ptq ¸`}l} 8 C n`1 2 I n`1 ptq, (3.36)
where I k : r0, T s Ñ R is defined for all k P N ˚by

I k ptq " ż T t ż T t 1 . . . ż T t k´1 1 ? t 1 ´t . . . ? t k ´tk´1 dt 1 . . . t k .
Noticing that I k ptq "

ż T t 1 ?
t 1 ´t I k´1 pt 1 qdt 1 for all k ě 2 and that I 1 ptq " 2 ?

T ´t, we find by induction that, for all n ě 2,

I n ptq " 2B 1 . . . B n´1 pT ´tq n{2
where the B k are given, for all k ě 1 by,

B k " ż 1 0 p1 ´uq k{2 u ´1{2 du " βp1{2, k{2 `1q " Γp1{2qΓpk{2 `1q Γpk{2 `3{2q
where β and Γ are Euler's functions. As a consequence, we get, for all k ě 1 and all t P r0, T s,

I k ptq " π k{2 Γpk{2 `1q pT ´tq k{2 .
In particular, lim nÑ`8 C n`1 2 I n`1 ptq " 0 for all t in r0, T s. Now we can compute

N ÿ k"1 C k 2 π k{2 Γpk{2 `1q pT ´tq k{2 " `8 ÿ k"1 C 2k 2 π k Γpk `1q pT ´tq k ``8 ÿ k"0 C 2k`1 2 π k`1{2 Γpk `1 `1{2q pT ´tq k`1{2 ď `8 ÿ k"1 C 2k 2 π k k! pT ´tq k ``8 ÿ k"0 C 2k`1 2 π k`1{2 k! pT ´tq k`1{2 ď e C 2 2 πpT ´tq ´1 `C2 ? π ? T ´te C 2 2 πpT ´tq ď p1 `C2 ? π ? T ´tqe C 2 2 πpT ´tq ´1.
Finally, we deduce from (3.36) that, for all t P r0, T s,

lptq ď C 1 p1 `C2 ? π ? T ´tqe C 2 2 πpT ´tq .
3.6.1 Proof of Theorem 3.1 Lemma 3.8. Assume that u P Cpr0, T s, E n q is a solution to the HJB equation (3.4) with f P Cpr0, T s, E n q and g P E n . Then

sup pt,xqPr0,T sˆR d |Dupt, xq| ď Cp ż T 0 }f ptq} 1 dt, }g} 1 q.
Proof. We use the classical Bernstein method. Let µ ą 0 and wpt, xq :" 1 2 e µt |Dupt, xq| 2 . Being f in Cpr0, T s, E n q, u is smooth in space and satisfies the HJB equation in the strong sense. Differentiating the equation with respect to x and taking the scalar product with }f ptq} 1 dt, }g} 1 q ą 0.

Lemma 3.9. Assume that u P Cpr0, T s, E n q is a solution to the HJB equation with data f P L 1 pr0, T s, E n q and g P E n and assume that u satisfies the estimate of the previous lemma then

sup tPr0,T s }uptq} n ď Cp ż T 0 }f ptq} n , }g} n q.
Proof. For all pt, xq P r0, T s ˆRd , it holds that But we can find a constant C " Cpsup tPr0,T s }uptq} k´1 q such that sup and we conclude by induction.

Following similar computations we can prove the following stability result.

Lemma 3.10. Assume that f 1 , f 2 P L 1 pr0, T s, E n q and g 1 , g 2 P E n . Suppose that, u 1 , u 2 P Cpr0, T s, E n q are the respective solutions the HJB equation with data pf 1 , g 1 q, pf 2 , g 2 q and satisfy the estimate of Lemma 3.9. Then

sup tPr0,T s }u 1 ptq ´u2 ptq} n ď Cp ż T 0 }f 1 ptq ´f2 ptq} n dt `}g 1 ´g2 } n q.
for some C " Cp

ż T 0 }f 1 ptq} n dt, ż T 0 }f 2 ptq} n dt, }g 1 } n , }g 2 } n q ą 0.
Proof. For all ps, xq P r0, T s ˆRd we can write for some C " Cp}u 1 psq} k , }u 2 psq} k q ą 0. The proof of the lemma follows from this observation and the same computations as the proof of Lemma 3.9.

Lemma 3.11. Assume that u P L 8 pr0, T s, E n q solves the HJB equation with data pf, gq P Cpr0, T s, E n q ˆEn`α then u belongs to Cpr0, T s, E n q.

Proof. Let us take k P 1, n . We fix h ą 0. For t P r0, T ´hs it holds

D k upt `h, xq ´Dk upt, xq " P T ´t´h D k gpxq ´PT´t D k gpxq `ż T t`h P s´t´h D k f psqpxqds ´ż T t P s´t D k f psqpxqds `ż T t`h
DP s´t´h D k´1 Hp., Dups, .qqpxqds ´ż T t DP s´t D k´1 Hp., Dups, .qqpxqds

" ∆ 1 `∆2 `∆3 .
We estimate the three differences as follows:

|∆ 1 | " |P T ´t´h D k gpxq ´PT´t D k gpxq| ď |D k gpxq ´Ph D k gpxq| ď h α{2 ||g|| k`α .
Now for the term involving f :

|∆ 2 | " | ż T t`h P s´t´h D k f psqpxqds ´ż T t P s´t D k f psqpxqds| " | ż T ´h t P s´t D k f ps `hqpxqds ´ż T t P s´t D k f psqpxqds| " | ż T ´h t P s´t pD k f ps `hq ´Dk f psqqpxqds ´ż T T ´h P s´t D k f psqpxqds| ď ż T ´h 0 }f ps `hq ´f psq} k ds `C? h sup tPr0,T s }f ptq} k´1 .
Finally for the term involving the Hamiltonian Being f in Cpr0, T s, E n q, the right-hand side converges to 0 when h goes to 0 and therefore lim hÑ0 sup tPr0,T ´hs }upt `hq ´uptq} n " 0 which concludes that u belongs to Cpr0, T s, E n q.

|∆ 3 | " | ż T t`
As a consequence, we get the existence of solutions from the classical case.

Proposition 3.9. Take f P L 1 pr0, T s, E n q and g P E n`α . Then there exists a unique solution in u P Cpr0, T s, E n q to the HJB equation with data pf, gq and it satisfies the estimate of Lemma 3.9.

Proof of Theorem 3.1. We take a sequence of smooth functions f m : r0, T s ˆRd Ñ R and g m : R d Ñ R converging respectively to f in L 1 pr0, T s, E n q and to g in E n`α . For each m, the existence of a strong solution u m P Cpr0, T s, E n q follows from Schauder theory and our a priori Lipschitz estimate. Thanks to the previous lemma, we know that u m is a Cauchy sequence in L 8 pr0, T s, E n q and therefore it converges in this space to some u. The subspace Cpr0, T s, E n q being closed in L 8 pr0, T s, E n q we have that u belongs to Cpr0, T s, E n q. We can also pass to the limit in the equation Proceeding exactly as in the proof of Theorem 3.1, we find that there exists a unique solution v P L 8 pr0, T s, E n q and it satisfies essup tPr0,T s }vptq} n ď Cp

u m pt,
ż T 0 }zptq} n dtq.
We can also proceed directly (without relying on the existence of solutions for smooth z) and follow Chapter 4.3.3. in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]. We recall the main steps here. Up to finding Lipschitz a priori estimates on the solutions using Bernstein method, as in Lemma 3.8 , we can pretend that H is Lipschitz continuous. Take u 1 P L 8 pr0, T s, E n q and define u " Φpu 1 q by upt, xq " P T ´tψ 2 pxq `ż T t P s´t ψ 2 psqpxqdνpsq `ż T t P s´t ϕ 1 psqpxqds ´ż T t P s´t rHp., Du 1 ps, .qqs pxqds.

Following the same computations as in Lemma 3.9 we can prove that, for all 1 ď k ď n, it holds

essup ps,xqPrt,T sˆR d |D k ups, xq| ď Cp1 `ż T t }u 1 psq} k ? s ´t dsq
for some C " Cp}ν}, sup tPr0,T s }ψ 1 ptq} n , sup tPr0,T s }ϕ 1 ptq} n , }ψ 1 } n , essup tPr0,T s }u 1 ptq} k´1 q ą 0. We use this to construct by induction pΛ 0 , pλ 1 , Λ 1 q, . . . , pλ n , Λ N qq depending only on the data p|ν|, sup tPr0,T s }ψ 1 ptq} n , sup tPr0,T s }ϕ 1 ptq} n , }ψ 1 } n q, such that

essup pt,xqPr0,T sˆR d |upt, xq| 1 `|x| ď Λ 0 , essup ps,xqPrt,T sˆR d |D k ups, xq| ď λ k e Λ k pT ´tq
for all 1 ď k ď n, whenever u 1 satisfies the same estimates. We let B be the subset of functions in L 8 pr0, T s, E n q satisfying these estimates. By construction B is stable under the action of Φ. Now if we take two functions u 1 1 , u 1 2 in B and if we define u 1 " Φpu 1 1 q, u 2 " Φpu 1 2 q, w " u 1 ´u2 and w 1 " u 1 1 ´u1

2 we find, following yet again the same computations, that

essup sPrt,T s }wptq} n ď C ż T t }w 1 psq} n ? s ´t ds (3.37)
for some C ą 0 depending only on the constants defining B. Thanks to estimate (3.37) we find that Φ is a contraction in L 1 e µt dt pr0, T s, E n q for µ large enough. Finally we take any u 1 P B and we build by induction u m`1 " Φpu m q. We know that u m converges in L 1 e µt dt pr0, T s, E n q to some u. However we can use inequality (3.37) once again to prove that u m actually converges in L 8 pr0, T s, E n q. The set B being closed in L 8 pr0, T s, E n q we can conclude that the limit u belongs to B and is a solution to the HJB equation.

The uniqueness of solutions is proved as in the proof of Theorem 3.1.

We let u : r0, T sˆR d Ñ R d be the unique element in its equivalence class of L 8 pr0, T s, E n q satisfying, for all pt, xq P r0, T s ˆRd Notice that, on the contrary to u, the function v is bounded on r0, T s ˆRd . We turn our attention to the time regularity. We take k ě 0 and t 1 , t 2 P r0, T s such that t 2 ě t 1 . We can write with C " Cpessup tPr0,T s }uptq} k`2 q ą 0. If we don't do the integration by parts in step (3.41) we find that }vpt 1 q ´vpt 2 q} k ď Cpt 2 ´t1 q for some C " Cpessup tPr0,T s }uptq} k`3 q ą 0. Being n ě 3 we get that v is bounded in C p1`αq{2,1`α pr0, T s ˆRd q for all α P p0, 1q. This concludes the proof of the lemma.

D k vpt 1 ,
Proof. Proof of Proposition 3.7. We take a sequence of non-negative functions ν m : r0, T s Ñ R in L 8 pr0, T sq such that ν m converges to ν is M `pr0, T sq. We also let u m be the solution to the associated HJB equation and define z m and v m according to 3. Using the previous lemma, we also have that the sequence pv m q is uniformly bounded in C 1`α 2 ,1`α pr0, T s ˆRd q for any α P p0, 1q. Up to a sub-sequence it converges (locally) to some r v in C 1`β 2 ,1`β pr0, T s ˆRd q for some β P p0, αq. Using Lebesgue dominated convergence theorem we can pass to the limit as m Ñ `8 in the equality v m pt, xq " ´ż T t P s´t rHp., Dv m ps, .q `Dz m ps, .qqs pxqds, we conclude that, for all pt, xq P r0, T s ˆRd r vpt, xq " ´ż T t P s´t rHp., Dr vps, .q `Dzps, .qqs pxqds.

If we let r u :" r v `z, we have that r u solves the HJB equation and, by uniqueness, r u " u in L 8 pr0, T s, E n q. Therefore vpt, xq " r vpt, xq for all pt, xq P r0, T s ˆRd and we conclude that v m converges locally uniformly to v in C 1`β 2 ,1`β pr0, T s ˆRd q.

Appendix

Existence of relaxed solutions

Proof of Proposition 3.1. Consider a weak solution of " dX t " αpt, X t qdt `?2dB t , X t"0 " X 0 " m 0 such that LpX t q " mptq, @t P r0, T s. The existence of such a solution is guaranteed by the fact that pα, mq solves the Fokker-Planck equation (see [START_REF] Trevisan | Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients[END_REF] and also Proposition 2. Taking s " 0 in the above computation also shows that

ż R d |x| 2 dmptq ď 2Ep|X t ´Xs | 2 q `2 ż R d |x| 2 dm 0 pxq ď C for another C " Cp ż R d |x| 2 dm 0 pxq, ż T 0 ż R d |αpt, xq| 2 dmptqpxqdtq ą 0.
Proof of Proposition 3.2. We set ω n " α n m n . By Cauchy-Schwarz inequality we find that the total variation |ω n | of ω n is uniformly bounded. Indeed we have

|ω n | " ż T 0 ż R d ˇˇˇd ω n dt b dm n ptq pt, xq ˇˇˇd m n ptqpxqdt ď ? T ˜ż T 0 ż R d ˇˇˇd ω n dt b dm n ptq pt, xq ˇˇˇ2 dm n ptqpxqdt ¸1{2 .
This estimate together with Proposition 3.1 allow us to use Banach-Alaoglu theorem on the one hand and Ascoli theorem on the other hand and deduce that for all r P p1, 2q, up to a subsequence, pm n , ω n q nPN converges in Cpr0, T s, P r pR d qqˆMpr0, T sˆR d , R d q to some element p r m, r ωq of Cpr0, T s, P r pR d qq ˆMpr0, T s ˆRd , R d q. It is straightforward that r mp0q " m 0 and the fact that p r m, r ωq satisfies the Fokker-Planck equation is a consequence of the weak-c onvergence of measures. Using Theorem 2.34 of [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] (see also Exemple 2.36) in [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]) we find that ω is absolutely continuous with respect to mptq b dt and

ż T 0 ż R d ˇˇˇd ω dt b dmptq pt, xq ˇˇˇ2 dmptqpxqdt ď lim inf nÑ`8 ż T 0 ż R d |α n pt, xq| 2 dm n ptqpxqdt.
By Proposition 3.1 again, this shows that m belongs to C 1{2 pr0, T s, P 2 pR d qq.

Now we give the proof of Lemma 3.1.

Proof of Lemma 3.1. This is precisely Theorem 2.4 in Chapter 2 and the result follows from Proposition 3.1 and Proposition 3.2. We consider a minimizing sequence pm n , ω n q satisfying (3.14) and such that, for all n P N, J ,δ pm n , ω n q ď inf J ,δ pm n , ω n q `1. By coercivity of H and therefore -by taking convex conjugates-of L we find that there is C 1 ą 0 such that, for all n P N,

ż R d ż T 0 ˇˇˇd ω n dt b dm n ptq pt, xq ˇˇˇ2 dm n ptqpxqdt ď C 1 . (3.42)
Using that pm n , ω n q satisfies the Fokker-Planck equation and m 0 belongs to P 2 pR d q we deduce from Proposition (3.2) that, for all r P p1, 2q, up to a subsequence, pm n , ω n q nPN converges in Cpr0, T s, P r pR d qq ˆMpr0, T s ˆRd , R d q to some element p r m, r ωq of Cpr0, T s, P 2 pR d qq Mpr0, T s ˆRd , R d q which satisfies the Fokker-Planck equation with initial position mp0q " m 0 . To conclude we use Theorem 2.34 of [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF] to prove that

Introduction

The goal of this chapter is to investigate the connection between the optimization problem:

inf pα,µq ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0 Fpµptqqdt `GpµpT qq (mfP) subject to $ ' ' & ' ' %
µ P Cpr0, T s, P 2 pR d qq, α P L 2 dtbµptq pr0, T s ˆRd , R d q B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd µp0q " µ 0 P P 2 pR d q Ψpµptqq ď 0 @t P r0, T s and a control problem for a large number N of interacting particles:

inf pα i,N t q 1ďiďN E P γ N « ż T 0 1 N N ÿ i"1 LpX i,N t , α i,N t qdt `ż T 0 Fpp µ N,x t qdt `Gpp µ N,x T q ff (NP) subject to $ & % dX i,N t " α i,N t dt `?2dB i,N t , pX 1,N 0 , . . . , X N,N 0
q " µ bN 0 under P, Ψpp µ N,x t q ď 0 for all t P r0, T s P γ N ´almost-surely.

In the latter problem, pB i,N q 1ďiďN are N independent standard Brownian motions supported on a probability space pΩ, F, Pq. We denoted by

p µ N,x t :" 1 N N ÿ i"1 δ X i,N
t the empirical measures and finally P γ N :" P " .|Ψpp µ N,x 0 q ď ´γN ı is the conditional probability with respect to the event ! Ψpp µ N,x 0 q ď ´γN ) for some suitable rate γ N ą 0 such that γ N Ñ 0 121 as N Ñ `8. The conditioning being necessary to ensure that the particles start from inside the constraint.

The former problem was thoroughly analyzed in Chapter 3 where we showed, in particular the existence of bounded, Lipschitz continuous optimal controls, for any initial position µ 0 P P 2 pR d q such that Ψpµ 0 q ă 0. The latter problem (NP) is, however very different in nature. Indeed, the state is a process pX 1,N t , . . . X N,N t q 0ďtďT valued in pR d q N , the empirical measures p µ N,x t are, by definition, random and the constraint has to be satisfied almostsurely. This type of constraint leads to new difficulties. Indeed, to dominate the effect of the diffusion, the controls cannot remain bounded and the value function associated to this problem is expected to blow-up near the boundary.

Without constraint, the connection between Problem (mfP) and Problems (NP) is by now well understood. Under more general structure conditions, Lacker proved in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] that the law of the empirical measures of weak solutions to the N -particle system converge to probability measures supported on the set of optimal solutions to the mean-field problem and therefore convergence of the value functions hold. Taking advantage of the regularizing effect of the diffusion and uniform in N Lipschitz and semi-concavity estimates for the value functions of the N -particles system, it was shown in [START_REF] Cardaliaguet | An algebraic convergence rate for the optimal control of Mckean-Vlasov dynamics[END_REF] that convergence actually holds with a rate (see Chapter 5). In the same setting, Cardaliaguet and Souganidis later proved in [START_REF] Cardaliaguet | Regularity of the value function and quantitative propagation of chaos for mean field control problems[END_REF] a propagation of chaos around "stable" solutions of the mean-field problem. Finally we mention that, under convexity assumptions on the mean-field costs F and G it is shown in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] that the value function associated to the mean-field control problem is a smooth (enough) function in the Wasserstein space. In this setting it is not difficult to prove that the convergence of the value functions holds with an optimal rate and we have quantitative propagation estimates for the optimal trajectories to the N -particles system toward the solution to the mean-field control problem.

We also mention that recent progresses were made in order to characterize the value function for the mean-field problem, in the general situation where it is not expected to be smooth. Similarly to the finite dimensional case, we expect the value function to be the unique viscosity solution (in some sense) to the dynamic programming equation. Different approaches have been taken in [START_REF] Burzoni | Viscosity Solutions for Controlled McKean-Vlasov Jump-Diffusions[END_REF][START_REF] Cecchin | Weak solutions to the master equation of potential mean field games[END_REF][START_REF] Conforti | Hamilton-Jacobi equations for controlled gradient flows: the comparison principle[END_REF][START_REF] Cosso | Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions[END_REF]. The most general result, so far, being [START_REF] Cosso | Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions[END_REF], where the authors rely on the approximation of the mean-field control problem by control problems for finite numbers of interacting particles.

Stochastic control problems with state constraint and non-degenerate diffusions were addressed in the seminal work [START_REF] Lasry | Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints -1. The model problem[END_REF] of Lasry and Lions. They showed that the blow-up behavior of the value function is directly related to the growth of the Hamiltonian and provided rates of divergence. This problem was later revisited by Leonori and Porretta in [START_REF] Leonori | The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint[END_REF] where the authors also prove the rate of divergence of the gradient of the value function.

In this chapter we prove the convergence of the value functions for the problems with almost-sure constraints toward the value function for the mean-field problem. Let us denote by Upµ 0 q, the value of Problem (mfP) and U N pµ 0 q, the value of Problem (NP) which are rigorously defined in Sections 4.1 and 4.2 respectively. Similarly to [START_REF] Cardaliaguet | An algebraic convergence rate for the optimal control of Mckean-Vlasov dynamics[END_REF] we proceed in two steps. On the one hand we prove that Upµ 0 q ď lim inf

N Ñ`8 U N pµ 0 q.
This boils down to finding weak limit points of sequences of nearly optimal weak solutions 123 to the N -particle problem. Once we know that U N pµ 0 q is bounded independently from N , this follows from the line of arguments of [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] for problems without constraint. On the other hand, proving that lim sup

N Ñ`8
U N pµ 0 q ď Upµ 0 q requires more care. Indeed an admissible control for the mean-field problem is, in general, not admissible for the particle system because of the almost-sure constraint. We also mention that, in contrast with [START_REF] Cardaliaguet | An algebraic convergence rate for the optimal control of Mckean-Vlasov dynamics[END_REF], the value function U N pµ 0 q is certainly not smooth as a function of µ 0 (at least because we introduced conditional expectations and because of the almostsure constraint) and therefore we cannot see it as a smooth sub-solution to the dynamic programming equation satisfied by U. Our strategy can be described as follows. Given an admissible control α for the mean-field control problem, we consider the particle system starting from an initial position pX 1,N 0 , . . . , X N,N N q " P µ bN 0 ,

X i,N t " X i,N 0 `ż t^τ N 0 αps, X i,N s qdt `ż t t^τ N β i,N t dt `?2B i,N t
where τ N :" inftt ě 0, Ψpp µ N,x t q ě ´γN 2 u and β i,N t is a feedback control designed so that,

P γ N -almost-surely 1 N N ÿ i"1 ˇˇX i,N t ´Xi,N τ N ˇˇ2 ď r 2 N @t ě τ N ,
where r N is a small radius depending on γ N which guarantees that, P γ N -almost-surely, Ψpp µ N,x t q ď 0, @t P r0, T s.

If α is bounded, Lipschitz continuous and taken so that the corresponding solution µ to " B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 (4.1)

satisfies Ψpµptqq ď ´δ, for all t P r0, T s, for some δ ą 0, we expect a strong convergence of p µ N,x t toward µptq for t P r0, τ N s and therefore τ N ^T must converge to T . The key step is to build pβ i,N t q 1ďiďN so that its contribution to the cost for the N -particle problem, vanishes as N Ñ `8. This can be done only if Ψpµ 0 q ă 0 and γ N does not converge too fast to 0. We also need to prove that it is enough to approximate admissible candidates pα, µq such that α is bounded, Lipschitz continuous with respect to the space variable and Ψpµptqq ď ´δ for all t P r0, T s, for some δ ą 0. Overall, our main result is the following. U N pµ 0 q " Upµ 0 q, whenever γpN q " N ´θ, with θ ă 1 d`8 .

4.1 The system of particles with almost-sure constraints

Assumptions

We first give the assumptions satisfied by L, F, G and Ψ. For U " F, G, Ψ, the map

U : P 2 pR d q Ñ R d satisfies
U is a bounded from below, C 1 map and δU δm belongs to CpP 2 pR d q, E 3`α q, (4.2)

where E 3`α is the subset of C 3 pR d q consisting of functions u such that sup

xPR d |upxq| 1 `|x| `sup xPR d |Dupxq| `sup xPR d |D 3 upxq| `sup x‰yPR d |D 3 upxq ´D3 upyq| |x ´y| α ă `8.
The Lagrangian L verifies Lpx, qq " sup pPR d t´p.q ´Hpx, pqu for all px, qq P R d ˆRd where H satisfies the following conditions for some C 0 ą 0.

• For all px, pq P R d ˆRd , C ´1 0 |p| 2 ´C0 ď Hpx, pq ď C 0 |p| 2 `C0 .

• H belongs to C 3 pR d ˆRd q.

• H and its derivatives are bounded on sets of the form R d ˆBp0, Rq for all R ą 0.

• For all px, pq P R d ˆRd , |D x Hpx, pq| ď C 0 p1 `|p|q.

• For all px, pq P R d ˆRd , 1

C 0 I d ď D pp Hpx, pq ď C 0 I d .
For the constraint, we also assume that Ψ is convex, that it satisfies the regularity condition

$ ' ' ' & ' ' ' % For all x P R d , m Þ Ñ δΨ δm pm, xq is C 1 with px, yq Þ Ñ δ 2 Ψ δm 2 pm,
x, yq in C 2 pR d ˆRd q for all m P P 2 pR d q and δ 2 Ψ δm 2 pm, x, yq and its derivatives being jointly continuous and bounded in P 2 pR d q ˆRd ˆRd . We also assume that there is at least one µ P P 2 pR d q such that Ψpµq ă 0. For convenience we put all of the above assumptions into Assume the above assumptions. (A)

Statement of the problem

Throughout this section we fix some µ 0 P P 2 pR d q such that Ψpµ 0 q ă 0. In its strong formulation, the N -state control problem is described as follows. We fix a filtered probability space pΩ, F, F, Pq satisfying the usual conditions and endowed with N independent adapted Brownian motions pB i,N t q i"1,...,N . We also assume that there are N independent F 0 -measurable initial positions pX i,N 0 q i"1,...,N distributed according to µ 0 and independent from the Brownian motions.

For some rate γ N ą 0 which goes to 0 as N Ñ `8 we denote P γ N the conditional probability with respect to the event

! Ψpp µ N,x 0 q ď ´γN
) .

The controller's problem is to minimize over controls pα i,N t q i"1,...,N adapted to the filtration generated by the Brownian motions and the initial positions

J N ppα i,N t q 1ďiďN q :" E P γ N « ż T 0 ˜1 N N ÿ i"1
LpX i,N t , α i,N t q `Fpp µ N,x t q ¸dt `Gpp µ N,x T q ff under the dynamics

X i,N t " X i,N 0 `ż t 0 α i,N s ds `?2B i,N s where p µ N,x t :" 1 N N ÿ i"1 δ X i,N t
and the particles are constrained to satisfy P γ N -almost surely the inequality Ψpp µ N,x t q ă 0, for all t P r0, T s.

We denote by U N pµ 0 q the value of the above problem. Notice that the initial positions are i.i.d under P but not under P γ N .

We also define

Ω N :" # px 1 , . . . , x N q P R dN , Ψp 1 N N ÿ i"1 δ x i q ă 0 + .
The condition (TransCondPsi) on the Wasserstein gradient of Ψ at the boundary ensures that the closure

Ω N of Ω N in pR d q N is Ω N " # px 1 , . . . , x N q P pR d q N , Ψp 1 N N ÿ i"1 δ x i q ď 0 + .
Similarly we define Ω 8 :" µ P P 2 pR d q, Ψpµq ă 0 ( and we have that Ω 8 :" µ P P 2 pR d q, Ψpµq ď 0 ( . Remark 4.1. Notice that it could very well happen that Ω N " ∅ for small values of N . However we neglect this detail since we always assume that there is some µ 0 P P 2 pR d q such that Ψpµ 0 q ă 0. Approximating µ 0 by atomic measures, we find that Ω N is not empty for N large enough.

The mean-field problem

For some µ 0 P P 2 pR d q such that Ψpµ 0 q ă 0, the problem is to minimize Jpα, µq :"

ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0
Fpµptqqdt `GpµpT qq over couples pµ, αq P Cpr0, T s, P 2 pR d qq ˆL2 dtbµptq pr0, T s ˆRd , R d q satisfying in the sense of distributions the Fokker-Planck equation " B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd µp0q " µ 0 , under the constraint that Ψpµptqq ď 0 for all t P r0, T s.

In Chapter 3, we showed that optimal controls for this problems exist and are bounded and Lipschitz continuous provided Ψpµ 0 q ă 0. We complete the analysis with a stability result with respect to the constraint. To this end we introduce, for all δ ą 0 small, U δ pµ 0 q the value of the same problem associated to the constraint Ψpµptqq ď ´δ for all t P r0, T s. Proposition 4.1. Assume Assumption (A) and assume as well that Ψpµ 0 q ă 0. Then it holds lim δÑ0 U δ pµ 0 q " Upµ 0 q.

Proof. Using the controllability result of Lemma 3.4, in Chapter 3, we know that U δ pµ 0 q is uniformly bounded for δ P r0, ´Ψpµ 0 q 2 s. By standard estimates, using the coercivity of L and the fact that µ 0 belongs to P 2 pR d q we find some R ą 0 such that, for all δ P r0, ´Ψpµ 0 q 2 s, for all t P r0, T s, As a consequence, we can apply Theorem 3.2 in Chapter 3 to U δ pµ 0 q and conclude that there is some 0 ą 0 such that, for all δ P r0, δ 0 s

U δ pµ 0 q " inf pα,µq ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0 Fpµptqqdt `GpµpT qq `1 0 "ż T 0 pΨpµptq `δq `qdt `pΨpµpT qq `δq ` (4.5)
where the infimum is taken over the couples pα, µq in Cpr0, T s, P 2 pR d qq ˆL2 dtbµptq pr0, T s Rd , R d q satisfying " B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 , but not necessarily the state constraint. Now consider pr µ, r αq an optimal solution for Upµ 0 q. On the one hand it is obvious that Upµ 0 q ď U δ pµ 0 q for any δ ě 0. On the other hand, for δ P r0, δ 0 s, using pr µ, r αq as a candidate in (4.5), it comes U δ pµ 0 q ´Upµ 0 q ď 1 0 "ż T 0 pΨpr µptqq `δq `qdt `pΨpr µpT qq `δq

` ď pT `1qδ 0 .
As a consequence, we have the limit lim δÑ0 U δ pµ 0 q " Upµ 0 q, which concludes the proof of the proposition.

Mean field limit

The main result of this section is to prove Theorem (4.1), that is the convergence of U N pµ 0 q to Upµ 0 q as N Ñ `8.

From mean-field to almost-sure constraint

In this section we prove the first inequality in Theorem (4.1).

Theorem 4.2. Let Assumption (A) hold. Assume further that µ 0 satisfies Ψpµ 0 q ă 0 and ż U N pµ 0 q ď Upµ 0 q, whenever γ N " N ´θ with θ ă 1 d`8 .

R d
Proof. To prove Theorem (4.2) we proceed as follows. First we fix δ ą 0 and we take α : r0, T s ˆRd Ñ R d to be an optimal control for U δ pµ 0 q. Using Theorem 3.4 in Chapter 3, we know that α is bounded and Lipschitz continuous in the space variable uniformly in time. We let µ be the corresponding trajectory, solution to " B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 .

In particular, Ψpµptqq ď ´δ for all t P r0, T s. We let pX 1,N t , . . . , X N,N t q 0ďtďT be the solution to

X i,N t " X i,N 0 `ż t^τ N 0 αps, X i,N s qdt `ż t t^τ N β i,N t dt `?2B i,N t
where τ N :" inftt ě 0, Ψpp µ N,x t q ě ´γN 2 u, with the convention inft∅u " `8, and β i,N t is the feedback control, defined for all t ě τ N ^T by

β i,N t " 4pX i,N t ´Xi,N τ N ^T q ř N i"1 |X i,N t ´Xi,N τ N ^T | 2 ´r2 N N ´2 d r 2 N pX i,N t ´Xi,N τ N ^T q,
with r N " γ N 4C Ψ and C Ψ a Lipschitz constant for Ψ with respect to d 2 . Lemma 4.1. P γ N -almost-surely, it holds that,

1 N N ÿ i"1 ˇˇX i,N t ´Xi,N τ N ^T ˇˇ2 ď r 2 N , @t ě τ N ^T.
Moreover, the following estimate holds

E P γ N « ż T τ N ^T 1 N N ÿ i"1 ˇˇβ i,N t ˇˇ2 dt ff ď 16d r 2 N N E P γ N " e T ´T ^τN ‰ `8d 2 r 2 N E P γ N rT ´T ^τN s .
We continue with the ongoing proof. We have taken r N and β N is such a way that P γ N -almost-surely Ψpp µ N,x t q ď ´γN 4 @t P r0, T s.

Indeed, by definition of τ N , P γ N -almost-surely, Ψpp µ N,x t q ď ´γN 2 for all t ď τ N , and, P γ N -almost-surely, by definition of r N and Lemma 4.1, it holds, whenever

t ě τ N ˇˇΨpp µ N,x t q ´Ψpp µ N,x τ N q ˇˇď C Ψ d 2 pp µ N,x t , p µ N,x τ N q ď C Ψ ? N ˜N ÿ i"1 |X i,N t ´Xi,N τ N | 2 ¸1{2 ď C Ψ ˆrN ď γ N
and, as a consequence, being Ψpp µ N,x τ N q " ´γN 2 , it holds that Ψpp µ N,x t q ď ´γN 4 . Therefore, we have an admissible control for U N pµ 0 q. Now, by standard propagation of chaos estimates (see [START_REF] Horowitz | Mean rates of convergence of empirical measures in the Wasserstein metric[END_REF]) it holds that

E « sup tPr0,T ^τN s d 2 2 pµptq, p µ N,x t q ff ď CN ´2 d`8 .
As a consequence, we get

P γ N rτ N ă T s " P γ N " Dt ă T, Ψpp µ N,x t q ě ´γN 2 ı ď P γ N " Dt ă τ N , Ψpp µ N,x t q ě ´3γ N 4  ď P γ N « sup tPr0,T ^τN s d 2 2 pp µ N,x t , µptqq ě 1 C 2 Ψ pδ ´3γ N 4 q 2 ff ,
where we use the facts that Ψpµptqq ď ´δ, for all t P r0, T s as well as |Ψpp µ N,x t q ´Ψpµptqq| ď C Ψ d 2 pp µ N,x t , µptqq. Using Markov's inequality we conclude that,

P γ N rτ N ă T s ď C 2 Ψ E P γ N " sup tPr0,T ^τN s d 2 2 pp µ N,x t , µptqq ı pδ ´3γ N 4 q 2 ď C 2 Ψ E " sup tPr0,T ^τN s d 2 2 pp µ N,x t , µptqq ı pδ ´3γ N 4 q 2 P " Ψpp µ N,x 0 q ď ´γN ı ď CN ´2 d`8 ,
for some C ą 0 independent of N . We deduce immediately that

E P γ N rT ´T ^τN s ď T P γ N rτ N ă T s ď CN ´2 d`8 .
Using Lemma (4.1), we get

E P γ N « ż T τ N ^T 1 N N ÿ i"1 |β i,N t | 2 ff ď C r 2 N N `C E P γ N rT ´T ^τN s r 2 N ď C N ´2 d`8 γ 2 N .
But we have chosen γ N so that lim

N Ñ`8 N ´2 d`8 γ 2 N " 0. Therefore it holds that lim N Ñ`8 E P γ N « ż T τ N ^T 1 N N ÿ i"1 |β i,N t | 2 ff " 0.
We easily deduce that, lim N Ñ`8

E P γ N « sup tPr0,T s d 2 2 pµptq, p µ N,x t q ff " 0.
As a consequence, α being bounded and F and G Lipschitz continuous with respect to d 1 ,

E P γ N « ż T ^τN 0 1 N N ÿ i"1 LpX i,N t , αpt, X i,N t qqdt `ż T ^τN 0 Fpp µ N,x t qdt ff `EP γ N « ż T T ^τN 1 N N ÿ i"1 LpX i,N t , β i,N t qqdt `ż T T ^τN Fpp µ N,x t qdt ff `EP γ N " Gpp µ N,x T q ı " ż T 0 ż R d Lpx, αpt, xqqdµptqpxqdt `ż T 0 Fpµptqqdt `GpµpT qq `op1q
as N Ñ `8. Finally, being α optimal for U δ pµ 0 q we have that lim sup N Ñ`8

U N pµ 0 q ď U δ pµ 0 q.

Yet, we have proved, in Proposition (4.1) that lim δÑ0 U δ pµ 0 q " Upµ 0 q and therefore, lim sup

N Ñ`8
U N pµ 0 q ď Upµ 0 q, which concludes the proof of the theorem.

Proof of Lemma (4.1) . For η ě 0 small, we introduce the stopping time

τ η :" inftt ě τ N ^T, 1 N N ÿ i"1 |X i,N t ´Xi,N τ N ^T | 2 ě r 2 N ´ηu,
with the convention that inf t∅u " `8.

For η ą 0 and T 1 ą T , we write B t " t pB 1,N t , . . . , B N,N t q and Y t " t pX i,N t , . . . , X N,N t q and apply Itô's lemma to get

´e´τ η ^T 1 logpr 2 N ´|Y τ η ^T 1 ´Yτ N ^T | 2 N q " ´e´τ N ^T logpr 2 N q `ż τ η ^T 1 τ N ^T e ´t logpr 2 N ´|Y t ´Yτ N ^T | 2 N qdt `ż τ η ^T 1 τ N ^T e ´t " 4pY t ´Yτ N ^T q |Y t ´Yτ N ^T | 2 ´r2 N N . 2pY t ´Yτ N ^T q N r 2 N ´|Y t ´Yτ N ^T | 2 ´2d r 2 N 2|Y t ´Yτ N ^T | 2 N r 2 N ´|Y t ´Yτ N ^T | 2  dt `ż τ η ^T 1 τ N ^T e ´t " 2dN N r 2 N ´|Y t ´Yτ N ^T | 2 `4|Y t ´Yτ N ^T | 2 pN r 2 N ´|Y t ´Yτ N ^T | 2 q 2  dt `?2 ż τ η ^T 1 τ N ^T 2pY t ´Yτ N ^T q N r 2 N ´|Y t ´Yτ N ^T | 2 .dB t ď ´4 ż τ η ^T 1 τ N ^T e ´t |Y t ´Yτ N ^T | 2 p|Y t ´Yτ N ^T | 2 ´r2 N N q 2 dt `ż τ η ^T 1 τ N ^T e ´t " ´4d r 2 N |Y t ´Yτ N ^T | 2 ´dN N r 2 N ´|Y t ´Yτ N ^T | 2  dt `?2 ż τ η ^T 1 τ N ^T 2pY t ´Yτ N ^T q N r 2 N ´|Y t ´Yτ N ^T | 2 .dB t
However, an elementary analysis reveals that

´4d r 2 N |Y t ´Yτ N ^T | 2 ´dN N r 2 N ´|Y t ´Yτ N ^T | 2 ď 2d r 2 N whenever 0 ď |Y t ´Yτ N ^T | 2 ă N r 2 N
. Therefore, we get, multiplying by e τ N ^T and taking expectations,

´EP γ N " e τ N ^T ´τ η ^T 1 logpr 2 N ´|Y τ η ^T 1 ´Yτ N ^T | 2 N q  `4E P γ N « ż τ η ^T 1 τ N ^T e τ N ^T ´t |Y t ´Yτ N ^T | 2 p|Y t ´Yτ N ^T | 2 ´r2 N N q 2 dt ff ď 2d r 2 N .
Letting T 1 Ñ `8, using the definition of τ η and Lebesgue dominated convergence theorem leads to

´logpηqE P γ N " e τn^T ´τ η 1 tτ η ă`8u ‰ `4E P γ N "ż τ η τ N ^T e τ N ^T ´t |Y t ´Yτ N ^T | 2 p|Y t ´Yτ N ^T | 2 ´r2 N N q 2 dt  ď 2d r 2 N .
(4.6) Notice that both terms in the left-hand side of 4.6 are non-negative for η ď 1. Letting η Ñ 0, we get, on the one hand that τ 0 " `8, P γ N -almost surely and, on the other hand, we obtain

4E P γ N "ż `8 τ N ^T e τ N ^T ´t |Y t ´Yτ N ^T | 2 p|Y t ´Yτ N ^T | 2 ´r2 N N q 2 dt  ď 2d r 2 N .
It follows that,

E P γ N « ż T τ N ^T 1 N ˇˇˇ4 pY t ´Yτ N q |Y t ´Yτ N | 2 ´r2 N N ´2 d r 2 N pY t ´Yτ N q ˇˇˇ2 dt ff ď 2E P γ N « ż T τ N ^T 1 N ˇˇˇ4 pY t ´Yτ N q |Y t ´Yτ N | 2 ´r2 N N ˇˇˇ2 dt ff `2E P γ N « ż T τ N ^T 1 N ˇˇˇ2 d r 2 N pY t ´Yτ N q ˇˇˇ2 dt ff ď 16d 2 r 2 N N E P γ N " e T ´T ^τN ‰ `8d 2 r 4 N E P γ N "ż T τ N ^T 1 N |Y t ´Yτ N | 2 dt  ď 16d 2 r 2 N N E P γ N " e T ´T ^τN ‰ `8d 2 r 2 N E P γ N rT ´T ^τN s ,
where we used, for the last inequality, the fact that, P γ N -almost-surely, for all t ě τ N ,

|Y t ´Yτ N | 2 ď N r 2 N .
This concludes the proof of the lemma.

From almost-sure constraint to mean-field constraint

To prove the second inequality we rely on compactness methods developed, in the context of Large Deviations by Budhiraja, Dupuis and Fischer [START_REF] Budhiraja | Large deviation properties of weakly interacting processes via weak convergence methods[END_REF] and, in the context of mean-field control, by Lacker [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] and Djete, Possamaï and Tan [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF]. To this end we need to introduce suitable weak formulations of the control problems. Let us first introduce some notations. We denote by C d :" Cpr0, T s, R d q the path space. The control space V is defined as the set of non-negative measures q over r0, T s ˆRd with the Lebesgue measure as time marginal and such that ż r0,T sˆR d |a| 2 dqpt, aq ă `8.

We denote by pX i,N , Λ i,N q the canonical process on pC d ˆVq N and define the empirical measures

p µ N :" 1 N N ÿ i"1 δ pX i,N ,Λ i,N q , p µ N,x t :" 1 N N ÿ i"1 δ X i,N t .
We define R N as the set of probabilities P N P P 2 ppC d ˆVq N q under which pX i,N 0 q i"1,...,N are iid with law µ 0 and

ϕpX 1,N t , . . . , X N,N t q ´N ÿ i"1 ż t 0 ż R d L N i ϕpX 1,N s , . . . , X N,N s , aqdΛ i,N s paqds
is a martingale under P N , for all smooth, compactly supported ϕ with L N i ϕpx 1 , . . . , x N , aq :" D x i ϕpx 1 , . . . , x N q.a `∆x i ϕpx 1 , . . . , x N q.

The control rule P N is also assumed to satisfy P γ N N pΨpp µ N t,x q ď 0, @t ě 0q " 1.

where we wrote, for simplicity P γ N N :" P N " .|Ψpp µ N,x 0 q ď ´γN ı . The N -state problem in its weak formulation is therefore to minimize over

P N P R N E P γ N N « ż T 0 ˜żR d 1 N N ÿ i"1
LpX i,N t , aqdΛ i,N t paq `Fpp µ N t,x q ¸dt `Gpp µ N,x T q ff where E P γ N N is the expectation under P γ N N . Similarly, for the mean-field problem we introduce the controlled martingale formulation. The control problem is described as follows. We let pX, Λq be the identity processes over pC d ˆVq and we look for probabilities m over C d ˆV such that X 0 is distributed according to µ 0 under m,

ϕpX t q ´ż t 0 ż R d
LϕpX s , aqdΛ s paqds is a martingale under m for all smooth compactly supported ϕ : R d Ñ R, with Lϕpx, aq " Dϕpxq.a `∆ϕpxq. The measure m is also assumed to satisfy the constraint ΨpX t #mq ď 0 @t ě 0.

We denote by R the set of such measures and we look for m P R which minimizes the cost function

Γpmq :" E m "ż T 0 ż R d LpX t , aqdΛ t paqdt  `ż T 0 FpX t #mqdt `GpX T #mq.
Before going on with the main result of this section, we make two remarks. The first one is that the value of the relaxed problem is no greater than U N pµ 0 q. Indeed any "strong" control induces a weak control with a lower cost. On the other hand, it is easier to show that the value of the relaxed mean-field problem is equal to Upµ 0 q. This follows for instance from the fact that Upµ 0 q is the value of an optimal control problem without constraint but with a strong enough penalization. And for problems without constraint, the equivalence between the different formulations is well known, see [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF].

We are ready to prove the desired inequality:

Theorem 4.3. Take P N a sequence of N -optimal solutions to the relaxed N -particles problem, for some sequence N Ñ 0. Then the sequence µ N #P γ N N is relatively compact in P p pP p pC d ˆVqq for every p P p1, 2q. Every limit point is supported on the set of solutions to the relaxed mean-field problem and it holds that Upµ 0 q ď lim inf N Ñ`8 U N pµ 0 q.

Proof. We will closely follow the steps of [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] and therefore we only highlight the differences due to the constraint. In light of [START_REF] Lacker | Mean field games via controlled martingale problems: Existence of Markovian equilibria[END_REF] Corollary B.2, to prove the pre-compactness of µ N #P γ N N , it suffices to prove that the mean measures 1 N N ÿ i"1 pX i,N , Λ i,N q#P γ N N are tight and to prove that sup

N E P γ N N 1 N N ÿ i"1 « sup tPr0,T s |X i,N t | 2 `ż T 0 ż R d |a| 2 dΛ i,N t paqdt ff ă `8. (4.7)
The tightness of the mean measures actually follows from (4.7) thanks to the compactness result of Proposition 3.5 in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] noticing that a martingale under P N remains a martingale under P γ N N . By standard estimates, it is enough to prove sup

N E P γ N N « 1 N N ÿ i"1 |X i,N 0 | 2 ff ă ` 8
as well as

sup

N E P γ N N « ż T 0 1 N N ÿ i"1 ż R d |a| 2 dΛ i,N t paqdt ff ă ` 8
in order to get (4.7). The former follows from

E P γ N N « 1 N N ÿ i"1 |X i,N 0 | 2 ff ď E P N " |X 1,N 0 | 2 ı P N " Ψpp µ N,x 0 q ď ´γN ı ď 2 ż R d |x| 2 dµ 0 pxq
which holds for N large enough since P N " Ψpp µ N,x 0 q ď ´γN ı Ñ 1 as N Ñ `8. The latter follows from the coercivity of L, the boundness of F, G and the fact that we took the P N as N -optimal solutions for the N -particle problem whose values are bounded independently from N (as proved in Theorem (4.2)). Now we take a limit point P P P p pP p pC d ˆVqq and prove that P is supported on the set of solutions to the mean field relaxed problem. First we have that p µ N,x 0 #P γ N N Ñ δ µ in P p pP p pR d qq. This follows from Glivenko-Cantelli law of large numbers since 1

N N ÿ i"1 E P γ N N " |X i,N 0 | 2
ı is bounded independently from N and, for all

f P C b pP p pR d qq, as N Ñ ` 8
ˇˇE P γ N N " f pp µ N,x 0 q ı ´EP N " f pp µ N,x 0 q ıˇˇˇď 2}f } 8 p1 ´PN " Ψpp µ N,x 0 q ď ´γN ı q Ñ 0.
Following [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] P γ N N pΨpp µ N,x t q ď 0q " 1.

Since P -almost surely m satisfies the martingale problem, we have that P -almost surely t Ñ X t #m is continuous and therefore we have that P ` m P P p pC d ˆVq, ΨpX t #mq ď 0 @t P r0, T sq (˘" 1.

where p µ

N,x t :" 1 N N ÿ i"1 δ X i,N t .
We assume that there is some x 0 P R d such that Ψpδ x 0 q ă 0. (4.9)

As a special case of Theorem (4.1) we have the following result.

Proposition 4.2. For all N ě 1, take x N 0 " px 0 , . . . , x 0 q such that Ψpδ x 0 q ă 0. Then it holds that

lim N Ñ`8 u N p0, x N 0 q " Upδ x 0 q.
Notice that, the initial position being deterministic and strictly inside the constraint, there is no need for conditioning as we did in the previous sections. Equivalently, P γ N " P for N large enough.

In the rest of this section we assume, for simplicity, that

The constraint tΨ ď 0u is contained inside the ball m P P 2 pR d q, d 1 pm, m 0 q ď R ( for some R ą 0. (

As a consequence, the constraints Ω N :"

! px 1 , . . . , x N q P R dN , Ψp 1 N ř N i"1 δ x i q ă 0 )
are bounded for all N ě 1. We also assume that δ 2 Ψ δm 2 has a linear derivative, with bounded and jointly continuous first order derivatives in the euclidean variables.

(4.11) Under Assumption (A) as well as these additional assumptions, for all N ě 1, the constraint Ω N is open, bounded and BΩ N is a manifold of class C 3 .

For pt, x N " px 1,N , . . . , x N,N qq P r0, T s ˆpR d q N we introduce the probability

v N pt, x N q :" Pp@s P r0, ts, Ψ ˜1 N N ÿ i"1 δ x i,N `?2B i,N t ¸ă 0q
where pB 1,N t , . . . , B N,N t q are N independent d-dimensional standard Brownian motions supported on some probability space pΩ, F, Pq.

We are precisely in the framework of [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF] section VI.6 and we can conclude that v N is C 1,2 in p0, T s ˆΩN and satisfies

$ & % B t v N ´∆v N " 0, in p0, T q ˆΩN v N " 0, in p0, T q ˆBΩ N v N " 1 in t0u ˆΩN .
Moreover, v N pt, xq ą 0 for all pt, x N q P p0, T s ˆΩN .

Proposition 4.3. For all x N P Ω N it holds

u N p0, x N q " ´2 N log v N pT, x N q,
where u N is the value function defined in (4.8).

Proof. For all pt, x N q P r0, T q ˆΩN we define w N pt, x N q :" ´2 N log v N pT ´t, x N q. We are going to proceed by verification to show that w N p0, x N q " u N p0, x N q in Ω N . For x N " px 1,N , . . . , x N,N q P Ω N , we define the following particle system

X N t :" x N ´ż t^τ 0 N Dw N pt, X N t qdt `?2 ż t^τ 0 dB N t " x N `ż t^τ 0 2 Dv N pT ´t, X N t q v N pT ´t, X N t q dt `?2 ż t^τ 0 dB N t ,
where B N t :" t pB 1,N t , . . . , B N,N t q and τ is the first exit time from Ω N :

τ :" inftt ě 0, X N t R Ω N u. For η ě 0 small, we introduce the stopping time τ η :" inftt ě 0, v N pT ´t, X N t q ď ηu. Notice that, by definition of v N , it holds that τ 0 " τ . Applying Itô's formula to log v N pT t, X N t q yields, for η ą 0,

log v N pT ´τ η ^T, X N τ η ^T q " log v N pT, x N q `ż τ η ^T 0 « ´Bt v N v N `2 ˇˇˇD v N v N ˇˇˇ2 `∆v N v N ´ˇˇˇD v N v N ˇˇˇ2 ff pT ´t, X N t qdt `ż τ η ^T 0 2 Dv N pT ´t, X N t q v N pT ´t, X N t q .dB N t " log v N pT, x N q `ż τ η ^T 0 ˇˇˇD v N pT ´t, X N t q v N pT ´t, X N t q ˇˇˇ2 dt `ż τ η ^T 0 2 Dv N pT ´t, X N t q v N pT ´t, X N t q .dB N t .
Taking expectations and recalling the definition of τ η we get logpηqPpτ η ď T q `Ppτ η ą T q ě log v N pT, x N q.

As a consequence, lim ηÑ0 Ppτ η ď T q " 0 and the control ´N Dw N pT ´t, X N t q is admissible. Let us show that it is optimal. Recalling the equation satisfied by v N , it holds that

$ & % ´Bt w N `N 2 |Dw N | 2 ´∆w N " 0, in p0, T q ˆΩN w N " `8, in p0, T q ˆBΩ N w N " 0 in tT u ˆΩN .
Let us take another admissible control α with the associated solution Y N to the SDE:

Y N t :" x N `ż t 0 α s ds `?2 ż t 0 dB N s .
Being α admissible, it holds that Y N t belongs to Ω N for all t P r0, T s almost surely. We can apply Itô's lemma to w N and get

0 " E " w N pT, Y N T q ‰ " w N p0, x N q `E "ż T 0 B t w N pt, Y N t q `αt .Dw N pt, Y N t q `∆w N pt, Y N t q  dt " w N p0, x N q `E "ż T 0 ˆαt .Dw N pt, Y N t q `N 2 |Dw N pt, Y N t q| 2 ˙dt  ě w N p0, x N q ´E "ż T 0 1 2N |α t | 2 dt



with equality if and only if α t " ´N Dw N pt, Y N t q. This means that the control ´N Dw N pt, Y N t q is optimal and the optimal value is given by w N p0, x N q which concludes the proof of the proposition.

Finally we obtain the following convergence.

Corollary 4.1. Let Assumption (A) as well as Assumptions (4.9), (4.10) and (4.11) hold. Assume that Ψpδ x 0 q ă 0 and write x N 0 " px 0 , . . . , x 0 q. Then it holds

lim N Ñ`8 2 N log v N pT, x N 0 q " ´Upδ x 0 q.
This is a very special case of the general result of Dawson of Gärtner, [START_REF] Dawson | Large deviations from the mckean-vlasov limit for weakly interacting diffusions[END_REF]. However the optimality conditions of Chapter 3 give a new way to compute the limit ´Upδ x 0 q.

Part III

A rate of convergence for the optimal control of McKean-Vlasov dynamics

Our results

To describe our result we need to introduce the map U : r0, T s ˆP2 pR d q Ñ R, where P 2 pR d q is the space of Borel measures on R d with a finite second moment, given, for pt 0 , m 0 q P r0, T s ˆP2 pR d q, by Upt 0 , m 0 q :" inf

α Er ż T t 0 `LpX t , α t q `FpLpX t |F B 0 t qq ˘`GpLpX T |F B 0 T qqs, (5.4)
where the infimum is taken over an appropriate set of admissible controls (this will be made precise later), F B 0 " pF B 0 t q 0ďtďT denotes the filtration generated by B 0 , LpX t |F B 0 t q is the law of X t conditioned upon F B 0 t , and

X t " X t 0 `ż t t 0 α s pX s qds `?2pB t ´Bt 0 q `?2a 0 pB 0 t ´B0 t 0 q, (5.5)
with B another Brownian motion, X t 0 a random initial condition with law m 0 and B 0 , B and X t 0 mutually independent.

Although it is known (more about this later in the introduction) that, as N tends to infinity, V N converges to U, the existing convergence results come without any rate. Our main result is the following algebraic convergence rate: there exists β P p0, 1s (depending on dimension only) and C ą 0 (depending on the data of the problem) such that, for any pt, xq P r0, T s ˆpR d q N one has:

ˇˇV N pt, m N x q ´Upt, m N x q ˇˇď CN ´β p1 `M 1{2 2 pm N x qq, (5.6)
where M 2 pm N x q " N ´1 ř N i"1 |x i | 2 is the second-order moment of the measure m N x . Although the exact value of β could be traced back through the computation, it is clearly not optimal. In particular, it is very far from the one obtained in the standard particle system. In the same way, even if some dependence with respect to a moment of the measure is expected, the dependence given here is probably far from sharp.

Background and related literature

The convergence of V N to U was shown by Lacker [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] in a very general framework and for suitable initial data but without common noise, that is a 0 " 0 in (5.2). Very recently, the results of [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] have been extended in Djete, Possamaï and Tan [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] to problems with a common noise and interaction through the controls. Beside [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF][START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF] several other papers have studied the question of the mean field limit of optimal control problems, for example Cavagnari, Lisini, Orrieri and Savaré [START_REF] Cavagnari | Lagrangian, Eulerian and Kantorovich formulations of multi-agent optimal control problems: Equivalence and Gamma-convergence[END_REF] and Fornasier, Lisini, Orrieri and Savaré [START_REF] Fornasier | Mean-field optimal control as Gamma-limit of finite agent controls[END_REF] investigate the problem without noise by Γ´convergence techniques. The recent contribution of Gangbo, Mayorga and Swiech [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] studies the mean field limit without idiosyncratic but with common noise using partial differential equations (PDE for short) techniques. This is The situation is different and much more difficult in the continuous state space. This might come as a surprise since the convergence rate for particle systems is very well understood; see, for instance, Fournier and Guillin, [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF]). The main difficulty we face is that, even though the optimal feedback in the particle system remains bounded independently of N (see Lemma 5.1), it cannot be expected to be uniformly continuous as a function of the empirical measure. Indeed this uniform continuity would imply the C 1 regularity of the limit U, which does not hold in general. So we have to find a way to show that, despite the fact that the controls played by each particle might be very different, a kind of concentration of measure takes place.

Strategy of the proof

A few words on the method of proof are now in order. Let us first point out that we do not rely on a propagation of chaos, which we cannot prove at this stage. Indeed, as for a given initial condition there might be several optimal trajectories for the limit problem, a propagation of chaos is not expected to hold without additional assumptions on the initial data. The main ingredients for the proof are, uniform in N , Lipschitz and semiconcavity estimates for V N , and a concentration inequality. To bound from above V N by U is relatively easy, because V N can be transformed into an approximate subsolution for the Hamilton-Jacobi equation (5.8). The opposite inequality is much trickier, because it seems impossible to transform an optimal control for the V N (in which the control depend on each particle) into a feedback for U. We overcome this difficulty by dividing the particles into subgroups in such a way that the optimal controls for the particles in each subgroup are close and show a propagation of chaos, based on a concentration inequality, for each subgroup. The proof being technical, we first show the result when there is no common noise (a 0 " 0) and, in a second step, extend the result to problems with a common noise.

Organization of the paper

The paper is organized as follows. In the rest of the introduction we fix notation. We state the assumptions and the main result in section 5.2. As the proof of the convergence rate is technical, we start in section 5.3 with the problem without common noise. Indeed this case contains the main ideas without the extra technicalities due to the common noise. We first give some estimates on V N and U (subsection 5.3.1), then show a first and relatively easy bound from above for V N in subsection 5.3.2. The main part of the proof, that is, the bound from below, which is the aim of subsection 5.3.3 requires a concentration inequality proved in subsection 5.3.4. We finally explain the adaptation of the proof to the case with common noise in section 5.4.

Notation

For x " px 1 , ..., x N q P pR d q N , m N

x P PpR d q denotes the empirical measure of x, that is, m N

x " 1 N ř N i"1 δ x i . We write I d for the identity matrix in R d , and B R for the ball in R d centered at the origin with radius R. If ϕ : r0, T s ˆRd Ñ R d is smooth enough, we denote by Dϕ, ∆ϕ and D 2 ϕ the derivatives with respect to space and by B t ϕ and B tt ϕ the derivatives with respect to time. Similarly, for V " Vpt, x 1 , ..., x N q : r0, T s ˆpR d q N Ñ R, we define the derivatives D x k V, ∆ x k V, B t V. We denote by PpR d q the set of Borel probability measures on R d and note that, if m P PpR d q has a density, for simplicity of notation, m is also be used to denote the density. Given m P PpR d q and p ě 1, we denote by M p pmq the p th moment of m, that is, M p pmq " ş R d |x| p dm, and by P p pR d q the set of m P PpR d q such that M p pR d q ă 8. We endow P p pR d q with the Wasserstein metric d p , defined by

d p p pm, m 1 q :" inf πPΠpm,m 1 q ż R d |x ´y| p dπpx, yq,
where Πpm, m 1 q is the set of all π P PpR d ˆRd q with marginals m and m 1 . We recall the duality formula

d 1 pm, m 1 q " sup φPL ż R d φdpm ´m1 q,
where L is the set of all 1-Lipschitz functions from R d to R. We write L R for the set of all 1-Lipschitz functions φ : B R Ă R d Ñ r´R, Rs. For any φ P L R , we denote by r φ the extension r φ : R d Ñ r´R, Rs (note that r φ is also 1-Lipschitz) given by r φpxq "

$ ' & ' % φpxq |x| ď R, 2R´|x| R φp R |x| xq R ă |x| ă 2R, 0 |x| ě 2R.
Finally, for U : P 1 pR d q Ñ R is smooth enough, δU δm : P 1 pR d q ˆR Ñ R denotes the linear functional derivative, which satisfies, for all m, m 1 P P 1 pR d q and all h P p0, 1q,

Upm 1 q ´Upmq " ż 1 0 ż R d δU δm pp1 ´hqm `hm 1 , xqpm 1 ´mqpdxqdh.
We use the standard convention ż R d δU δm pm, xqmpdxq " 0 for all m P P 1 pR d q. If δU δm is differentiable with respect to the space variable, we define the L-derivative of U by D m U pm, xq " D x δU δm pm, xq. Higher order derivatives are defined similarly. We refer to [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF][START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF] for the properties of the L-derivatives.

Assumptions and main result 5.2.1 Assumptions

We now state our standing assumptions on the maps H, F and G, which constitute the data of our problem. We keep in mind that L : R d ˆRd Ñ R is a Legendre transform of H with respect to the last variable: Lpx, aq " sup pPR d r´a ¨p ´Hpx, pqs.

We assume that

#

There exist constants c, C ą 0 such that ´C `c|p| 2 ď Hpx, pq ď C `1 c |p| 2 for all px, pq P R d ˆRd ,

(5.10)

H : R d ˆRd Ñ R is of class C 2 , ( 5
F : P 1 pR d q Ñ R is of class C 2 with F, D m F, D 2 ym F and D 2 mm F uniformly bounded, (5.15
) and, finally, G : P 1 pR d q Ñ R is of class C 4 with all derivatives up to order 4 uniformly bounded.

(5.16) For simplicity, in what follows we put together all the assumptions above in assume that (5.10), (5.11), (5.12), (5.13), (5.14), (5.15) and (5.16) hold, (5.17)

Remark 5.1. We make the following comments regarding (5.17).

(i) The strict convexity of H with respect to the gradient variable is standard in optimal control. In particular, it implies that L has the same regularity as H.

(ii) Although (5.13), which is used to obtain, independent of N , Lipschitz estimates on the value function V N (see Lemma 5.1), is more restrictive, but we do not know it is possible to avoid it. It is satisfied, for instance, by a Hamiltonian of the form Hpx, pq " |p| 2 `V pxq ¨p for some smooth and globally Lipschitz continuous vector field V : R d Ñ R d .

(iii) The fact that the "full" Hamiltonian px, p, mq Ñ Hpx, pq ´Fpmq has a separate form is not completely necessary. Some (small) extensions are possible, but we have decided to keep it in a separate form in order to avoid unnecessary technicalities.

(iv) The uniform bounds on D m F and D m G imply that both maps are Lipschitz continuous in P 1 pR d q. The additional smoothness is used to obtain, independent of N , semiconcavity estimates on the value function V N (see Lemma 5.3)

Formulation of the problem

For concreteness, we fix throughout the paper a filtered probability space pΩ, F, F " pFq tě0 , Pq satisfying the usual conditions and hosting independent d-dimensional Brownian motions B 0 and pB k q kPN .

Definition of V N

The definition of V N and the relevant quantities/functions were given and explained in the introduction-see (5.1), (5.2) and (5.3).

As explained in the introduction, it is well known that under (5.17), V N is the unique classical solution to the Hamilton-Jacobi equation (5.7) and that the infimum in (5.1) is achieved (in feedback form) by the function α " pα k q N k"1 : r0, T s ˆpR d q N Ñ R N given by α k pt, xq " ´Dp Hpx k , N D x k V N pt, xqq.

(5.19)

The definition of U without common noise Suppose now that a 0 " 0. To define U, we find it more intuitive to work with closed-loop controls, and to view the problem in terms of deterministic control of the associated Fokker-Planck equation. For fixed pt 0 , m 0 q P rt 0 , T s ˆP2 pR d q, let Apt 0 , m 0 q to be the set of pairs pm, αq with m " pm t q tPrt 0 ,T s " pmpt, ¨qq tPrt 0 ,T s P C 0 prt 0 , T s; P 2 pR d qq, α : rt 0 , T s ˆRd Ñ R d measurable and such that Then we define U : r0, T s ˆP2 pR d q Ñ R by Upt 0 , m 0 q " inf pm,αqPApt 0 ,m 0 q ! ż T t 0 `żR d Lpx, αpt, xqqmpt, dxq `Fpm t qqdt `Gpm T q ) (5.20)

One advantage to using this deterministic formulation of the McKean-Vlasov control problem is that, at least in the absence of common noise, the dynamic programming principle is straightforward. In particular, we can assert the following dynamic programming principle, which will be useful in what follows.

Proposition 5.1. Assume (5.17). Then, for any 0 ď t 0 ď t 1 ď T , Upt 0 , m 0 q " inf pm,αqPApt 0 ,m 0 q ! ż t 1 t 0 `żR d Lpx, αpt, xqqm t pdxq `Fpm t qqdt `Upt 1 , m t 1 q

) .

The definition of U with common noise

To define U when a 0 ą 0, we once again use a form of closed-loop formulation, but this time the relevant Fokker-Planck equation becomes stochastic and we work with a notion of weak solution.

For fixed pt 0 , m 0 q P r0, T s ˆP2 pR d q, we define a control rule R P Apt 0 , m 0 q to be a tuple R " pΩ, F, F, P, W, m, αq, where 1. pΩ, F, F " pF t q 0ďtďT , Pq is a filtered probability space supporting the d-dimensional Brownian motion W , 2. α " pα t q t 0 ďtďT is a F-progressively measurable taking values in L 8 pR d ; R d q and such that α is uniformly bounded, in the sense that

Now we define

Upt 0 , m 0 q " inf RPApt 0 ,m 0 q E P " ż T t 0 `żR d Lpx, α t pxqqm t pdxq `Fpm t qqdt `Gpm T q ı .

(5.23)

The connection to the informal description (5.4) of U is that, if α is a bounded L 8 pR d ; R d qvalued process defined on some filtered probability space probability space pΩ, F, F " pF t q 0ďtďT , Pq supporting independent Brownian motions B and W , α is a adapted to the filtration of W and X is a strong solution to the McKean-Vlasov equation X t " X t 0 `ż t t 0 α s pX s qds `?2pB 0 t ´B0 t 0 q `?2a 0 pW 0 t ´W 0 t 0 q, (5.24) then pΩ, F, F W , W, m, αq P Apt 0 , m 0 q, where m t " LpX t |W q, that is, m is the conditional law of X given the filtration of the Brownian motion W .

As in the case a 0 " 0, we have the following dynamic programming principle.

Proposition 5.2. Assume (5.17). Then, for any 0 ď t 0 ă t 1 ď T , for U defined by (5.23), we have Upt 0 , m 0 q " inf pm,αqPApt 0 ,m 0 q

E P "ż t 1 t 0 p ż R d
Lpx, α t pxqqm t pdxq `Fpm t qqdt `Upt 1 , m t 1 q  .

Unlike in the case without common noise, where the control problem is deterministic and thus the dynamic programming principle is straightforward, in the common noise case we will need to use some machinery from Djete, Possamaï and Tan [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] and Lacker, Shkolnikov and Zhang [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] to verify that the dynamic programming principle holds in this setting. To streamline the presentation, we present the proof of Proposition 5.2 as well as of some other technical results from [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF][START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF][START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] in the Appendix.

Remark 5.2. We could have defined U using (5.23) when a 0 " 0 as well, and, in the end, it would be possible (thanks in part to Lemma 5.2 below) to prove that this is equivalent to the definition (5.20). We chose to define things separately with and without common noise mostly to avoid some unnecessary technicalities and to simplify the presentation for the reader interested in the case without common noise. The only mathematical reason for splitting up the definitions is that for technical reasons it is convenient to work with L 8 feedback controls in the case of common noise, whereas without common noise we have no difficulty working with square-integrable controls.

The main result

With V N defined by (5.1), U defined by (5.20), if a 0 " 0, or (5.23), if a 0 ą 0, we have the following result.

Theorem 5.1. Suppose that Assumption (5.17) holds. Then there exists β P p0, 1s, which depends only on d and C ą 0 depending on the data such that, for any pt, xq P r0, T sˆpR d q N , ˇˇV N pt, xq ´Upt, m N

x q ˇˇď CN ´β p1 `M2 pm N x qq.

For the convenience of the reader we repeat here the strategy of the proof. We detail in section 5.3 the proof of Theorem 5.1, the adaptation to the case a 0 ą 0 being the aim of section 5.4. The proof of Theorem 5.1 requires several steps: We first provide uniform regularity estimates on V N (Lipschitz and semiconcavity estimates, see Lemma 5.1 and 5.3). Then we show how to bound from above V N by U plus an error term (Proposition 5.3). This Thus, for some constant C depending on T , on the regularity of L, F and G and on }Dα} 8 , Upt 0 , m 1 q ď ż T ď Upt 0 , m 0 q `Cd 1 pm 1 , m 0 q. This establishes the estimate |Upt 0 , m 0 q ´Upt 0 , m 0 q| ď Cd 1 pm 0 , m 0 q.

(5.27)

Finally, we fix s 0 ă t 0 , and we choose pm, αq optimal in the definition of Ups 0 , m 0 q. By dynamic programming (Proposition 5.1), we have Ups 0 , m 0 q " ż t 0 s 0 ´żR d Lpx, αpt, xqqm t pdxq `Fpm t q ¯dt `Upt 0 , m t 0 q, and so |Ups 0 , m 0 q ´Upt 0 , m 0 q| ď | ż t 0 s 0 ´żR d Lpx, αpt, xqqm t pdxq `Fpm t q ¯dt| `|Upt 0 , m t 0 q ´Upt 0 , m 0 q| ď Cpt 0 ´s0 q `Cd 1 pm t 0 , m 0 q ď Cpt 0 ´s0 q `Cpt 0 ´s0 q 1{2 ď Cpt 0 ´s0 q 1{2 , where we have used (5.27) and the boundedness of α, together with the fact that Assumption 5.10 implies a similar inequality for L. This completes the proof.

The key estimate on V N is given in the following Lemma.

Lemma 5.3. Assume (5.17). There exists an independent of N constant C,such that, for any N ě 1 and any ξ " pξ i q P pR d q N and ξ 0 P R,

N ÿ i,j"1 D 2 x i x j V N pt, xqξ i ¨ξj `2 N ÿ i"1 D 2 x i t V N pt, xq ¨ξi ξ 0 `D2 tt V N pt, xqpξ 0 q 2 ď C N N ÿ i"1 |ξ i | 2 `Cpξ 0 q 2 .
(5.28)

Remark 5.4. Inequality (5.28) plays a crucial role in the proof of Lemma 5.9 below. Since V N converges to U, it follows that (5.28) implies the semi-concavity of the extension r U : r0, T s ˆL2 pp r Ω, r F, r Pq; R d q Ñ R defined, for X P L 2 p r Ω, R d q, by r Upt, Xq :" Upt, LpXqq, where p r Ω, r F, r Pq is a fixed atomless probability space and LpXq is the law of the random variable X.

Fix pt 0 , m 0 q P r0, T q ˆP2 pR d q and let α ˚be optimal in the definition of Upt 0 , m 0 q. Using Lemma (5.4) together with a standard verification argument, for example, using Itô's formula in [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF], we see that p V N pt 0 , m 0 q ď inf αPApt 0 ,m 0 q ! ż T t 0 `żR d Lpx, αpt, xqqm t pdxq `p F N pm t q ¯dt `p G N pm T qs ) and, hence, p V N pt 0 , m 0 q ď ż T t 0 `żR d Lpx, α ˚pt, xqq `p F N pm t q ¯dt `p G N pm T q.

(5.31)

Since, in view of Lemma 5.2, α ˚is uniformly bounded by a constant independent of N , an easy computation shows that the corresponding state process satisfies So, by the optimality of α ˚, we can use (5.31) together with the estimates of p F N and p G to obtain p V N pt 0 , m 0 q ď Er ż T t 0 `LpX t , α t q `FpLpX t qq ¯dt `GpLpX T qqs `Cp1 `M 1{2 2 pm 0 qqN ´β ď Upt 0 , m 0 q `Cp1 `M 1{2 2 pm 0 qqN ´β . Fix now x 0 P pR d q N . Then the Lipschitz estimate on V N and the same argument as above yield ˇˇV N pt 0 , x 0 q ´p V N pt 0 , m N x 0 q ˇˇď Cp1 `M 1{2 2 pm N x 0 qqN ´β . Putting together the last two estimates gives (5.30).

The main estimate

The main step of the proof is to show the opposite inequality. Proposition 5.4. There exists a constant β P p0, 1s, which depends only on the dimension, and a constant C ą 0, which depends on the data, such that, for any N ě 1 and any pt, xq P r0, T s ˆpR d q N , Upt, m N x q ´VN pt, xq ď CN ´β p1 `1

N N ÿ i"1 |x i | 2 q.
(5.32) Fix j P t1, . . . , Ju, set α k " α j if k P C j and let

X k t 0 `τ " x k 0 `τ α k `?2B k τ and Y k s 0 `τ " y k 0 `τ α k `?2B k τ , m j Y s 0 `τ " 1 n j ÿ kPC j δ Y k s 0 `τ
and m j X t 0 `τ "

1 n j ÿ kPC j δ X k t 0 `τ ,
consider the solution m j to # B t m j ´∆m j `αj ¨Dm j " 0 in ps 0 , T q ˆRd , m j ps 0 , ¨q " m j y 0 in R d , and, finally, set mpsq " 1 N ř jPJ n j m j psq. We state next the concentration inequality we need for the proof. We postpone the proof of Lemma 5.7 to subsection 5.3.4 except for the one of the third inequality, since it contains an argument is needed for the ongoing proof.

Proof of the third inequality in Lemma 5.7. Using the first two inequalities of Lemma 5.7 as well as the Cauchy-Schwarz inequality, the concavity of the maps n Ñ n 1´β and n Ñ n 1´2β , the fact that ř j n j " N , and the assumption that β P p0, 1{2q, and recalling that |J| " Cδ ´d and the estimate of M 2 pmps 0 qq in Lemma 5.5 we obtain the following string of inequalities which prove the claim.

Since the α j are uniformly bounded, the map Lp¨, α j q is uniformly Lipschitz independently of j. Hence, using Lemma 5.7 and Lemma 5.5, we find Note that the last inequality we used exactly the same argument as for the proof given above of the third inequality of Lemma 5.7.

ż s 0 `h s 0 ż R d J ÿ j"1
Hence, recalling the optimality of px 0 , y 0 q and using the equation for V N , we get 0 ě pe s 0 `h ´es 0 qpUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ´Cδ ´dβ p1 `λ´1 2 q Using the Lipschitz regularity of F and Lemma 5.7 to deal with the difference of the F and Lemma 5.5 to deal with the term in ř i |y i 0 |, we find 0 ě e s 0 hpUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ´Cδ ´dβ p1 `λ´1 2 q

h β N β ´Cλh 1{2 N ´1 N ÿ i"1
h β N β ´Cλ 1{2 h 1{2 ´Cθh ´Ch 2 ´es 0 `hE « 1 N ż t 0 `h t 0 N ÿ k"1
pLpX k s , α k q `αk ¨pN D x k Vps, X s qq `HpX k s , N D x k Vps, X s qqdsqds ff .

The regularity of L and H and the uniform boundedness of the α k and of N D x k V N allow to infer that 0 ě e s 0 hpUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ´Cδ ´dβ p1 `λ´1 2 q

h β N β ´Cλ 1{2 h 1{2 ´Cθh ´Ch 2 ´es 0 `hE « 1 N ż t 0 `h t 0 N ÿ k"1
pLpx k 0 , α k qds `αk ¨pN D x k Vps, X s qq `Hpx k 0 , N D x k Vps, X s qqqds

ff ´Ch 3{2 .
where in the last line we choose r β even smaller if necessary. With this choice of r β, we have now established that (5.36) holds for all values of N .

Finally, we conclude that, for all pt, xq P r0, T s ˆpR d q N , e t pUpt, m N

x q ´VN pt, xqq ď e s 0 pUps 0 , m N y 0 q ´VN pt 0 , x 0 qq `λ 2N

N ÿ i"1 |x i | 2
ď CN ´minp r β,α 3 q p1 `1

N N ÿ i"1 |x i | 2 q.
Before proving the various lemmas used in the proof of Proposition 5.4, we complete the proof of the main result.

Proof of Theorem 5.1. Combining Proposition 5.3 and Proposition 5.4 we know that there exist β P p0, 1s (depending on dimension) and C ą 0 such that, for any pt, xq P r0, T sˆpR d q N , ˇˇUpt, m N

x q ´VN pt, xqq ˇˇď CN ´β p1 `M 1{2 2 pm N x q `M2 pm N x qq ď CN ´β p1 `M2 pm N x qq.

We continue with the proofs of the several auxiliary results sated earlier.

Proof of Lemma 5.5. The proof of the first statement is an immediate consequence of the uniform bound on U and V N and of the Lipschitz estimate for V N .

Proof of Lemma 5.8. For K P N and any nonnegative integrable functions m 1 0 , . . . , m K 0 on R d such that ř K k"1 m k 0 P PpR d q, let U K pt 0 , m 1 0 , . . . , m K 0 q :" inf pm 1 ,β 1 q,...,pm K ,β K q

ż T t 0 p ż R d K ÿ k"1 Lpx, β k pt, xq m k pt, dxq qm k pt, xqdx `Fp K ÿ k"1 m k ptqqqdt `Gp K ÿ k"1 m k pT qq,
where the infimum is taken over the tuple of measures pm k , β k q (the β k being a vector measure) with β k ăă m k such that pm k , β k q solve in the sense of distributions B t m k ´∆m k `divpβ k q " 0 in pt 0 , T s ˆRd and m k pt 0 q " m k 0 in R d .

We establish next that U K pt 0 , m 1 0 , . . . , m K 0 q " Upt 0 , m 1 0 `¨¨¨`m K 0 q, and the result will then follow from Proposition 5.1.

Since obviously U K pt 0 , m 1 0 , . . . , m K 0 q ď Upt 0 , m 1 0 `¨¨¨`m K 0 q, next we concentrate on the reverse inequality.

Proof of the concentration inequality

To complete the proof of Proposition 5.4, we are need to show Lemma 5.7. For this, it is convenient to introduce a few more facts.

Let Lp , Rq be the -covering number of L R with respect to the L 8 -distance, that is, Lp , Rq " inftk P N : there exist φ 1 , ..., φ k P L R such that for all φ P L R , }φ ´φj } L 8 ă for some ju.

It is known (see, for example, [START_REF] Tikhomirov | Entropy and -Capacity of Sets In Functional Spaces[END_REF]) that Lp , 1q ď exptC ´du, (5.40) and, after a rescaling argument, Lp , Rq ď exptC `R ˘du.

(5.41) Indeed, if tφ 1 , ..., φ n u P L is {R-dense in L, then t r φ 1 , ..., r φ n u is -dense in L R , where r φ i pxq " Rφp x R q. Thus (5.41) follows from (5.40). We need two preliminary estimates and note that, without loss of generality, we can take t 0 " 0 in what follows. Finally, we recall the notation after Lemma 5.5. We may now apply Hoeffding's inequality (see, for example, Proposition 2.5 in [START_REF] Wainwright | High-Dimensional Statistics: A Non-Asymptotic Viewpoint[END_REF]) to complete the proof. Lemma 5.11. There exists a constant C such that, for any j P t1, ..., , Ju and R ą 0,

Er sup φPL R ż R d r φ `mj phq ´mj Y h ˘s ď Cp1 `R d d`2 qpn j q ´1 d`2 h 1 d`2 .
Proof. We fix ą 0 and use the estimate on Lp , Rq to choose K ď exptC `R ˘du and φ 1 , ..., φ K in L R such that, for each φ P L R , there exists k P t1, ..., Ku such that }φ ´φk } L 8 pB R q ă , and hence

› › › r φ ´r φ k › › › L 8 pR d q ď .
Then, using Lemma 5.10 and the upper bound on K, for any x ą , we have Further computations reveal that there is a constant C such that x ą 2 as soon as

x ě C R d d`2 h 1 d`2 γ 1 d`2 pn j q 1 d`2
.

(5.44)

By choosing γ even smaller, and deduce, in view of (5.42) and (5.43), that, for some constant C and all R, x as in (5.44) Finally, we give the proof of the concentration inequality.

Proof of Lemma 5.7. Throughout this argument, C denotes a positive constant which, although changing from line to line, depends only on d, T , and sup j |α j |.

Next, we prove (5.34) in the case t 0 " 0. We fix R ą 0, and note that, any ψ P L normalized with ψp0q " 0, can be written as ψ " r φ `ϕ, with φ P L R and |ϕ| ď |x|1 B c R . Thus, for any h P p0, 1s, we get Optimizing in R, that is, taking R " pn j q 1 2d`4 h ´1 2d`4 a 1 `M2 pm j p0qq, gives the result with β " 1 2d`4 .

The proof of Theorem 5.1 with a common noise

We now show that the method developed above can be adapted to problems with a common noise, that is, when a 0 is positive. Recall that V N and U are defined by (5.1) and (5.23) respectively.

Proof of Theorem 5.1 when a 0 ą 0. Since the proof follows closely the one in the case a 0 " 0, here we emphasize and explain the main differences.

We first note that the estimates of Lemma 5.1 and 5.3 remain valid (with the same proof), that is, there exists C ą 0 such that

}V N } 8 `N sup j }D x j V N } 8 `}B t V N } 8 ď C,
and, for any pt, xq P r0, T s ˆpR d q N , pξ i q i"1,...,N P pR d q N and ξ 0 P R,

N ÿ i,j"1 D 2 x i x j V N pt, xqξ i ¨ξj `2 N ÿ i"1 D 2 x i t V N pt, xqξ i ξ 0 `D2 tt V N pt, xqpξ 0 q 2 ď C N N ÿ i"1 |ξ i | 2 `Cpξ 0 q 2 . 1 n j ÿ kPC j δ Y k s 0 `τ ,
and m j be the solution to dm j t " " p1 `a0 q∆m j t ´αj ¨Dm j t ‰ `?2a 0 Dm j t ¨dB 0 t in pt 0 , T s ˆRd and m j t 0 " m j y 0 in R d . Finally, we set m s " N ´1 ř jPJ n j m j s , and claim that, for all h ě 0 and j P t1, . . . , Ju,

E " d 1 pm j s 0 `h, m j Y s 0 `h q ı ď Cp1
`M 1{2 2 pm j s 0 qq h β pn j q β , (5.46) and E " d 1 pm s 0 `h, m N X t 0 `h q ı ď Cθ `Cδ ´dβ p1 `λ´1 2 q h β N β .

(5.47)

The proof follows from Lemma 5.7. Indeed, to establish (5.46), we first note that the process pm t q tPrs´0,T s solves (5.22) in the sense of distribution (with B 0 replacing W ) if and only if the process r m t " pId ´?2a 0 pB 0 t ´Bt 0 qq7m t solves P´a.s. in the (classical) sense of distributions, with r α t pxq " α t px `?2a 0 pB 0 t ´B0 t 0 q, the equation

d
2α 0 B 0 h q ı " E " d 1 p r m j s 0 `h, m j r Y s 0 `h q ı ,
and so (5.46) holds. The proof for (5.47) is similar.

We proceed with the proof by noticing that the Dynamic Programming in Lemma 5.8 still holds but with an expectation, since now the measures are random, and with Proposition 5.2 replacing Proposition 5.1.

Moreover, since the conclusion of Lemma 5.9 also holds as already pointed out, we can argue as in the proof of Proposition 5.4 (the time-regularity provided by Lemma 5.12 replacing that in Lemma 5.2) that Upt, m N x q ´VN pt, xqq ď C

1 N β p1 `1 N N ÿ i"1 |x i | 2 q.
The conclusion then follows as in the proof of Theorem 5.1.

Appendix

The purpose of this appendix is to adapt some technical results from [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF] and [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] to our setting. Most importantly, we need to infer the dynamic programming principle (Proposition 5.2) in our setting from the dynamic programming principle which is stated in Theorem 3.1 of [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF]. Most of the arguments here are straightforward adaptations of the superposition and mimicking results achieved in [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF], and so the proofs are only sketched.

Following Definition 2.1 in [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF] and Definition 2.3 [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] we define, for each pt 0 , m 0 q P r0, T s ˆP2 pR d q, the set of weak controls A w pt 0 , m 0 q to be the set of tuples R " pΩ, F, P, F " pF t q 0ďtďT , G " pGq 0ďtďT , X, B, W, m, αq such that 1. pΩ, F, Pq is a probability space equipped with filtrations G, F such that, for all 0 ď t ď T , G t Ă F t and F t _ F B T K G T |G t .

2. X " pX t q 0ďtďT is a continuous, F-adapted R d valued process.

3. α " pα t q t 0 ďtďT is a bounded, F-predictable process taking values in R d .

4. pB, W q is a R d ˆRd -valued standard F Brownian motion, W is G-adapted, and F t _ σpBq K G T .

5. m " pm t q t 0 ďtďT is a G-predictable process taking values in P 2 pR d q and such that m t " LpX t |G t q for dP b ds-a.e. ps, ωq P rt, T s ˆΩ.

6. the state equation X t " X t 0 `ż t t 0 α s ds `?2pB t ´Bt 0 q `?2a 0 pW t ´Wt 0 q, LpX t 0 q " m 0 holds for all t 0 ď t ď T .

We also define U w pt 0 , m 0 q :" inf RPAwpt 0 ,m 0 q E P r ż T t 0 pLpX t , α t q `Fpm t qqdt `Gpm T qs

In our context, a superposition principle is a result asserting the following: given a control rule R " pΩ, F, F, P, W, m, αq P A P Rpt 0 , m 0 q, we can find an extension p r Ω, r F, Gq of pΩ, F, Fq hosting another Brownian motion B independent of F and a process X such that dX t " α t pX t qdt `?2dB t `?2a 0 dW t such that m t " LpX t |F t q. We refer to [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] for details. The superposition results of [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF] are useful to us because we need to apply some technical results from [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF][START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF], and the superposition results allow us to check that our formulation is equivalent to the one used in [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF][START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF].

In what follows, for technical reasons, that is, to have the coercivity condition on the cost appearing in Assumption 2.1 of [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF], we will work with a truncated version of the weak Proof of Proposition 5.2. We combine Theorem 3.1 of [START_REF] Fabrice | Extended mean field control problem: a propagation of chaos result[END_REF] with Proposition 5.5 to conclude that, for all 0 ď t 0 ď t 1 ď T and any R ě R 0 , Upt 0 , m 0 q " U R pt 0 , m 0 q " U R W pt 0 , m 0 q " inf RPA R W pt 0 ,m 0 q E P r ż t 1 t 0 pLpX t , α t q `Fpm t qqdt `UR W pt 1 , m t 1 qs " inf RPA R W pt 0 ,m 0 q E P r ż t 1 t 0 pLpX t , α t q `Fpm t qqdt `UR pt 1 , m t 1 qs " inf RPA R W pt 0 ,m 0 q E P r ż t 1 t 0 pLpX t , α t q `Fpm t qqdt `Upt 1 , m t 1 qs.

Since R can be arbitrarily large, it is easy to see that the above imply Upt 0 , m 0 q " inf RPAwpt 0 ,m 0 q E P r ż t 1 t 0 pLpX t , α t q `Fpm t qqdt `Upt 1 , m t 1 qs.

To get from here to Upt 0 , m 0 q " inf RPApt 0 ,m 0 q E P " ż t 1 t 0 `żR d Lpx, α t pxqqm t pdxq `Fpm t q ˘dt `Upt 1 , m t 1 q ı , we again use the superposition and adapt arguments results from [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF].

  0 pxq, where the supremum runs over the couples pλ, φq P R `ˆC 1,2 b pr0, T s ˆRd q satisfying " ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq ď f 1 2 pt, xq in r0, T s ˆRd φpT, xq ď λhpxq `g1 pxq in R d , and where f 1 2 : r0, T s ˆRd Ñ R and g 1 : R d Ñ R are such that f 2 pt, mq "

  continuous, bounded and has one linear derivative in m. The first order functional derivative δf 2 δm : r0, T s ˆP1 pR d q ˆRd Ñ R is globally Lipschitz continuous, bounded and x Ñ δf 2 δm pt, m, xq belongs to C 3`α b pR d q with bounds uniform in pt, mq. (e) g is continuous, bounded and has one functional derivative in m such that x Ñ δg δm pm, xq belongs to C 3`α b pR d q with bounds uniform in m.

42CHAPTER 2 . 2 .

 22 STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW Assumptions on the Hamiltonian (a) H is C 1 in pt, x, p, M q. The partial derivatives B x H, B p H and B M H are Lipschitz in r0, T s ˆRd ˆBp0, Rq ˆBp0, Rq for all R ą 0.

  Then the Hamilton-Jacobi-Bellman equation " ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq " f 1 2 pt, xq in r0, T s ˆRd φpT, xq " g 1 pxq in R d admits a unique strong solution φ P C

ż R d |x| r 2 2 ď 1 . 1 `

 2211 dmptqpxq ă M 2 1 for some new M 2 1 ą 0. The uniform estimate on |ω| follows by Hölder's inequality |ω|pr0, T s ˆRd q ď ˆż T 0 ż R d dmptqpxqdt ˙1{r 2 ˜ż T 0 ż R d ˇˇˇd ω dt b dm pt, xq ˇˇˇr 2 dmptqpxqdt ¸1{r T 1{r 2 M 1{r 2 Finally, mptq b dt-almost everywhere S p ˆdW dt b dm pt, xq ˙P rΛ ´, Λ `s which means that |W |pr0, T s ˆRd q ď ? dΛ `. The claim follows taking M " M 2

  xqdmpxq and gpmq " ż R d g 1 pxqdmpxq with g 1 P C 3`α b pR d q and f 1 2 satisfying the assumptions of Theorem 2.1. Let us introduce a dual problem for RP. Definition 2.2 (Dual Problem). The dual problem is : sup pλ,φqPR `ˆA,φPHJ ´pλh`gq ż R d φp0, xqm 0 pdxq, (DP)

"

  ´Bt upt, xq `Hpt, x, Dupt, xq, D 2 upt, xqq " f 1 2 pt, xq in r0, T s ˆRd upT, xq " g 1 pxq in R d (2.21) admits a unique strong solution u P C s ˆRd q when H satisfies Assumptions (1), (2), g 1 belongs to C 3`α b pR d q and f 1 2 P C b pr0, T s, C 3`α b pR d qq is Hölder continuous in time, uniformly in x.

  pr0, T s ˆRd q.

70CHAPTER 2 .

 2 STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW

B

  ˚:" tb P B, Lpa ˚, bq ď C ˚u is not empty and compact in B, b Ñ Lpa, bq is lower semicontinuous in B ˚for every a P A.

  t ϕpt, xq `Dϕpt, xq.αpt, xq `∆ϕpt, xqs dmptqpxqdt " 0. (3.3) Using an approximation argument similar to [129] Remark 2.3, we can extend the class of test functions to C 1,2

Proposition 3 . 1 .

 31 Assume that m P Cpr0, T s, P 2 pR d qq and α P L 2 mptqbdt `r0, T s ˆRd , R d satisfy the Fokker-Planck equation (3.2), starting from the initial position m 0 P P 2 pR d q then, sup tPr0,T s ż R d |x| 2 dmptqpxq `sup t‰s d 2 2 pmptq, mpsqq |t ´s| ď C for some C " Cp ż R d |x| 2 dm 0 pxq, ż T 0 ż R d |αpt, xq| 2 dmptqpxqdtq ą 0.

  1 and it satisfies the estimate sup tPr0,T s }uptq} n ď Cp ż T 0 }f ptq} n dt, }g} n q.

First

Remark 3 . 4 .

 34 Ideally we would like to consider constraints of the form Ψpmq "

" 0 with r m 1 , r m 1 q

 11 xqdmpxq " 0 for all m P P 2 pR d q and all C 1 map U . But the inequality ´Hpx, Dr upt, xqq ´dr ω dt b d r mptq pt, xq.Dr upt, xq ď Lpx, dr ω dt b d r m pt, xqq holds, with equality if and only if dr ω dt b d r m pt, xq " ´Dp Hpx, Dr upt, xqq. 0 pxq ď J l ,δ p r m, r ωq with equality if and only if dr ω dt b d r m pt, xq " ´Dp Hpx, Dr upt, xqq, dtb r mptq-almost everywhere. Now if we consider r m 1 solution to B t r m 1 ´divpD p Hpx, Dr upt, xqq r m 1 q `∆ r m 1 p0q " m 0 , a similar computation shows that ż R d r up0, xqdm 0 pxq " J l ,δ p´D p Hpx, Dr upt, xqq r m 1 which means that the cost ż R d r up0, xqdm 0 pxq can indeed be reached and, by minamility of pr ω, r mq we get ż R d r up0, xqdm 0 pxq " inf pω,mq J l ,δ (3.24) and r ω " ´Dp Hpx, Dr upt, xqq r mptq b dt.

  ´yqm y pt, xqdm 0 pyq where m y ptq is the solution to "

Proposition 3 . 5 . 2 dt 2

 3522 Suppose that pm, u, λ, βq is a solution of (3.11) for some , δ ą 0. Thenthe map t Þ Ñ Ψpmptqq is C 1 in r0, T s and C 2 in r0, T s Ş tt : Ψpmptqq ‰ 0uwith derivatives given by d dt Ψpmptqq " ´żR d D m Ψpmptq, xq.D p Hpx, Dupt, xqqdmptqpxq `żR d div x D m Ψpmptq, xqdmptqpxq and d Ψpmptqq " λptq ż R d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Ψpmptq, xqdmptqpxq `F pDuptq, D 2 uptq, D∆uptq, mptqq

  b d r mptq pt, xq.Dupt, xq `Hpx, Dupt, xqq  d r mptqpxqdt with equality if and only if dr ω dt b d r mptq " ´Dp Hpx, Duq, dt b d r mptq ´ae.

  d up0, xqdm 0 pxq and we can conclude that the infimum of the linearized problem is indeed ż R d up0, xqdm 0 pxq, it is achieved at p r m, r ωq and (3.34) holds true. Collecting the equations satisfied by u and r m, relation (3.34) as well as the exclusion conditions of Lemma 3.6, we get the optimality conditions for the constrained problem.

2 e

 2 e µt Dupt, xq gives ´Bt wpt, xq `Dwpt, xq.D p Hpx, Dupt, xqq ´∆wpt, xq " ´µwpt, xq ´Dx Hpx, Dupt, xqq.e µt Dupt, xq `Df pt, xq.e µt Dupt, xq ´eµt |D 2 upt, xq| 2 . Now, by assumption on H, |D x Hpx, Dupt, xqq| ď C 0 p1 `|Dupt, xq|q and therefore, for µ " 2C 0 , ´Bt wpt, xq `Dwpt, xq.D p Hpx, Dupt, xqq ´∆wpt, xq ď C 0 e µt |Dupt, xq| `Df pt, xq.e µt Dupt, xq ď ? 2e C 0 T pC 0 `}f ptq} 1 q sup ps,yqPr0,T sˆR d a wps, yq. By comparison between w and the obvious super-solution pt, xq Þ Ñ 1 2C 0 T }g} 2 1 `?2e C 0 T sup ps,yqPr0,T sˆR d a wps, yq ż T t pC 0 `}f psq} 1 q ds we deduce that, for all pt, xq P r0, T s ˆRd , wpt, xq ď Cp1 `sup ps,yqPr0,T sˆR d a wps, yqq for some C " Cp ż T 0 }f ptq} 1 dt, }g} 1 q ą 0. And therefore, sup pt,xqPr0,T sˆR d |Dwpt, xq| ď C for another constant C " Cp ż T 0

  xPR d |D k´1 Hpx, Dups, xqq| ď Cp1 `sup xPR d |D k ups, xq|q and therefore, by Grönwall's lemma 3.7, sup pt,xqPr0,T sˆR d |D k upt, xq| ď Cp}g} k , ż T 0 }f ptq} k dt, sup tPr0,T s }uptq} k´1 q

  38 and 3.39 respectively with u replaced by u m . Being ν m in L 8 pr0, T sq, z m , v m and u m are continuous in time and space. Clearly it is enough to prove the convergence of z m toward z, Dz m toward Dz and v m toward v. Assume that νpttuq " 0. In this case, we have, for all x P R d z m pt, xq Ñ zpt, xq, Dz m pt, xq Ñ Dzpt, xq, as m Ñ `8.

Theorem 4 . 1 .

 41 Let Assumption (A) (introduced below) hold. Assume further that µ 0 P P 2 pR d q satisfies Ψpµ 0 q ă 0 and ż R d |x| d`5 dµ 0 pxq ă `8. Then lim N Ñ`8

  Ψpm, xq| 2 dmpxq ‰ 0, whenever Ψpmq " 0 (TransCondPsi)

ż R d |x| 2

 2 dµ δ ptqdt ă R 2 whenever pµ δ ptqq 0ďtďT is optimal for U δ pµ 0 q. Now we use Assumption (TransCondPsi) aswell as the continuity of µ Þ Ñ Ψpµq and µ Þ Ñ ż R d |D m Ψpµq| 2 dµpxq with respect to d 1 , to conclude that there exists some η R ą 0 such that ż R d |D m Ψpµ, xq| 2 dµpxq ě η R , whenever |Ψpµq| ď η R and ż R d |x| 2 dµpxq ă 2R 2 . If we define δ 0 :" η R 4 , it is plain to check that ż R d |D m Ψpµ, xq| 2 dµpxq ě η R whenever δ P r0, δ 0 s, |Ψpµq `δ| ď η R 4 and ż R d |x| 2 dµpxq ď 2R 2 .

|x| d`5 dµ 0 pxq ă ` 8 .

 8 Then it holds that lim sup N Ñ`8

1 .

 1 m solves (in the sense of distributions) the Fokker-Planck equation B t m " ∆m ´divpmαq in pt 0 , T s ˆRd and m t 0 " m 0 , |αpt, xq| 2 mpt, dxqdt ă 8.

  Lpx, α t pxqqmpt, dxq `Fpmptqqqdt `GpmpT qq `C sup tPrt 0 ,T s d 1 pµptq, mptqq

sup tPrt 0

 0 ,T s ż R d |x| 2 mpt, dxq ď p1 `CT q ż R d |x| 2 m 0 pdxq `CT.It then follows from the Lipschitz continuity of F with respect tod 1 that p F N pmptqq ď Fpmptqq`C ż pR d q N d 1 pm N x , mptqq N ź j"1mpt, dx j q ď Fpmptqq`Cp1`M 1{2 2 pm 0 qqN ´β and, in the same way, p G N pmpT qq ď Gpmptqq `Cp1 `M 1{2 2 pm 0 qqN ´β .

Lemma 5 . 7 .d 1 d 1 " d 1

 57111 There exist positive constants β P p0, 1{2q. which depending on d, and C, which depends only on sup j |α j |, d and T , such that, for all h ě 0, E " d 1 pm j ps 0 `hq, m j pm j ps 0 `hq, m j pmps 0 `hq, m N Y s 0`h q ı ď Cδ ´dβ p1 `λ´1 2 q h β N β and E pmps 0 `hq, m N X t 0 `h q ı ď Cθ `Cδ ´dβ p1 `λ´1 2 q h β N β .

  k q `αk ¨pN D x k Vps, X s qq `HpX k s , N D x k Vps, X s qqqds ff .

Lemma 5 . 10 ." ´nj x 2 sh 0 Dups, Y k s qdB k s ą xs ď

 51020 There exists a constant C ą 0 such that, for any φ P L and any j P t1, ..., Ju, we have Pr ż R d φpm j phq ´mj Y h q ą xs ď exp Ch * . Proof. Let u the solution of " ´Bt u ´∆u ´αj ¨Du " 0 in p0, hq ˆRd , uphq " φ in R d ; note that, since }Dφ} ď 1, }Du} 8 ď 1. Using Itô's formula and the equation for m j , we get ż The random variables h ´1{2 ş h 0 Dups, Y k s qdB k s are independent and sub-Gaussian, uniformly in k. Indeed, viewing h ´1{2 ş 0 Dup¨, Y k qdB k as a time-changed Brownian motion, we have that h ´1{2 ş h 0 Dupt, Y k t qdB k t pdq " B τ , where B is a standard Brownian motion and τ ď 1 is a stopping time (we use here that }Du} 8 ď 1). In particular, Pr ż Pr sup 0ďtď1 |B t | ą h 1{2 xs, from which the claim follows easily.

  Hpt, x, Dφpt, xq, D 2 φpt, xqq " ´bpt, x, αpt, xqq.Dφpt, xq ´σt σpt, x, αps, xqq.D 2 φpt, xq (1.26) ´f1 pt, x, αpt, xqqfor some solution pλ, φ, mq of the system of optimality conditions:

	$ ' ' ' ' '	´Bt upt, xq `Hpt, x, Dupt, xq, D 2 upt, xqq "	δf 2 δm	pt, mptq, xq	in p0, T q ˆRd
	' ' '	B
	'
	&
	'
	'
	'
	'
	'
	'
	'
	'
	'
	%

t m ´divpB p Hpt, x, Dupt, xq, D 2 upt, xqqmq `ÿ i,j B 2 ij ppB M Hpt, x, Dupt, xq, D

  2

  . By a linearization procedure akin to what is done in Chapter 2, we prove, in Theorem 3.5 that optimal solutions pα, mq exist and satisfy α " ´Bp Hp., Duq for some solution pu, m, λ, βq of

	$ '	´Bt upt, xq `Hpx, Dupt, xqq ´∆upt, xq
	' ' ' ' & '	" B t m ´divpB p Hpx, Dupt, xqqmq ´∆m " 0 λptq δΨ δm pmptq, xq `δF δm	pmptq, xq	in p0, T q ˆRd , in p0, T q ˆRd ,
	' ' ' ' %	upT, xq "	β δ	δΨ δm	pmpT q, xq	`δG δm	pmpT q, xq in R d ,	mp0q " m 0 .

  2 pR d qq :

	$ ' ' ' ' ' ' ' ' '	´Bt upt, xq `Hpt, x, Dupt, xq, D 2 upt, xqq " B t m ´divpB p Hpt, x, Dupt, xq, D 2 upt, xqqmq	δf 2 δm	pt, mptq, xq	in r0, T s ˆRd (2.1a)
	' & '	`ÿ i,j	B 2 ij ppB M Hpt, x, Dupt, xq, D 2 upt, xqqq ij mq " 0	in r0, T s ˆRd (2.1b)
	'
	' ' ' ' ' ' ' '	upT, xq " λ	δΨ δm	pmpT q, xq	`δg δm	pmpT q, xq in R d , mp0q " m 0	(2.1c)
	%	λΨpmpT qq " 0, ΨpmpT qq ď 0, λ ě 0,		(2.1d)
	where Hpt, x, p, M q :" sup aPA t´bpt, x, aq.p ´σt σpt, x, aq.M ´f1 pt, x, aqu is the Hamiltonian
	of the system. The forward equation, Equation 2.1b is a Fokker-Planck equation which
	describes the evolution of the probability distribution m of the optimally controlled process.
	The backward equation, Equation 2.1a is an Hamilton-Jacobi-Bellman equation satisfied by
	the adjoint state u. The nonnegative parameter λ is the Lagrange multiplier associated to
	the terminal constraint. The forward and backward equations are coupled through the source
	term for the HJB equation, the terminal condition for the HJB equation and the exclusion
	condition λΨpmpT qq " 0.
	Our main result, Theorem 2.2 states that, under suitable growth and regularity assump-
	tions, optimal Markov policies α P L 0 pr0, T s ˆRd , Aq exist and satisfy :
	αpt, xq P argmax aPA ´bpt, x, aq.Dupt, xq ´σt σpt, x, aq.D 2 upt, xq ´f1 pt, x, aq	(
	for some solution pλ, u, mq of the above system of PDEs. Notice that we do not a priori require
						ż
	Ψ to be a convex function. When Ψpmq "
						R d

  continuous in both variables together with all derivatives D r t D s x φ with 2r `s ď n. Moreover, }φ}n`α 2 ,n`α is bounded, where

	}φ}n`α 2 ,n`α :"	ÿ 2r`sďn	}D r t D s x φ} 8	`ÿ 2r`s"n	tPr0,T s sup	}D r t D s x φpt, .q} α
	`ÿ 0ăn`α´2r´să2	sup xPR d	}D r t D s x φp., xq}n`α´2r´s

2

.

  Theorem 12.1.10. For every competitor pm, ω, W q such that

	Lpt, x,	dω dt b dm	pt, xq,	dW dt b dm
			Lpt, x,	dω dt b dm	pt, xq,	dt b dt dW	pt, xqq "	ż
	and
	ˆdω dt b dm	pt, xq,	dt b dm dW	pt, xq	˙" ˆżA	bpt, x, aqdq A pt, xqpaq,	ż

pt, xqq ă `8 one can find a measurable function q A : r0, T s Rd Ñ P 1 pAq such that, for every pt, xq P r0, T s ˆRd one has

A f 1 pt, x, aqdq A pt, xqpaq, A σ t σpt, x, aqdq A pt, xqpaq ˙.

  2.1, we know that mptqbdt-almost-everywhere : Λ ´Id ď dW dt b dm ď Λ `Id . Let pΩ, F, F, P, pX, Bqq be a weak solution to the SDE s ă t ď T , with M r 2 and C r 2 positive constants depending only on r 2 we have

	dX t "	dω dt b dm	pt, X t qdt `c2	dW dt b dm	pt, X t qdB t
	with LpX t q " mptq for all t P r0, T s. The existence of such a solution is ensured by the fact
	that m solves the FPE with coefficients all 0 ď	dω dt b dm	,	dW dt b dm	(see Proposition 2.1). Now, for

  [START_REF] Achdou | Mean Field Type Control with Congestion[END_REF] we can proceed similarly and prove that L 1 ppλ, φq, mq ď lim inf L 1 ppλ, φq, m k q and L 3 ppλ, φq, nq ď lim inf

	kÑ`8	kÑ`8

L 3 ppλ, φq, n k q, for every pλ, φq P R d ˆA1 . Let us detail this point for L 3 since the same argument works for L 1 . If pλ, φq belongs to R `ˆA 1 then lim inf kÑ`8

  Dϕ ˚p., .q, D 2 ϕ ˚p., .q ˘´f 1 2 p., .q} 8 `1and C 2 " ´||g 1 || 8 `C1 ´1. Then we let empty and that there exists some M ą 0 such that any pm, ω, W, nq P B ˚satisfies Estimate (2.11). We deduce that B ˚is (strongly) bounded and using (2.12) we see that B ˚is weakly-˚compact. Now we can use (2.11) and (2.12) once again to show that for all C ą 0 and all pλ, φq P R `ˆA 1 , HJ ´pλh `g1 q for some pλ, φq P R `ˆA 1 if and only if " ´Bt φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqq ď f 1 2 pt, xq in r0, T s ˆRd , φpT, xq ď λhpxq `g1 pxq in R d .

	2.2. A RELAXED PROBLEM: OPTIMAL CONTROL OF THE FOKKER-PLANCK EQUATION57
	and	inf nPM `pR d q	L 3 ppλ, φq, nq "	"	0 ´8 otherwise. if φpT, xq ď λhpxq `g1 pxq in ˆRd ,
	so we can conclude that
			C ˚:"	sup	inf	Lpλ, φq, pm, ω, W, nqq	`1
					pλ,φqPR `ˆA 1	pm,ω,W,nqPB
	pm,ω,W,nqPB and we check that pλ,φqPR `ˆA 1 inf sup	Lppm, ω, W, nq, pλ, φqq "	pλ,φqPR `ˆA 1 sup	pm,ω,W,nqPB inf	Lppm, ω, W, nq, pλ, φqq
						ż
	where φ P Finally, we get min B Therefore it holds that pm,ω,W qPK		"	sup pλ,φqPR `ˆA 1 ,φPHJ ´pλh`g 1 q	R d	φp0, xqm 0 pdxq,
	3: Min/Max argument Now we are going to use the Von Neumann Theorem 2.6 to show that inf pm,ω,W,nqPB Lppλ, φq, pm, ω, W, nqq " inf mPM `pr0,T sˆR d q L 1 ppλ, φq, mq `inf nPM `pR d q	L 3 ppλ, φq, nq
						`żR d	φp0, xqdm 0 pxq
	inf pm,ω,W,nqPB but we have pλ,φqPR `ˆA 1 sup	Lppφ, λq, pm, ω, W, nqq "	sup
	inf				´8 otherwise

pλ,φqPR `ˆA 1 inf pm,ω,W,nqPB Lppλ, φq, pm, ω, W, nqq.

To check that the hypothesis of the theorem are satisfied, we define ϕ ˚pt, xq :" a 1 `|x| 2 pt T ´1q and φ ˚pt, xq :" ´a1 `|x| 2 `C1 ¯pt ´T ´1q `C2 , where C 1 " }H `., ., ˚:" tpm, n, ω, W q P B such that Lpp0, φ ˚q, pm, ω, W, nqq ď C ˚u is not B ˚X tpm, n, ω, W q P B such that Lppλ, φq, pm, ω, W, nqq ď Cu is (possibly empty and) compact. Therefore we can apply the Von Neumann theorem, Theorem 2.6 to show that inf pm,ω,W,nqPB sup pλ,φqPR `ˆA 1 Lppφ, λq, pm, ω, W, nqq " sup pλ,φqPR `ˆA 1 inf pm,ω,W,nqPB Lppλ, φq, pm, ω, W, nqq. Step 4: Computation of the dual problem Let pλ, φq P R `ˆA 1 be fixed and consider the problem inf pm,ω,W,nqPB Lppλ, φq, pm, ω, W, nqq. Recall the definitions of L 1 , L 2 and L 3 in Step 2 of the proof and observe first that, for fixed pm, nq, inf pω,W q L 2 ppλ, φq, pm, ω, W qq " 0 with the infimum being achieved if and only if, " ω " ´Bp Hpt, x, Dφpt, xq, D 2 φpt, xqqm, W " ´BM Hpt, x, Dφpt, xq, D 2 φpt, xqqm. mPM `pr0,T sˆR d q L 1 ppλ, φq, mq " " 0 if ´Bt φ `Hpt, x, Dφ, D 2 φq ď f 1 2 pt, xq in r0, T s ˆRd ,

  58CHAPTER 2. STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW Using the inequations satisfied by φ and the definition of L we get after reorganizing the terms

		λ	ˆ´ż R d	hpxqdmpT qpxq ˙ď J RP pm, ω, W q	´żR d	φp0, xqdm 0 pxq.	(2.13)
	Consider pλ, φq a candidate for the dual problem. Since pm, ω, W q satisfies the Fokker-Planck
	equation we have, taking φ as a test function
	ż			ż
	φpT, xqdmpT qpxq "	φp0, xqdm 0 pxq
	R d			R d
	`ż T	ż
	0	R d

d hpxqdmpT qpxq ă 0. " B t φpt, xq `dω dm b dt pt, xq.Dφpt, xq `dW dm b dt pt, xq.D 2 φpt, xq  dmptqpxqdt.

  Proof of Corollary 2.1. Let pφ, λq P A and pm, ω, W q P K points where the primal and the dual problems are achieved. One has ż R d φp0, xqdm 0 pxq " J RP pm, ω, W q. Given the constraint on φ and the fact that m is non-negative we get Yet, pm, ω, W q solves the Fokker-Planck equation and φpT, xq ď λhpxq `g1 pxq for all x P R d so

	2.2. A RELAXED PROBLEM: OPTIMAL CONTROL OF THE FOKKER-PLANCK EQUATION59
				ż	φp0, xqdm 0 pxq	´ż T	ż	p´B t φpt, xq `Hpt, x, Dφpt, xq, D 2 φpt, xqqqdmptqpxqdt
				R d						0	R d
										ě	ż T 0	ż R d	" L ˆt, x,	dω dt b dm	pt, xq,	dW dt b dm	pt, xq ˙ dmptqpxqdt
										`żR d	g 1 pxqdmpT qpxq.
	λ	ż	hpxqdmpT qpxq	´ż T	ż	Dφpt, xq.dωpt, xq	´ż T	ż	D 2 φpt, xq.dW pt, xq
		R d							0	R d	0	R d
	ě	ż T 0	ż R d	" L ˆt, x,	dω dt b dm	pt, xq,	dW dt b dm
							ż
	Remember that	hpxqdmpT qpxq ď 0 and λ ě 0 so
							R d
				ż T 0	ż R d	rL ˆt, x,	dω dt b dm	pt, xq,	dW dt b dm	pt, xq ˙`Hpt, x, Dφpt, xq, D 2 φpt, xqq
				´Dφpt, xq.	dω dt b dm	pt, xq ´D2 φpt, xq.	dW dt b dm	pt, xqqsdmptqpxqdt
						ż
				ď λ	hpxqdmpT qpxq ď 0.
						R d
					´Dφpt, xq.	dω dt b dm	pt, xq ´D2 φpt, xq.	xqq r mptq b dt dW dt b dm pt, xq
	and p r λ, r φ, r mq satisfies the optimality conditions " L ˆt, x, dω dt b dm pt, xq, dW dt b dm pt, xq ˙`Hpt, x, Dφpt, xq, D 2 φpt, xqq
	$ ' and since H is differentiable, mptq b dt-ae it holds ´Bt r φpt, xq `Hpt, x, D r φpt, xq, D 2 r φpt, xqq " f 1 2 pt, xq ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' % B t r m ´divpB p Hpt, x, D r φpt, xq, D 2 r φpt, xqq r mq `ÿ i,j B 2 ij ppB M Hpt, x, D r φpt, xq, D 2 r λ ż hpxqd r mpT qpxq " 0, ż hpxqd r mpT qpxq ď 0, r λ ě 0. $ ' & ' dω dt b dm pt, xq " ´Bp Hpt, x, Dφpt, xq, D 2 φpt, xqq in r0, T s ˆRd dW φpt, xqqq r % dt b dm	(2.15)
				R d					R d

ij r mq " 0 in r0, T s ˆRd r φpT, xq " r λhpxq `g1 pxq in R d , r mp0q " m 0 pt, xq ˙`Hpt, x, Dφpt, xq, D 2 φpt, xqq  dmptqpxqdt. But, by definition of L, the integrand is always nonnegative. So, mptq b dt-ae we have pt, xq " ´BM Hpt, x, Dφpt, xq, D 2 φpt, xqq.

Finally, since all the inequalities at the beginning of this proof are actually equalities, we get the necessary conditions for optimality. 60CHAPTER 2. STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW 2.3 Proof of the Main Results

  particular, x ‰ y. Since t ‰ T for β ď β 0 , we can apply the maximum principle for semi-continuous functions from[START_REF] Michael | User's guide to viscosity solutions of second order partial differential equations[END_REF]. Let ϕ β pt, x, yq " K|x ´y| `β 1 t . Computing the various derivatives for |x ´y| ą 0 gives B t ϕ β pt, x, yq " ´β t 2 B x ϕ β pt, x, yq " K |x´y| px ´yq B y ϕ β pt, x, yq " K |x´y| py ´xq D 2 ϕ β pt, x, yq " K

	and, in $
	'
	'
	'
	'
	&
	' ' ' '		ˆId ´Id	˙´K	ˆpx ´yq b px ´yq ´px ´yq b px ´yq ˙.
	%	|x´y|	´Id I d	|x´y| 3	´px ´yq b px ´yq px ´yq b px ´yq
	In particular, if N :" px ´yq b px ´yq, then N ě 0 (rank one symetric matrice with positive
	trace) and thus it is elementary to show that	ˆN ´N N ´N	˙ě 0. Therefore, it holds
				D 2 ϕ β pt, x, yq ď	|x ´y| K	´Id I d ˆId ´Id	˙.
			$ &	´ν, K x´y |x´y| , X ¯P P	2,`u pt, xq
			%	´ν `β t 2 , K x´y |x´y| ´2βy, Y ¯P P	2,´u pt, yq
	and
		ˆX 0 ´pY `2βI d q 0 ˙ď 3	K |x ´y|	ˆId ´Id ´Id I d
	H ˆt, x, K	x |x ´y| ´y	, X ˙´f 1 2 pt, xq ď ν ď H ˆt, y, K	x |x ´y| ´y	´2βy, Y ˙´f 1 2 pt, yq.
					|x ´y| ě	δ 2K	,	(2.24)

xq ´upt, yq ´K|x ´y| ď 2K|x ´y| ´K|x ´y| ď K|x ´y|, which leads to the lower bound Now, from the maximum principle, we get ν P R, X, Y P S d pRq such that ˙. Observe that |ν| is bounded by L T the time -Lipschitz constant of u and thus |ν| is bounded independently of K. Now we use the equation satisfied by u to get

  1 2 pt, yq ě ν ´Cβ, where C " 2Λ `?d. Thus, l being C 1 , there exists λ P r0, 1s such that From inequality (2.26) we are going to obtain a lower bound on |Z λ | " a TrpZ 2 λ q and from inequality (2.27) we are going to obtain an upper bound on |Z λ |. Combining the two bounds will get a contradiction for K large enough. First we exploit (2.26). It gives us

	lpλq " ν ´Cλβ,	(2.26)
	l 1 pλq ě ´Cβ.	(2.27)

  1 2 e µt |Du n | 2 for some µ ą 0 and we get, after differentiating the HJB equation and taking scalar product with e µt Du n : ´Bt w n pt, xq `Bp Hpt, x, Du n pt, xq, D 2 u n pt, xqq.Dw n pt, xq `BM Hpt, x, Du n pt, xq, D 2 u n pt, xqq.D 2 w n pt, xq " Df n pt, xq.Du n pt, xqe µt ´Bx Hpt, x, Du n pt, xq, D 2 u n pt, xqq.Du n pt, xqe µt`eµt B M Hpt, x, Du n pt, xq, D 2 u n pt, xqq.pD 2 u n pt, xqq 2 ´1 2 µe µt |Du n pt, xq| 2 ´|D 2 u n pt, xq| 2 ´1 2 µe µt |Du n pt, xq| 2 ,where we used the growth assumption (2(f)ii) on B x H, Assumption (2e) and the uniform ellipticity of H. Now we can choose µ " µp}Df } 8 , C BxH , Λ ´q ą 0 such that the right-hand side of the above expression is bounded by above and, by the maximum principle for parabolic equations we get that }Du n } 8 ď C for some C " Cp}Dg 1 } 8 , }Df 1 2 } 8 , C DxH , Λ ´q ą 0. Now we let v n :" B t u n . By differentiating the HJB equation with respect to time we get that v n solves ´Bt v n pt, xq `Bp Hpt, x, Du n pt, xq, D 2 u n pt, xqq.Dv n pt, xq `BM Hpt, x, Du n pt, xq, D 2 u n pt, xqq.D 2 v n pt, xq " ´Bt Hpt, x, Du n pt, xq, D 2 u n pt, xqq `Bt f 1pnq 2 pt, xq v n pT, xq " HpT, x, Dg 1 pxq, D 2 g 1 pxqq ´f 1pnq 2 pT, xq.

	$
	&
	%

ď Df 1pnq 2 pt, xq.Du n p, xqe µt `CBxH p1 `|Du n pt, xq| `|D 2 u n pt, xq|qe µt |Du n pt, xq| ´eµt Λ

  1. (Comparison). Let u be a bounded continuous viscosity sub-solution to (2.30) and v be a bounded continuous viscosity super-solution to (2.30) such that, x Ñ upT, xq is uniformly continuous and such that, for all x P R d : upT, xq ď vpT, xq. Then for all pt, xq P r0, T s ˆRd , upt, xq ď vpt, xq.2. (Existence of solutions). Assume further that g 1 belongs to C 2 b pR d q. Then, for all K ě 2}Dg 1 } 8 there exists a unique bounded, continuous viscosity solution u to (2.30) such that upT, xq " g 1 pxq over R d .3. (Regularity). Assume as well that f 12 is Lipschitz continuous in the time variable. Then, there is L ą 0 (independent from K) such that |upt, xq ´ups, yq| ď L|t ´s| `K|x ´y|, for all pt, xq, ps, yq in r0, T s ˆRd .

Proof. (Comparison) Let ą 0 be a fixed parameter, and define u pt, xq " upt, xq´ pT ´tq. We are going to show that u pt, xq ď vpt, xq for all pt, xq P r0, T sˆR d . Suppose on the contrary that δ :" sup pt,xqPr0,T sˆR d u pt, xq ´vpt, xq ą 0. Let α ą 0 and β ą 0 be small parameters and φ α,β pt, x, yq :" u pt, xq ´p1 `α2 qvpt, yq ´|x ´y| 2 α ´β|y| 2 ´β t .

  Recalling the definition of H and the Lipschitz regularity of b and f 1 with respect to the space variable we get Now we use estimate (2.31) and the Lipschitz regularity of σ with respect to the space variable to deduce that, for all a P A, σ t σpt, x, aq.X ´σt σpt, y, aq.Y " Tr ˆσt σpt, x, aq σpt, x, aq t σpt, y, aq σ t σpt, y, aq σpt, y, aq t σpt, x, aq

	72CHAPTER 2. STOCHASTIC CONTROL WITH TERMINAL CONSTRAINT IN LAW
	Now we use the inequations satisfied by u and v to get
	$ ' & ' % Notice in particular that maxt´ν ` `Hpt, x, H ˆt, y, 2 αµ px ´yq 2 α ´2 β px ´yq, Xq ´f 1 2 pt, xq, µ y, Y µ ´2β µ I d ˙´Hpt, y, 2 α |x ´y| ´Ku ď 0 2 px ´yq, Y q α maxt´ν µ ´β µt 2 `Hpt, y, 2 αµ px ´yq ´2β µ yq, Y µ ´2β µ I d q ´f 1 2 pt, yq, | " H ˆt, y, 2 αµ px ´yq ´2β µ y, Y µ ´2β µ I d ˙´H ˆt, y, 2 α px ´yq, 2 αµ ˙ˆX 0 px ´yq 0 ´Y ď ´2 β y| ´Ku ě 0. Ḣ Y µ ´2β I d µ µ (2.32) 2 α 6 α Tr ˆσt σpt, x, aq σpt, x, aq t σpt, y, aq σ t σpt, y, aq σpt, y, aq t σpt, x, aq ˙ˆI d ´Id ´Id I d ď ˆt, y, 2 α px ´yq, Y µ ´2β µ I d ˙´Hpt, y, 2 px ´yq, Y q α |x´y| ď K. Since µ ą 1, we get for β sufficiently small (depending on α) that | 2 αµ px ´yq ´2 β µ y| ď K and therefore we deduce from (2.32) the two inequalities # H `t, x, 2 α px ´yq, X ˘´f 1 2 pt, xq ď ν ´ H ´t, y, 2 αµ px ´yq ´2β µ y, Y µ ´2β µ I d ¯´f 1 2 pt, yq ě ν µ . 6 α |σpt, x, aq ´σpt, y, aq| 2 ď ď C K ˆˇˇˇ2 αµ px ´yq ´2β µ y ´2 α px ´yq ˇˇˇ`ˇˇˇY µ ´2β I d ´Y ˇˇˇď µ C |x ´y| 2 , ď C K `α|x ´y| `β|y| `β `α2 |Y | α C K pα|x ´y| `β|y| `β `αq (2.36) and therefore, for some constant C ą 0, it holds Finally, using the boundness and the Lipschitz continuity with respect to x of f 1 2 we obtain Multiplying the second inequality by µ and substracting the two inequalities leads to ď µH ˆt, y, 2 αµ px ´yq ´2 β µ y, Y µ ´2β µ I d ˙´H ˆt, x, 2 α px ´yq, X μf Hpt, y, 2 α px ´yq, Y q ´Hpt, x, 2 α px ´yq, Xq ď C ˆ2 α from (2.33), (2.34), (2.35) and (2.36) |x ´y| 2 `|x ´y| ˙. (2.34) It remains to estimate ď C K ˆ1 α |x ´y| 2 `|x ´y| `α2 `α `α|x ´y| `β|y| `β˙.
	1 2 pt, yq `f 1 2 pt, xq.			(2.33)
	Recalling that µ " 1 `α2 we rewrite (2.33) as follows µH ˆt, y, 2 αµ px ´yq ´2 β µ y, Y µ ´2β µ I d ˙´Hpt, y,	2 α	px ´yq, Y q
	ď Hpt, y, " α 2 H ˆt, y, 2 α px ´yq, Y q ´Hpt, x, 2 αµ px ´yq ´2 β µ y, 2 Y px ´yq, Xq `f 1 2 pt, xq ´f 1 Ḣ µ ´2β µ I d 2 pt, yq α `µH ˆt, y, 2 αµ px ´yq ´2β µ y, Y µ ´2β µ I d ˙´Hpt, y, 2 α px ´yq, Y q ˆt, y, 2 αµ px ´yq ´2β µ y, Y µ ´2β µ I d ˙´Hpt, y, 2 α px ´yq, Y q.
	`f 1 2 pt, yq ´µf 1 2 pt, yq	2,`u	pt, xq
	pν `β t 2 , 2 α px ´yq, Y q P P	2,´p p1 `α2 qvpt, yq `β|y| 2 q
	and
	´6 α Yet we have the equivalence ˆν, 2 α px ´yq, X ˙P P I 2d ď 2,`u ˆX 0 0 ´Y ˙ď 6 α pt, xq ô ˆν ´ , ˆId ´Id ´Id I d 2 α px ´yq, X ˙P P ˙. and, letting µ :" 1 `α2 , it holds as well ˆν `β t 2 , 2 α px ´yq, Y ˙P P 2,´`µ vpt, yq `β|y| 2 ô (2.31) 2,`u pt, xq ˆν µ `β µt 2 , 2 αµ px ´yq ´2 β µ y, Y µ ´2β µ I d ˙P P 2,´v pt, yq. Hpt, y, 2 α px ´yq, Y q ´Hpt, x, 2 α H ˆt, y, 2 αµ px ´yq ´2 β µ y, Y µ ´2β µ I d ˙ď H ˆt, y, 2 αµ px ´yq ´2β µ y, 0 ˙`C ˇˇˇY µ ´2β I d ˇˇď µ px ´yq, Xq " sup aPA " ´bpt, y, aq. * C K p1 `|Y |q , 2 α px ´yq ´σt σpt, y, aq.Y ´f1 pt, y, aq ´sup aPA " ´bpt, x, aq. * where we use the fact that ˇˇˇ2 αµ px ´yq ´2β y ˇˇˇď K which comes from (2.32). To estimate µ 2 α px ´yq ´σt σpt, x, aq.X ´f1 pt, x, aq ď sup aPA " `bpt, x, aq ´bpt, y, aq ˘. 2 px ´yq * `sup f 1 pt, x, aq ´f1 pt, y, aq ( |Y | we use (both sides of) the matrix inequality (2.31) to deduce that ´6 α I d ď Y ď 6 I d and α therefore ~Y ~ď 6 α . As a consequence we get α aPA `sup α 2 H ˆt, y, 2 αµ px ´yq ´2 β µ y, Y µ ´2β µ I we find

aPA σ t σpt, x, aq.X ´σt σpt, y, aq.Y ( ď C ˆ2 α |x ´y| 2 `|x ´y| ˙`sup aPA σ t σpt, x, aq.X ´σt σpt, y, aq.Y ( . On the one hand, using the boundness of B M H (see Remark (2.4)) and the growth condition (2b) on Hpt, x, p, 0q, we get the following, where C K ą 0 depends from now on K ą 0 but not on α and β and may change from line to line, d ˙ď C K pα 2 `αq. (2.35) On the other hand, using again the boundness of B M H as well as Assumption (2d) on B p H

  1 pxq ´Cg 1 pT ´sq and g 1 pxq `Cg 1 pT ´sq are respectively viscosity super-solution and subsolution to(2.37). By comparison we have that |upT ´s, xq ´g1 pxq| ď C g 1 s for all s P r0, T s. If we fix s P r0, T s and define for all pt, xq P rs, T s ˆRd , vpt, xq " upt ´s, xq it is plain to check that v `pt, xq :" vpt, xq ´C1 st and v ´pt, xq :" vpt, xq `C1 st are respectively sub and super-solutions to (2.37) over rs, T s ˆRd , where C 1 is such that |Hpt ´s, x, p, M q ´Hpt, x, p, M q| `|f1 2 pt ´s, xq ´f 1 2 pt, xq| ď C 1 s for all s P r0, T s, all t P rs, T s and all px, p, M q P R d ˆRd ˆSd pRq. By comparison we find that for all s P r0, T s, all t P rs, T s and all x P R d , upt, xq ´v`p t, xq ď sup

		upT, xq ´v`p T, xq
		xPR d
	upT, xq " g 1 pxq	2 uq ´f 1 2 pt, xq, |Du| ´Ku " 0 in p0, T q ˆRd , in R d	(2.37)

(Regularity) Being u a solution to (2.37), it is straightforward that u is K-Lipschitz continuous with respect to the space variable. We now prove the time regularity. As previously observed, for C g 1 ě sup pt,xq |Hpt, x, Dg 1 pt, xq, D 2 g 1 pt, xqq ´f 1 2 pt, xq|, g

  thesis), that solutions pm, αq of Problem (uP) exist and satisfy αpt, xq " ´Dp Hpx, Dupt, xqq with pm, uq solution to the Mean-Field Game (MFG) system of partial differential equations

	For convenience we put all the assumptions concerning H into
				Assume AH2, AH3, AH4, AH5, AH6.	(AH)
	Notice that, taking convex conjugates, we see that L satisfies a coercivity condition
	similar to Assumption (AH2): for all px, qq P R d ˆRd ,
				1 4C 0	|q| 2 ´C0 ď Lpx, qq ď	C 0 4	|q| 2 `C0 ,
	and the first term in the total cost J looks very much like a kinetic energy.
	A typical example of functions satisfying the condition (Ureg) is the class of cylindrical
	functions of the form
			Fpmq " F	ˆżR d	f 1 pxqdmpxq, . . . ,	R d ż	f k pxqdmpxq ˙,
	$ ' ' ' &	´Bt upt, xq `Hpx, Dupt, xqq ´∆upt, xq " B t m ´divpD p Hpx, Dupt, xqqmq ´∆m " 0 δF δm	pmptq, xq in p0, T q ˆRd , in p0, T q ˆRd ,
	' ' ' %	upT, xq "	δG δm	pmpT q, xq in R d ,	mp0q " m 0 ,
						)
		"	H and its derivatives are bounded on sets of the form R d ˆBp0, Rq for all R ą 0.	(AH4)
			For all px, pq P R d ˆRd , |D x Hpx, pq| ď C 0 p1 `|p|q.	(AH5)
		$
		&
						(AH6)

%

There is some µ ą 0 such that, for all px, pq P R d ˆRd ,

1 µ I d ď D pp Hpx, pq ď µI d .

where F and the f i , 1 ď i ď k are smooth with bounded derivatives. Assumption (Ureg) also implies that pm, xq Ñ D m Upm, xq is uniformly bounded in P 2 pR d q ˆRd and therefore, a simple application of Kantorovitch-Rubinstein duality for d 1 proves that U is Lipschitz continuous with respect to this distance. Under the above assumptions on F, G and H it is well-known (see

[START_REF] Briani | Stable solutions in potential mean field game systems[END_REF]

or Chapter 2 of this

  .[START_REF] Bogachev | Fokker-Planck-Kolmogorov Equations[END_REF]) ωq satisfying(3.14). Once again, we set J l ,δ pm, ωq " `8 if ω is not absolutely continuous with respect to dt b mptq.Proof. To avoid uniqueness issues we modify the cost function J ,δ so that the new problem reads

		J l ,δ pω, mq :"	ż T 0	ż R d	Lpx,	dω dt b dmptq	pt, xqqdmptqpxqdt
			`ż T 0	ż R d	"	λptq δΨ δm	p r mptq, xq	`δF δm	 mptq, xq p r	dmptqpxqdt
			`żR d	"	β δ	δΨ δm	p r mpT q, xq	`δG δm	p r mpT q, xq,		dmpT qpxq
	over the pairs pm, inf	" J ,δ pm, ωq	`ż T	 mptqqdt qpmptq, r	.	(3.19)
										0
	Lemma 3.2. Let p r m, r ωq be a fixed solution to Problem (P ,δ ). Then there exist λ P L 8 pr0, T sq
	and β P R `satisfying
	λptq "	$ & %	0 λptq P r0, 1s if Ψp r if Ψp r mptqq ă 0, mptqq " 0, 1 if Ψp r (3.17) mptqq ą 0,	β	$ & %	" 0 " 1 P r0, 1s if Ψp r if Ψp r mpT qq ă 0, if Ψp r mpT qq ą 0, mpT qq " 0,	(3.18)
	such that p r m, r ωq minimizes

  Remark 2.3 shows that the integration by part formula (3.3) holds for u m and therefore, we get

	ż
	R d	u m pt 2 , xqdmpt 2 qpxq	´żR d	u m pt 1 , xqdmpt 1 qpxq
			ż t 2	ż
			" rB " t 1 R d ż t 2 ż rαpt, xq.Du
			t 1	R d

t u m pt, xq `αpt, xq.Du m pt, xq `∆u m pt, xqs dmptqpxqdt m pt, xq `Hpx, Du m pt, xqq ´fm pt, xqs dmptqpxqdt

  [START_REF] Achdou | Mean field games and applications: Numerical aspects[END_REF]. We consider r u P Cpr0, T s, E n q solution to

	$ '	´Bt r upt, xq `Hpx, Dr upt, xqq ´∆r upt, xq
	' ' & ' ' ' %	r upT, xq "	β δ	δm δΨ	" p r mpT q, xq λptq δΨ δm `δG p r mptq, xq δm p r mpT q, xq `δF δm	in R d , mptq, xq p r	in p0, T q ˆRd ,	(3.23)
	-the existence of such a solution is guaranteed by Theorem 3.1-and we proceed by veri-
	fication. We use Lemma 3.3 to get
	ż R d	r up0, xqdm 0 pxq "	´ż T 0	ż R d	" Hpx, Dr upt, xqq	`dr ω dt b d r m	pt, xq.Dr upt, xq		d r mptqdt.

  belongs to P 2 pR d q for all t P r0, T s . On the other hand, by convexity of Ψ we get for all t P r0, T s,

	we get
		´ż T 0	ż R d	λptq δΨ δm	pmptq, xqdmptqpxqdt	´żR d	β δ	δΨ δm	pmpT q, xqdmpT qpxq
									ż T	ż		"	
							ď	0	R d	Lpx, αpt, xqq	`δF δm	pmptq, xq	dmptqpxqdt
							`żR d	δG δm	pmpT q, xqdmpT qpxq	´żR d	up0, xqdm 0 pxq.	(3.26)
	On the one hand -using (3.24) in the proof of Theorem 3.5 and the notations therein-
		ż
	we have that to p r m, r ωq and therefore J ,δ up0, xqdm 0 pxq " J ,δ l p r m, r ωq. But the linearized costs cancel when applied R d m, r ωq " Jp r m, r ωq. And since L, F and G are bounded from l p r ż
	below we get a lower bound on up0, xqdm ż R d R d δΨ δm pmptq, xqdmptqpxq ď Ψpmptqq ´Ψpmptqq
												ď ´θ ´Ψpmptqq
		xq.Dupt, xq `Hpx, Dupt, xqq	´λptq δΨ δm	pmptq, xq	´δF δm	pmptq, xq		dmptqpxqdt
	"	ż R d	"	β δ	δΨ δm	pmpT q, xq	`δG δm	 pmpT q, xq	dmpT qpxq	´żR d	up0, xqdm 0 pxq.

Now, reorganizing the terms and using the fact that, by definition of L, we have for all pt, xq in r0, T s ˆRd αpt, xq.Dupt, xq `Hpx, Dupt, xqq ě ´Lpx, αpt, xqq, 0 pxq independent of and δ. The other terms in the right-hand side of (3.26) are also bounded from above since Jpα, mq ă `8 and since x Þ Ñ δF δm pm, xq and x Þ Ñ δG δm pm, xq are bounded in E n with bounds uniform in m and mptq

  Being u in Cpr0, T s, E n q and m in Cpr0, T s, P 2 pR d qq we get that t Þ Ñ Ψpmptqq is C 1 with Ψpmptq, xq.D p Hpx, Dupt, xqqdmptqpxq `żR d div x D m Ψpmptq, xqdmptqpxq. Now we assume that Ψpmptqq ‰ 0. We denote by lpt, xq the integrand lpt, xq :" ´Dm Ψpmptq, xq.D p Hpx, Dupt, xqq `div x D m Ψpmptq, xq

	d dt	Ψpmptqq "	´żR d	D m

xq.D p Hpx, Dups, xqqdmpsqpxqds `ż t 0 ż R d div x D m Ψpmpsq, xqdmpsqpxqds.

  APsiC2) so we can apply Proposition 3.3 to D m Ψpmptq, xq and div x D m Ψpmptq, xq and deduce that l belongs to C 1,2b ppt 1 , t 2 q ˆRd q and therefore t Þ Ñ d dt Ψpmptqq is differentiable at t with d 2 dt 2 Ψpmptqq " rB t lpt, xq ´Dp Hpx, Dupt, xqq.Dlpt, xq `∆lpt, xqs dmptqpxq. Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Ψpmptq, xqdmptqpxq

	ż R d Computing B t l leads to B t lpt, xq " ´d dt ´Dm Ψpmptq, xq.D 2 D m Ψpmptq, xq.D p Hpx, Dupt, xqq pp Hpx, Dupt, xqqB t Dupt, xq `d dt div x D m Ψpmptq, xq " ´d dt D m Ψpmptq, xq.D p Hpx, Dupt, xqq `d dt divD m Ψpmptq, xq ´Dm Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD 2 upt, xqD p Hpx, Dupt, xqq ´Dm Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD x Hpx, Dupt, xqq `Dm Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD∆upt, xq `λptq D and therefore d 2 dt 2 Ψpmptqq " λptq ż R d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Ψpmptq, xqdmptqpxq `F pDuptq, D 2 uptq, D∆uptq, mptqq with F pDuptq,D 2 uptq, D∆uptq, mptqq " ż R d r´D p Hpx, Dupt, xqq.Dlpt, xq `∆lpt, xqs dmptqpxq ´żR d d dt D d 2 R d dt 2 Ψpmptqq " ż λptq D m

t upt, xq " Hpx, Dupt, xqq ´∆upt, xq ´λptq δΨ δm pmptq, xq ´δF δm pmptq, xq and u belongs to Cpr0, T s, E n q with n ě 3. This means that B t u is differentiable with respect to x with ´Bt Dupt, xq `Dx Hpx, Dupt, xqq `D2 upt, xqD p Hpx, Dupt, xqq ´D∆upt, xq " λptq D m Ψpmptq, xq `Dm Fpmptq, xq. But m solves the Fokker-Planck equation, Ψ satisfies Assumptions (Ureg) and (m Ψpmptq, xqD 2 pp Hpx, Dupt, xqq.D m Ψpmptq, xq `Dm Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Fpmptq, xq, m Ψpmptq, xq.D p Hpx, Dupt, xqqdmptqpxq `żR d d dt div x D m Ψpmptq, xqdmptqpxq ´żR d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD 2 upt, xqD p Hpx, Dupt, xqqdmptqpxq ´żR d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD x Hpx, Dupt, xqqdmptqpxq `żR d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD∆upt, xqdmptqpxq `żR d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Fpmptq, xqdmptqpxq. Remark 3.7. An explicit formula for Dl, ∆l or F is not necessary for our purpose however a tedious but straightforward computation leads to

  xq.D p Hpx, Dupt, xqqdmptqpxq `żR d D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD m Fpmptq, xqdmptqpxq Ψpmptq, x, yqD p Hpx, Dupt, xq.D p Hpy, Dupt, yqqdmptqpxqdmptqpyq `żR d D x D m Ψpmptq, xqD p Hpx, Dupt, xqq.D p Hpx, Dupt, xqqdmptqpxq xqD 2 upt, xq.D 2 upt, xqD 2 pp B p i Hpx, Dupt, xqqdmptqpxq

		ż
	`żR d mm ´2 ż R d D 2
	R i"1 ż R d pmptq, ´żR d D x i δΨ δm D m Ψpmptq, xq. Ý Ñ ∆ x D p Hpx, Dupt, xqqdmptqpxq
	´2 ż R d	D x D m Ψpmptq, xqD 2 xp Hpx, Dupt, xqqdmptqpxq
	´żR d	D m Ψpmptq, xq.D 2 pp Hpx, Dupt, xqqD x Hpx, Dupt, xqqdmptqpxq
	`żR d	D 2 xp Hpx, Dupt, xqqD

d D x D m Ψpmptq, xq.D 2 upt, xqD 2 pp Hpx, Dupt, xqqdmptqpxq ´n ÿ m Ψpmptq, xq.D p Hpx, Dupt, xqqdmptqpxq

  Cpsup pt,xqPr0,T sˆR d q |Dupt, xq|q ą 0. Above with use the fact that sup xPR d |P t gpxq| ď sup xPR d |gpxq| for a bounded function g and sup xPR d Now we proceed with higher order derivatives and we argue by induction. Take k ě 2 and assume that we have shown that Using the inequality sup xPR d |DP t gpxq| ď C ? t sup xPR d |gpxq| we get |D k upt, xq| ď |P T ´tD k gpxq|

	|upt, xq| ď |P T ´tgpxq|	`ż T	|P s´t f psqpxq|ds	`ż T	|P s´t rHp., Dups, .qqs pxq|ds
				t			t
	ď 2 ? T }g} 0 p1 `|x|q	`2? T	ż T	}f psq} 0 p1 `|x|qds `Cp1 `sup	|Dups, xq|q
							t	pt,xqPr0,T sˆR d
	for some C " |Ptgpxq| 1`|x| ď 2 ? T sup xPR d	|gpxq| 1`|x| for a function
	g with linear growth. Since u is assumed to satisfy the Lipschitz estimate of the previous
	lemma 3.8, it holds that
							ż T
				sup	}uptq} 1 ď Cp	}f ptq} 1 dt, }gptq} 1 q.
				tPr0,T s	0
							ż T
			sup	}uptq} k´1 ď Cp	}f ptq} k´1 dt, }gptq} k´1 q.
			tPr0,T s			0
				`ż T	|P s´t D k f psqpxq|ds	`ż T	|DP s´t D k´1 rHp., Dups, .qqs pxq|ds
						t	t
	ď }g} k	`ż T t	}f psq} k ds	`C ż T t	sup xPR d |D k´1 rHpx, Dups, xqqs | ? s ´t	ds.

  Hpx, Du 1 ps, xqq ´Hpx, Du 2 ps, xqq " pDu 1 ps, xq ´Du 2 ps, xqq.

	ż 1
	D
	0

p Hpx, rDu 1 ps, xq `p1 ´rqDu 2 ps, xqqdr and deduce that, for all k ě 1, sup xPR d |D k´1 rHpx, Du 1 ps, xqq ´Hpx, Du 2 ps, xqqs | ď C}u 1 psq ´u2 psq} k

h

  DP s´t´h D k´1 Hp., Dups, .qqpxqds ´ż T

	`Cpessup tPr0,T s }uptq} k q ? h		? s	´t	ds
	ď Cpessup tPr0,T s }uptq} k qp	? h	`ż T ´h t	}ups `hq ´upsq} k ? s ´t	dsq.

t DP s´t D k´1 Hp., Dups, .qqpxqds| " | ż T ´h t DP s´t D k´1 rHp., Dups `h, .q ´Hp., Dups, .qqs pxqds ´ż T T ´h DP s´t D k´1 rHp., Dups, .qqs pxqds| ď Cpessup tPr0,T s }uptq} k q ż T ´h t sup xPR d |D k ups `h, xq ´Dk ups, xq| Using again Grönwall Lemma 3.7, we get, for all t P r0, T s, }upt `hq ´uptq} n ď Cpessup tPr0,T s }uptq} n qph α{2 }g} n`α `ż T ´h 0 }f ps `hq ´f psq} n ds `?h sup tPr0,T s }f ptq} n´1 q.

  P t´s rHp., Du m ps, .qqs pxqds to conclude that u is a solution.The uniqueness of solutions is a straightforward consequence of the stability estimate of the previous lemma. 3.6.2 Proof of Theorem 3.6. . Proof of Theorem 3.6 We can remark that u is a solution (3.27) if and only if v :" u´z is a solution to

		xq " P T ´tg m pxq	`ż T	P s´t f m psqpxqds	´ż T
				t		t
	Proofwhere	zpt, xq :" P T ´tψ 2 pxq	`ż T	P s´t ψ 1 ps, xqdνpsq	`ż T
					t

"

´Bt v `Hpx, Dv `Dzq ´∆v " 0 in r0, T s ˆRd , vpT, xq " 0 in R d .

t P s´t ϕ 1 psqpxqds.

  Lemma 3.12. The function v belongs to L 8 pr0, T s, E n q Ş C ,1`α pr0, T s ˆRd q for any α P p0, 1q and satisfies the following estimate essup pt,xqPr0,T sˆR d |vpt, xq| `essup tPr0,T s }vptq} n Cp|ν|, }ψ 2 } n , sup tPr0,T s }ψ 1 ptq} n , sup tPr0,T s }ϕ 1 ptq} n q ą 0.Proof. The estimate on essup tPr0,T s }vptq} n follows from the fact that v " u´z and the analog estimate for u. The L 8 estimate is also straightforward since

							1`α
			2 `essup s‰t }vptq ´vpsq} n´2 ? t ´s `essup t‰s	}vptq ´vpsq} n´3 |t ´s|	ď C,
	for some C " |vpt, xq| ď	ż T	sup	|Hpx, Dups, xq|ds ď Cp sup	}uptq} 1 q.
				t	xPR d	tPr0,T s
	upt, xq " P T ´tψ 2 pxq	`ż T	1 pt,T s psqP s´t ψ 1 psqpxqdνpsq	`ż T	P s´t ϕ 1 psqpxqds
					0		t
	´ż T
	We also define, for all pt, xq P r0, T s ˆRd
	zpt, xq :" P T ´tψ 2 pxq	`ż T	1 pt,T s psqP s´t ψ 1 ps, xqdνpsq	`ż T	P s´t ϕ 1 psqpxqds	(3.38)
				0			t
	and
	vpt, xq :"	´ż T

t P s´t rHp., Dups, .qqs pxqds. t P s´t rHp., Dups, .qqs pxqds. (3.39)

  xq ´Dk vpt 2 , xq " D k rP s´t 1 rHp., Dups, .qqs pxqs ds D k rP s´t 1 rHp., Dups, .qqs pxq ´Ps´t 2 rHp., Dups, .qqs pxqs ds. (3.40) On the one hand, for all ps, xq P rt 1 , t 2 s ˆRd , |D k rP s´t 1 rHp., Dups, .qqs pxqs | ď Cpessup tPr0,T s }uptq} k`1 q. On the other hand, for all s P rt 2 , T s, P s´t 1 rHp., Dups, .qqs pxq ´Ps´t 2 rHp., Dups, .qqs pxq " ´ż t 2 D k rP s´t 1 rHp., Dups, .qqs pxq ´Ps´t 2 rHp., Dups, .qqs pxqs rP s´t 1 rHp., Dups, .qqs pxq ´Ps´t 2 rHp., Dups, .qqs pxqdss| Cpsup tPr0,T s }uptq} k`2 q ą 0. We conclude from (3.40) that essup xPR d |D k vpt 1 , xq ´Dk vpt 2 , xq| ď Cpessup tPr0,T s }uptq} k`1 qpt 2 ´t1 q

	for some C " `Cpessup tPr0,T s }uptq} k`2 q ? t 2 ´t1 ď C ? t 2 ´t1
				ż t 2
					t 1
	`ż T
		t 2
					t 1	d dt	P s´t rHp., Dups, .qqs pxqdt
			ż t 2	ż t 2
		"	∆P s´t rHp., Dups, .qqs pxqdt "	P s´t ∆ rHp., Dups, .qqs pxqdt,
			t 1	t 1
	and therefore,
					"	´ż t 2	DP s´t D k´1 ∆ rHp., Dups, .qqs pxqdt	(3.41)
					t 1
	from which it follows that,
		ż T
	sup	|D k
	xPR d	t 2
		ď C	ż T t 2	ż t 2 t 1	1 s ? ´tdtds ď 2CpT ´t2 q ? t 2 ´t1 ,

  1 in Chapter 2). Using Jensen inequality, we get for t, s P r0, T s with s ă t xq| 2 dmptqpxqdtq ą 0 since d 2 pmpsq, mptqq ď Ep|X t ´Xs | 2 q 1{2 .

	Ep|X t ´Xs | 2 q ď 2E	«	ˇˇˇż	t	αpu, X u qdu ˇˇˇ2	ff	`4E	"	|B t ´Bs | 2 ‰
				s
		ď 2pt ´sq 2 E	"ż t s	|αpu, X u q| 2 du t ´s	`4pt ´sq
						ż T	ż
		ď 2pt ´sq				|αpt, xq| 2 dmptqpxqdt `4pt ´sq
						0		R d
	and therefore	d 2 pmpsq, mptqq ď C	? t	´s
	ż T	ż
	for some C " Cp	|αpt,
	0	R d

  Proposition 5.2 we have that P is supported on the set of measures solution to the martingale problem. It remains to show that the constraint is satisfied P -almost surely at the limit. By continuity of Ψ, for all t P r0, T s it holds that

	P	`	m P P p pC d ˆVq, ΨpX t #mq ď 0q (˘ě	lim sup
				N Ñ`8

  Hpx, pq ě c R I d for all px, pq P R d ˆBR , Hpx, pq| ď Cp|p| `1q for all px, pq P R d ˆRd ,

	.11)
	$
	' H is locally strictly convex with respect to the last variable,
	'
	&
	' ' % that is, for any R ą 0, there exists c R ą 0 such that D 2 pp (5.12)
	# there exists a constant C ą 0 such that |D x (5.13)
	# for any R ą 0, there exists C R ą 0 such that
	|D 2 xx Hpx, pq| `|D 2

xp Hpx, pq| ď C R for all px, pq P R d ˆBR ,

(5.14)

  Indeed, let |a| ď R, x P R d and p " D a Lpx, aq. Then, by (5.11), we have Lpx, aq " ´a ¨p ´Hpx, pq. By (5.10), we have

	Lpx, aq ě ´R|p| ´C `p1{cq|p| 2
	while
	Lpx, aq ď sup p 1 t´a ¨p1 `C ´c|p 1 | 2 u ď C	`R2 4c	,
	which implies (5.18).

. (v) As L is the Legendre transform of H, Assumptions (5.10) and (5.11) imply that # for any R ą 0, there exists C R ą 0 such that |D a Lpx, aq| ď C R for all px, aq P R d ˆBR (5.18)

  }α t } L 8 pR d ;R d q satisfies the stochastic McKean-Vlasov equation # dm t pxq " rp1 `a0 q∆m t pxq ´divpm t α t pxqqs dt `?2a 0 Dm t pxq ¨dW t in pt 0 , T s ˆRdm t 0 " m 0 in R d .(5.22)The last condition means that, P´a.s., for any smooth test function φ P C 8 c pr0, T s ˆRd q with a compact support and for any t P rt 0 , T s one has, pB t φ s pxq `αs pxq ¨Dφ s pxq `p1 `a0 q∆φ s pxqqm s pdxqds Dφ s pxqm s pdxq ¨dW s .

					›	›
					›	›
					› ›	sup	› ›	ă 8.	(5.21)
					› tPrt 0 ,T s	› L 8 pΩq
	3. m ż φ t pxqm t pdxq "	ż	φ 0 pxqm 0 pdxq	`ż t	ż
	R d	R d				t 0	R d
		`ż t	2a 0 ?	ż
			t 0		R d

  Lpx, α j qm j ps, xqdxds

	1 N n j ď E « ż s 0 `h s 0 ÿ j"1 p ÿ kPC j J	1 N	LpX k t 0 ´s0	`s, α j q	N `C 1	n j d 1 pm j psq, m j X t 0 ´s0	`s qqds	ff
	ď E	« ż t 0 t 0	`h	N ÿ k"1	1 N	LpX k s , α k qds	ff	`Cθh	`C J ÿ j"1	1 N	n j p1	`M 1{2 2 pm j s 0 qq	h β pn j q β
	ď E	« ż t 0 t 0	`h	N ÿ k"1	1 N	LpX k s , α k qds	ff	`Cθh `Cδ ´dβ p1 `λ´1 2 q	h β N β .

  We fix a small positive parameter γ, and note that, if" γ ´1 d Rh 1{d x ´2{d pn j q ´1{d ,

	ż										ż
	Pr sup φPL R	R d	r φpm j phq ´mj Y h q ą xs ď Pr Dk such that	R d	r φ k pm j phq ´mj Y h q ą x ´ s
			ď	K ÿ k"1	Pr ż R d	r φ k pm j phq ´mj Y h q ą x ´ s ď exp	" C	`R ˘d	´nj px ´ q 2 Ch	*	. (5.42)
	then
				R exp	" C	`R ˘d	´nj x 2 Ch	*	" R exp	" Cγ	n j x 2 h	´nj px ´ q 2 Ch	*	.	(5.43)

  ,

			P r sup φPL R	ż R d	r φpm j phq ´mj Y h q ą xs ď exp	" ´nj x 2 Ch	*	.
	It follows that
	Er sup φPL R	ż R d	r φpm j phq ´mj Y h qs ď	ż Cp R d h n j q 0	1 d`2	1dx	`ż 8 Cp R d h n j q	1 d`2	expt	´nj x 2 Ch	udx
					ď Cp1	`R d d`2 qpn j q ´1 d`2 h	1 d`2 .

  Erd 1 pm j phq, m j Y h qs " Ersup Erd 1 pm j phq, m j Y h qs ď Cp1 `R d d`2 qpn j q ´1 d`2 h

	ż
	φPL	R d	φpm j phq ´mj Y h qs
		ż						ż
	ď Er sup φPL R	R d	r φpm j phq ´mj Y h qs	`żR d	|x|1 B c R m j phq `Er	R d	|x|1 B c R m j Y h s
	ď Er sup φPL R	ż	R d	r φpm j phq ´mj Y h qs	`M2 pm j phqq R	Y h qs `ErM 2 pm j R
	ď Er sup φPL R	ż R d	r φpm j phq ´mj Y h qs	`C p1 `M2 pm j p0qqq R	.	(5.45)
	Using Lemma 5.11, we find that

1 d`2 `C p1 `M2 pm j p0qqq R ď Cp1 `Rqpn j q ´1 d`2 h 1 d`2 `C p1 `M2 pm j p0qqq R .

  r m t pxq " r∆ r m t pxq ´divp r m t r α t pxqqs dt in rt 0 , T s ˆRd r m t 0 " m 0 in R d ,(5.48)

		Next, we consider
				r m j t 0 `τ " pId ´?2a 0 B 0 τ q7m j t 0 `τ and r Y k t 0	`τ " Y k t 0	`τ ´?2a 0 B 0 τ ,
	and notice that r m j and r Y k solve the same equations as in Lemma 5.7, and, hence, (5.46)
	holds with r m j t 0	`h replacing m j t 0	`h and m j r Y t 0	`h replacing m j Y t 0	`h .
		Since
												m j t 0 `r " r m j t 0 `τ ‹ δ ? 2α 0 Bτ ,
	and
		m j Y t 0	`τ "	1 n j	ÿ kPC j	δ Y k r t 0	`τ `?2a o B 0 τ "	´1 n j	ÿ kPC j	δ Y k r s 0	`τ ¯‹ δ ? 2a o B 0 τ " r m j Y s 0	`τ ‹ δ ? 2a o B 0 τ ,
	we can conclude that
	E	" d 1 pm j s 0	`h, m j Y s 0	`h q	ı	" E	" d 1 pm j s 0	`h ˚δ? 2α 0 B 0 h	, m j Y s 0	`h ˚δ?

Remerciements

months during the Fall 2021. Finalement je remercie du fond du coeur ma famille, mes amis et Apolline pour leur soutien tout au long de ce travail.

The rate γ N is certainly not optimal but it is sufficient to ensure the convergence of U N pµ 0 q toward Upµ 0 q. Let us also notice that Pp ! Ψpp µ N,x 0 q ď ´γN ) q Ñ 1 as N Ñ `8 as soon as γ N Ñ 0 since Ψpµ 0 q ă 0 and pX 1,N 0 , . . . , X N,N 0 q is distributed according to µ bN 0 under P.

Connection with large deviations for non-interacting diffusions. Our results are closely related to the large deviations principle for (non-interacting) diffusions. Indeed if we consider the probability v N pt, x N q :" Pp@s P r0, ts, Ψ ˜1 N

where x N " px 1,N , . . . , x N,N q and pB 1,N t , . . . , B N,N t q are N independent d-dimensional standard Brownian motions supported on some probability space pΩ, F, Pq. Then v N satisfies the heat equation

where Ω N :" ! px 1,N , . . . , x N,N q P R dN , Ψp 1 N ř N i"1 δ x i,N q ă 0 )

. If we make a logarithmic transformation and let u N pt, x N q " ´2 N log v N pT ´t, x N q, we obtain that u N solves $ & % ´Bt u N `N 2 |Du N | 2 ´∆u N " 0, in p0, T q ˆΩN , u N " `8, in p0, T q ˆBΩ N , u N " 0 in tT u ˆΩN , which is the dynamic programming equation for Problem (NP) when F " G " 0 and Lpx, qq " 1 2 |q| 2 for all px, qq P R d ˆRd . In section 4.4 we discuss this rigorously. Notice that this method to obtain estimates on the probability v N by making a logarithmic transformation and studying the stochastic control problem corresponding to the resulting Hamilton-Jacobi-Bellman equation is reminiscent of [START_REF] Fleming | Exit probabilities and optimal stochastic control[END_REF].

The rest of the chapter is organized as follows. First we introduce, in Section 4.1 the N -particle problem and explain what boundary behavior can be expected. Then we briefly recall the main features of the mean-field problem and provide a stability result with respect to the constraint in Section 4.2. We then prove the convergence results in Section 4.3. We start with Theorem 4.2 which gives the more difficult lim sup N Ñ`8 U N pµ 0 q ď Upµ 0 q and finish with the reverse Upµ 0 q ď lim inf N Ñ`8 U N pµ 0 q in Theorem 4.3. Finally in Section 4. [START_REF] Achdou | Mean Field Type Control with Congestion (II): An Augmented Lagrangian Method[END_REF] we discuss an application of our result to large deviations theory.

Heuristic concerning the boundary behavior. If we denote by u N pt, x 1,N , . . . , x N,N q the value of the same problem but this time initialized at a deterministic position px 1,N , . . . , x N,N q P Ω N at time t P r0, T s, we expect u N to satisfy

δ x i,N q in r0, T q ˆΩN , u N " `8 in r0, T q ˆBΩ N , u N pT, x 1,N , . . . , x N,N q " Gp 1

Well-posedness of the above equation with such a boundary condition as well as its rigorous connection with the control problem is a challenging question.

In [START_REF] Lasry | Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints -1. The model problem[END_REF], Lasry and Lions studied explosive solutions to

where Ω is the interior of a compact domain of R d with a smooth boundary and p ą 1. In case f is smooth and bounded, they proved that W and the constraint X t P Ω @t ě 0, P ´almost-surely.

Leonori and Porretta further proved, in [START_REF] Leonori | The boundary behavior of blow-up solutions related to a stochastic control problem with state constraint[END_REF], refinements of these results together with the blow-up behavior of the gradient of the solution. In particular, for the quadratic case which is of interest to us, they proved that v ´logpdpxqq is bounded and Lipschitz over Ω. If we translate these results to our setting, keeping track of the dimension and using Ψ to measure the distance to the boundary, this suggests (at least in the infinite horizon case) that

Even in the infinite horizon setting we don't try to replicate the aforementioned results because the connection between the solution to the HJB equation and the value function in the quadratic case is not clear. However they give us a good idea of the expected behavior of the system near the boundary.

The fact that P is supported on the set of optimal solutions of the mean-field relaxed problem follows from the lower semi-continuity of the cost functional as proved in [START_REF] Lacker | Limit theory for controlled mckean-vlasov dynamics[END_REF]

U N pµ 0 q ď Upµ 0 q and we already pointed out, before stating Theorem (4.3), that Upµ 0 q " inf mPR Γpmq. Finally we have that Upµ 0 q ď lim inf

which concludes the proof of the theorem.

Application to Large Deviations

Throughout this section we take F " G " 0 and Lpx, qq " 1 2 |q| 2 for all px, qq P R d ˆRd . In this setting, Upµ 0 q reads Upµ 0 q " inf pα,µq

where the infimum is taken over the couples pα, µq satisfying $ ' ' & ' ' % µ P Cpr0, T s, P 2 pR d qq, α P L 2 dtbµptq pr0, T s ˆRd , R d q B t µ `divpαµq ´∆µ " 0 in p0, T q ˆRd , µp0q " µ 0 P P 2 pR d q, Ψpµptqq ď 0 @t P r0, T s.

For all N ě 1, we fix a filtered probability space pΩ, F, F, Pq supporting N d-dimensional independent standard Brownian motions pB 1,N t , . . . , B N,N t q. For x N " px 1,N , . . . , x N,N q in Ω N , we define u N p0, x N q as follows u N p0, x N q :" inf

ff where the infimum is taken over controls pα i,N q i"1,...,N adapted to the filtration generated by the Brownian motions, such that the dynamics

satisfy, P-almost surely the inequality Ψpp µ N,x t q ď 0, for all t P r0, T s, Chapter 5

A rate of convergence for the mean-field limit

This chapter is based on a joint work with Pierre Cardaliaguet, Joe Jackson and Panagiotis Souganidis.

Introduction

We consider an optimal control problem with a large number of particles. The value function for this optimization problem reads

where T ą 0 is a finite horizon, t 0 P r0, T s is the initial time, and x 0 " px 1 0 , . . . , x N 0 q P pR d q N is the initial position of the N particles. The infimum is taken over the set A N of progressively measurable pR d q N -valued processes α " pα k q N k"1 in L 2 pr0, T s ˆΩ; pR d q N q and X " pX 1 , . . . , X N q solves, for each k P t1, . . . , N u,

The pB k q kě0 are independent d-dimensional Brownian motions defined on the fixed filtered probability space pΩ, F, F, Pq satisfying the usual conditions, and L 2 pr0, T s ˆΩ; pR d q N q denotes the set of square-integrable and progressively measurable processes taking values in pR d q N . Here δ x is the Dirac mass at x, and the empirical measure m N Xt is given by

(5.

3)

The cost function L : R d ˆRd Ñ R is supposed to be convex in the second variable and smooth while the maps F, G : P 1 pR d q Ñ R are assumed to be smooth and bounded over the space P 1 pR d q of Borel measures on R d with a finite first moment (precise assumptions will be given in Section 5.2). The constant a 0 ě 0 is the level of the common noise, and the pB k q kě1 are viewed as independent or idiosyncratic noises.

possible thanks to the fact that V N solves the Hamilton-Jacobi equation

Hpx j , N D x j V N pt, xqq " Fpm N x q in p0, T q ˆpR d q N

V N pT, xq " Gpm N x q in pR d q N (5.7) where Hpx, pq " sup

r´p ¨α ´Lpx, αqs, while U is expected to solve (in some sense) the infinite dimensional Hamilton-Jacobi equation

trpD 2 mm Upt, x, m, y, y 1 qqmpdyqmpdy 1 q `żR d Hpy, D m Upt, m, yqqmpdyq " Fpmq in p0, T q ˆP2 pR d q UpT, mq " Gpmq in P 2 pR d q (5.8)

For the definition of the derivatives D m U and D 2 mm U we refer to books of Cardaliaguet, Delarue, Lasry and Lions [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF] and Carmona and Delarue [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF].

One of the reasons for introducing the value functions is that they provide optimal feedbacks for the optimization problems. For the particle system, this optimal feedback is given (rigorously) by α i pt, xq " ´Dp Hpx i , N D x i V N pt, xqq, while for the limit system it takes the form (at least formally)

The difficulty in the PDE analysis of [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] is that, in the absence of the idiosyncratic noise, the value functions V N are not smooth in general. Thus in [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] the equation (5.7) has to be interpreted in the viscosity sense. A suitable notion of viscosity solution for the infinite dimensional Hamilton-Jacobi equation (5.8) without idiosyncratic noise is introduced in [START_REF] Gangbo | Finite Dimensional Approximations of Hamilton-Jacobi-Bellman Equations in Spaces of Probability Measures[END_REF] , and then is proven that V N converges to this viscosity solution. In the presence of idiosyncratic noise the notion of viscosity solution to (5.8) is not understood yet and we will not try to use this approach.

Our results continued

While the existing results mentioned above demonstrate the convergence of V N to U under many different technical hypotheses and using a variety of techniques, none provides a rate of convergence. Our main result fills this gap in the literature, by providing a rate of convergence of V N to U in the presence of both idiosyncratic and common noise.

The primary challenge we face is related to the (lack of) regularity of U. Indeed, if U is a smooth solution solution to (5.8), then the projections U N : r0, T s ˆpR d q N Ñ R given by U N pt, xq " Upt, m N

x q are smooth solutions of the Hamilton-Jacobi equation

) with

If D mm U is bounded, then we see immediately that |E n | " Op1{N q. Thus U N solves the same equation as V N , up to a term of order Op1{N q. By a comparison argument, we conclude that |U ´V| " Op1{N q, that is, there exists a constant C such that, for all t P r0, T s and

See also [START_REF] Germain | Rate of convergence for particle approximation of PDEs in Wasserstein space[END_REF] for more on what convergence results can be obtained once (5.8) has a sufficiently smooth solution. This argument is similar to the approach taken in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF][START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications II[END_REF] to study the convergence problem in the context of mean field games (see Lasry and Lions [START_REF] Lasry | Mean field games[END_REF]) in situations where a classical solution to the so-called master equation is known to exist (see Bayraktar and Cohen [START_REF] Bayraktar | Analysis of a finite state many player game using its master equation[END_REF] and Cecchin and Pelino [START_REF] Cecchin | Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation[END_REF] for related results). In this setting, convergence is related to the propagation of chaos for the optimal trajectories of the game.

Of course, the simple argument outlined above works only when the value function U is smooth. For instance this would be is the case if the maps F and G were convex and sufficiently smooth (see the discussion in Chap. 3.7 of [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[END_REF]). However, we will not assume such a convexity property and the map U is expected to present discontinuities in its first order derivative; for instance, an example can be found in Briani and Cardaliaguet [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF]. Because of this, the techniques in [START_REF] Cardaliaguet | The master equation and the convergence problem in mean field games[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[END_REF] break down.

When the value function is not smooth, the convergence rate has been studied primarily in the case of finite state space (see Kolokoltsov [START_REF] Vassili | Nonlinear Markov Games on a Finite State Space (Mean-field and Binary Interactions)[END_REF] and Cechin [START_REF] Cecchin | Finite state N-agent and mean field control problems[END_REF]). In this finite state space setting, the convergence rate is of order 1{ ? N . Indeed, as explained in [START_REF] Cecchin | Finite state N-agent and mean field control problems[END_REF], the particle system is then a kind of discretization of the continuous McKean-Vlasov equation. estimate is relatively easy and boils down to transforming the map V N into a subsolution of the HJ equation (5.8). The converse estimate is more involve and is the aim of Proposition 5.4. The technical reason is that we found no way to embed U into the equation for V N as a subsolution. Actually, since U is semiconcave, it is naturally a supersolution of that equation and the remaining term is a priori large. We overcome this issue by using locally optimal feedback of the N ´problem for the continuous one, the main difficulty being to compare the empirical measure in the N ´problem to the solution of the Fokker-Planck equation. This step is by no means trivial and relies on a key concentration result (Lemma 5.7).

The proof of Theorem 5.1 without common noise

Throughout this section we assume that a 0 " 0.

Some regularity estimates

Throughout the proof we use systematically the fact that V N is the unique solution of the uniformly parabolic backward PDE (5.7) and, therefore, is smooth. We first start with regularity estimates for V N . Lemma 5.1. Assume (5.17). There exists a constant C ą 0 such that, for any N ě 1,

Remark 5.3. The estimate on D x j V N implies that the optimal feedback of the problem, given by α k pt, xq " ´Dp Hpx i , N D x j V N pt, xqq remains uniformly bounded.

Proof. The bound on V N is obvious. We note that w

(5.25) It follows from the maximum principle that N }D x j V N } 8 is bounded uniformly in N and j.

CHAPTER 5. A RATE OF CONVERGENCE FOR THE MEAN-FIELD LIMIT

In the same way,

and the uniform bound on }B t V N } 8 follows again from the maximum principle.

Lemma 5.2. Assume (5.17). There is a constant C such that U satisfies |Upt 0 , m 0 q ´Ups 0 , m 0 q| ď C ´|t 0 ´s0 | 1{2 `d1 pm, mq ¯, @ t 0 , s 0 P r0, T s, m 0 , m 0 P P 2 pR d q.

Moreover, there exists a constant C ą 0 such that, if pt 0 , m 0 q P r0, T s ˆP1 pR d q and pm, αq is optimal in the definition of Upt 0 , m 0 q in (5.4), then }α} 8 ď C.

Proof. The result is standard so we only sketch the argument and refer to [START_REF] Briani | Stable solutions in potential mean field game systems[END_REF] for more details. Fix pt 0 , m 0 q P r0, T s ˆP1 pR d q. It follows from (5.17) that there exists at least a pair pm, αq optimal in the definition of Upt 0 , m 0 q. Moreover, for such optimal pair pm, αq, there exists a map u : rt 0 , T s ˆRd Ñ R with α t pxq " ´Dp Hpx, Dupt, xqq and such that pu, mq solves the system Fix m 1 P P 1 pR d q and let µ be the solution to B t µ ´∆µ `divpµαq " 0 in pt 0 , T q ˆRd with µpt 0 q " m 1 .

It is easy to check that there exists C depending on }Dα} 8 and on T such that sup tPrt 0 ,T s

ω " ř N i,j"0 ω i,j and σ k "

A straightforward computation gives

Denote by γ the right-hand-side of the equality above. Recalling that H is strongly convex in the p variable and that N B x k V N is bounded, we have, for all 1 ď k ď N ,

We can use again the Lipschitz bounds on V N and (5.14) to deduce that

Next, fix pt 0 , x 0 q and consider the weak solution m N to

Integrating the equation satisfied by r ω against m N we deduce that, for all pt 0 , x 0 q P r0, T s pR

In order to bound the right-hand side of the inequality above, we first note that, by the equation satisfied by V N , we have

where F N pxq :" Fpm N x q and G N pxq :" Gpm N x q, and, similarly,

Recalling the expressions of the derivatives of F N and G N in function of the derivatives of F and G in Proposition 5.35 of [START_REF] Carmona | Probabilistic theory of mean field games with applications I[END_REF], we obtain, after a tedious but straightforward computation that, under our standing assumptions on F and G, the quantity sup x }r ωpT, xq} 8 is bounded by

The easy estimate

The second step in the proof of Theorem 5.1 is an upper bound of V N in terms of U. Our strategy will be to first compare U to p V N , where p V N pt, mq :"

(5.29)

We start with a Lemma, whose proof is a straightforward computation which is essentially the same as the one carried out in the proof of Proposition 3.1 in Cardaliaguet and Masoero [START_REF] Cardaliaguet | Weak KAM theory for potential MFG[END_REF]. Hence, we omit the details.

Lemma 5.4. Let p V N be given by (5.29). Then p V N is smooth and satisfies the inequality

where p F N pmq "

mpdx j q and p G N pmq "

Next we prove the easier inequality in Theorem (5.1).

Proposition 5.3. There exist constants C (depending on the data) and β (depending only on d) such that for any pt, x 0 q P r0, T s ˆpR

(5.30)

Proof. Theorem 1 in [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF] gives constants C and β depending only on d such that, for any m P P 2 pR d q and for all N P N,

Proof. Following a viscosity solutions-type argument, we double the variables and, for θ, λ P p0, 1q, we set M :" max pt,xq,ps,yqPr0,T sˆpR d q N e s pUps, m N y q´V N pt, xqq´1 2θN

We denote by ppt 0 , x 0 q, ps 0 , y 0 qq a maximum point in the expression above and we continue estimating in the next lemma the error related to the penalization Lemma 5.5. There exists a constant C such that, for any i P t1, . . . , N u,

We postpone the proof of the lemma and continue with the ongoing proof.

As pointed out in the introduction, the main difficulty is that it does not seem possible, at least to us, how to transform an optimal control for the V N (in which the control depend on each particle) into a feedback for U. We overcome this difficulty by dividing the players into subgroups in such a way that the optimal controls for the agents in each subgroup are close and showing a propagation of chaos-type result for each subgroup using a a concentration inequality.

We begin explaining how to create the subgroups based on an appropriate partition of t1, . . . , N u. Since we will use it again in the next section, state and prove the following lemma.

Lemma 5.6. For each δ ą 0 there exist a constant C depending only on the data, and a partition pC j q jPt1,...,Ju of t1, . . . , N u and α j P R d for j " 1, . . . , J such that J ď Cδ ´d and, for all k P C j , ˇˇHpx k 0 , N D x k V N pt 0 , x 0 qq `αj ¨pN D x k V N pt 0 , x 0 qq `Lpx k 0 , α j q ˇˇď Cδ.

(5.33)

Proof. Let p α k pt, xq " ´Dp Hpx k , N D x k V N pt, xqqbe the optimal feedback for particle k, and recall (see Remark 5.3),that there exists R depending only on the data such that |p α k pt, xq| ď R.

Given δ ą 0, we can find a δ-covering of B R Ă R d consisting of J ď Cδ ´d balls of radius δ entered at pα j q jPt1,...,Ju Ă B R .

Then, we choose the partition pC j q jP1,...J such that, for each k P C j , |p α k pt, xq ´αj | ď δ. It follows that, for each k P C j ,

where we have used 5.18.

We proceed with the ongoing proof.

The Lipschitz regularity of U in Lemma 5.2 and the definition of X t and Y t give

To proceed we need a dynamic programming-type argument, which is stated next. Its proof is postponed for later in the paper.

Lemma 5.8. With the notation above, we have

Using Itô's formula for V N we find M ě e s 0 `hU ps 0 , m N y 0 q ´es 0 `h ż s 0 `h

and, in view of (5.33),

(5.35) The semiconcavity of V N and the penalization by the term in θ give the next lemma. The proof is postponed to end of the section. Lemma 5.9. For any pt, xq P r0, T s ˆpR

We continue with the ongoing proof. Inserting the estimate of Lemma 5.9 in (5.35), we obtain 0 ě e s 0 hpUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ´Cδ ´dβ p1 `λ´1 2 q

ě e s 0 hpUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ´Cδ ´dβ p1 `λ´1 2 q

Dividing by h we find, for each choice of θ, λ, δ ą 0 and 0 ă h ď pT ´s0 q ^pT ´t0 q, the estimate

We take θ " h α 1 , δ " p λ ´1{2 h β´1 N β q α 2 , λ " N ´α3 and h " N ´α4 . By fixing appropriate choices of α 1 , α 2 , α 3 and finally α 4 we deduce that the estimate e s 0 pUps 0 , m N y 0 q ´VN pt 0 , x 0 qq ď CN ´r β (5.36) holds for some r β " r βpβq P p0, 1{2q and for all values of N such that h " N ´α4 ď pT ´s0 q pT ´t0 q. For those values of N such that h " N ´α4 ě pT ´s0 q ^pT ´t0 q, we have by Lemma 5.5 that pT ´s0 q _ pT ´s0 q ď h `Cθ, and so using Lemmas 5.1 and 5.2, we have |Ups 0 , m N y 0 q ´VN pt 0 , x 0 q| ď |Ups 0 , m N y 0 q ´Gpm N y 0 q| `|Gpm N y 0 q ´Gpm N x 0 q| `|Gpm N x 0 q ´VN pt 0 , x 0 q| ď Cph `θq 1{2 `Cθ `Cph `θq ď CN ´r β , Fix ε ą 0, let pm 1 , β 1 , . . . , m K , β K q be ε´optimal for U K pt 0 , m 1 0 , . . . , m K 0 q, and set β " ř N k"1 β K and mptq " ř N k"1 m k ptq. Then pm, βq solves B t m ´∆m `divpβq " 0 in pt 0 , T s ˆRd and mpt 0 q " m 0 in R d .

and we have

where the second inequality follows from the convexity of the map pβ, mq Ñ mLpx, β m q and the third one by the definition of U.

Proof of Lemma 5.9. Set p k " D x k Vpt 0 , x 0 q and p t " B t Vpt 0 , x 0 q. Then, in view of Lemma 5.3, we have, for any pt, xq, pt 0 , x 0 q P r0, T s ˆpR

The optimality of pt 0 , x 0 , s 0 , y 0 q also gives, for any pt, xq,

pt 0 ´s0 q 2 `VN pt 0 , x 0 q. (5.37)

From (5.37), we conclude that

Furthermore, rearranging (5.37) yields

and, after some elementary manipulations,

Assuming that θ ď p2Cq ´1, tt follows that wpt, xq " V N pt 0 , x 0 q ´VN pt, xq

is convex and satisfies

Thus, for any pt, xq and any ps, yq, we have

Letting y k " x k 0 `1 2 θN D x k wpt, xq and s " t 0 `1 2 θB t wpt, xq in the inequality above, we obtain

D x k wpt, xq ¨px k ´xk 0 q `Bt wpt, xqpt ´t0 q, (5.38) and, after using the Cauchy-Schwarz inequality,

Recalling the definition of w and that

and |B t wpt, xq| " | ´Bt V N pt 0 , x 0 q `2Cpt ´t0 q| ď C. Returning to (5.39), we have

from which we deduce the result by the definition of p k .

We note for later use that this implies that the conclusion of Lemma 5.9 still holds, because it relies only on the above estimates.

However, the proof of Lemma 5.2 does not adapt to the case a 0 ą 0. Hence, we need a new argument which relies on some results of [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF].

In particular, we have the following analogue of Lemma 5.2. Lemma 5.12. Assume (5.17). There exists a constant C ą 0 depending only on the data such that, for all s, t P r0, T s and all m, m 1 P P 4 pR d q |Ups, mq ´Upt, m 1 q| ď C ´d1 pm, m 1 q `|t ´s| ¯.

Moreover, there exists a constant C ą 0 such that, for any ą 0 and any pt 0 , m 0 q P r0, T s ˆP2 pR d q, there exists an -optimal control rule R " pΩ, F, F, W, m, αq P Apt 0 , m 0 q for Upt 0 , m 0 q such that }α} 8 ď C.

Proof. Fix R ą 0 and let V N,R and U R denote the values of the problems defining V N and U when controls are restricted to the ball B R Ă R d .

More precisely, define A N,R to be the set of α " pα k q N k"1 such that |α k | ď R for each R, and A R pt 0 , m 0 q to be the set of pΩ, F, F, P, m, αq P Apt 0 , m 0 q such that |α| ď R. Then define V N,R exactly as in (5.1) but with A N,R replacing A and define U R exactly as in (5.23) but with A R pt 0 , m 0 q replacing A R . Then Proposition 5.5 and Theorem 3.6 of [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] give lim N Ñ8

V N,R pt, x N q " U R pt, mq where x N " px 1 , ..., x N q, m P P 4 pR d q and x 1 , x 2 , ... P R d are such that sup

It follows from Lemma 5.1 and Lemma 5.13, that there is R 0 ą 0 such that V N,R 0 " V N and U R 0 " U, and so we infer that, for all x i and m as above, lim N Ñ8

V N pt, x N q " Upt, mq.

Hence, the uniform regularity on V N established in (5.2), which, as noted above, holds equally well when a 0 ą 0, is enough to conclude that, for some C ą 0, |Upt, mq ´Upt, m 1 q| ď Cd 1 pm, m 1 q for all m, m 1 P P 4 pR d q.

Finally, for any ą 0 and any pt 0 , m 0 q, we can choose an -optimal pair pm, αq for U R 0 , and that this control is also -optimal for U. This completes the proof.

Let p V N be defined in Lemma 5.3. Then it is easily checked that p V N is smooth and satisfies, with p F N and p G N as in Lemma 5.3,

Then, as in the proof of Lemma 5.3, it is possible to use Itô's formula for conditional measures (see, for example, [START_REF] Carmona | Probabilistic Theory of Mean Field Games with Applications II[END_REF]) to infer that, for any solution pm, αq to (5.22),

Using the same argument as in the proof of Lemma 5.3 with Lemma 5.12 replacing Lemma 5.7, we arrive at

x 0 qqN ´β . We now turn to the opposite inequality. As before, for θ, λ P p0, 1q,

and denote by ppt 0 , x 0 q, ps 0 , y 0 qq a maximum point in the expression above.

As in Lemma 5.5 we have

Next, for δ ą 0, we use the partition pC j q jPt1,...,Ju be a partition of t1, . . . , N u constructed in Lemma 5.6.

We set α k " α j if k P C j , and let

formulation defined here. Namely, we define A R W pt 0 , m 0 q just as A w pt 0 , m 0 q, but with the controls α required to take values in B R Ă R d . Then, we use U R w pt 0 , m 0 q :" inf RPA R w pt 0 ,m 0 q E P r ż T t 0 pLpX t , α t q `Fpm t qqdt `Gpm T qs

We also truncate the original form of the problem, by defining U R just like U, but with controls α required to take values in B R Ă R d .

The following can be obtained using the superposition and following results of [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF], as in the proof of Theorem 8.3 of [START_REF] Lacker | Superposition and mimicking theorems for conditional McKean-Vlasov equations[END_REF].

It is also useful to note that our regularity result Lemma 5.1, which holds also in the case a 0 ą 0, can be used to infer that U R " U for all R ě R 0 . Lemma 5.13. There exists R 0 depending on the data such that, for each R ě R 0 , U R " U.

Proof. Theorems 3.1 and 3.6 in [START_REF] Mao Fabrice Djete | McKean-Vlasov optimal control: limit theory and equivalence between different formulations[END_REF] together with Proposition 5.5 allow to conclude that, for each R ą 0, we have the following form of convergence of V N,R to U R . For all t P r0, T s, µ P P 4 pR d q and x i P R d such that sup

m in P 2 pR d q we have, for x N " px 1 , ..., x N q P pR d q N , lim N Ñ8

V R,N pt, x N q " U R pt, mq.

(5.49)

Next, notice that, by (5.1) (see Remark 5.3), there is R 0 depending only on the data such that, for all R ě R 0 , V N,R " V N . Thus (5.49) actually gives, for all R ě R 0 , lim N Ñ8

V N pt, x N q " U R pt, mq

It follows that

U " U R 0 .

Indeed, clearly U ď U R 0 .

For the other inequality, for any pt 0 , m 0 q, we can choose R " pΩ, F, F, P, W, m, αq to be -optimal in the definition of Upt 0 , m 0 q. Since α is bounded by hypothesis, there exist R ě R 0 such that R P A R pt 0 , m 0 q, and, hence, U R 0 pt 0 , m 0 q " U R pt 0 , m 0 q ď Upt 0 , m 0 q ` . Letting Ñ 0 gives Upt 0 , m 0 q " U R 0 pt 0 , m 0 q. Now, we turn to the dynamic programming principle, that is, Proposition 5.2.

ABSTRACT

We analyse mean-field stochastic control problems under constraints in law. The goal is to characterize optimal solutions thanks to a mean-field game system of partial differential equations. We start with a problem with terminal constraint and go on with a problem with constraint at all time. For this second problem we prove the time regularity of optimal controls under suitable geometric assumptions on the constraint. We also explain how this problem arises as limit of control problems for a large number of interacting agents with symmetric, almost-sure constraints. Finally we investigate quantitatively the mean-field limit for stochastic control problems without constraint and prove that this convergence holds with a rate.

MOTS CLÉS

Contrôle Stochastique; Théorie des Jeux à Champ Moyen; Optimisation sous Contraintes; Limite de Champ Moyen RÉSUMÉ On étudie des problèmes de contrôle stochastique de type champ-moyen avec des contraintes sur la loi du processus. Le but est de caractériser les stratégies optimales grâce à un système couplé d'équations aux dérivées partielles qui est celui de la théorie des jeux à champ-moyen. On étudie tout d'abord un problème avec contrainte terminale puis un problème avec contrainte à chaque instant. Pour ce second problème on étudie la régularité en temps des contrôles optimaux sous des conditions géométriques sur la contrainte. On montre également comment ce problème apparaît comme limite de problèmes de contrôle où un grand nombre d'agents sont soumis à des contraintes symétriques et presque-sûres. Dans une dernière partie, on étudie quantitativement, en obtenant un taux de convergence, la limite champ-moyen pour des problèmes de contrôle stochastique sans contrainte.

KEYWORDS

Stochastic Control; Mean-Field Control; Mean-Field Games; Mean-Field Limit; Constrained Optimization