Jean-François Chassagneux ˚

François Delarue

P Rt

A PROBABILISTIC APPROACH TO CLASSICAL SOLUTIONS OF THE MASTER EQUATION FOR LARGE POPULATION EQUILIBRIA

Keywords: Master equation, McKean-Vlasov SDEs, forward-backward systems, decoupling eld, Wasserstein space, master equation MSC Classication (2000): Primary 93E20, secondary 60H30, 60K35. Proposition 2.2. Given pt, x, µq

We analyze a class of nonlinear partial dierential equations (PDEs) dened on R d ˆP2pR d q, where P2pR d q is the Wasserstein space of probability measures on R d with a nite second-order moment. We show that such equations admit a classical solutions for suciently small time intervals. Under additional constraints, we prove that their solution can be extended to arbitrary large intervals. These nonlinear PDEs arise in the recent developments in the theory of large population stochastic control.

More precisely they are the so-called master equations corresponding to asymptotic equilibria for a large population of controlled players with mean-eld interaction and subject to minimization constraints. The results in the paper are deduced by exploiting this connection. In particular, we study the dierentiability with respect to the initial condition of the ow generated by a forward-backward stochastic system of McKean-Vlasov type. As a byproduct, we prove that the decoupling eld generated by the forward-backward system is a classical solution of the corresponding master equation. Finally, we give several applications to mean-eld games and to the control of McKean-Vlasov diusion processes.

Introduction

The theory of large population stochastic control describes asymptotic equilibria among a large population of controlled players with mean eld interaction and subject to minimization constraints. It has received a lot of interest since the earlier works on mean-eld games of Lasry and Lions [START_REF] Lasry | Jeux à champ moyen I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen II. Horizon ni et contrôle optimal[END_REF][START_REF] Lasry | Mean eld games[END_REF] and of Huang, Caines and Malhamé [START_REF] Huang | Large population stochastic dynamic games: closedloop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF]. Mean-eld game theory is the branch of large population stochastic control theory that corresponds to the case when equilibria inside the population are understood in the sense of Nash and thus describe consensus between the players that make the best decision they can, taking into account the current states of the others in the game. We cover this class of control problems in Section 5.2. There are other types of large population equilibria in the literature yielding dierent types of asymptotic control problems. As an example, the case when players obey a common policy controlled by a single center of decision is investigated in [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF][START_REF] Carmona | Control of McKean-Vlasov versus Mean Field Games[END_REF]. We cover this distinct control problem in Section 5.3.

Lasry and Lions described equilibria by means of a fully-coupled forward-backward system consisting of two partial dierential equations: a (forward) Fokker-Planck equation describing the dynamics of the population and a (backward) Hamilton-Jacobi-Bellman equation describing the optimization constraints. In his seminal lectures at ˚Department of Mathematics, Imperial College London. j.chassagneux@imperial.ac.uk : Department of Mathematics, Imperial College London. d.crisan@imperial.ac.uk ; Laboratoire Jean-Alexandre Dieudonné, Université de Nice Sophia-Antipolis. delarue@unice.fr 1 the Collège de France, Lions noticed that the ow of measures solving the Fokker-Planck equation (that is the forward part of the system) can be interpreted as the characteristic trajectories of a nonlinear PDE. The equilibrium of a large population of players with mean eld interaction is characterized through a nonlinear partial dierential equation set on an enlarged state space that contains both the private position of a typical player and the distribution of the population. The solution of the PDE contains all the necessary information to entirely describe the equilibria of the game and, on the model of the Chapman-Kolmogorov equation for the evolution of a Markov semi-group, it is called the master equation of the game. This equation has the form 1 B t upt, x, µq " Aupt, x, µq `f `x, upt, x, µq, Bupt, x, µq, ν ˘`ż

R d " Cupt, x, µq ‰ p¨qdµp¨q, (1.1)
for t ą 0 and px, µq P R d ˆP2 pR d q, where P 2 pR d q is the Wasserstein space of probability measures on R d with a nite second-order moment. In (1.1), ν is the image of µ by the mapping R d Q x Þ Ñ px, upt, x, µqq; moreover, A and B are dierential operators that dierentiate in the x variable, respectively at the second and rst order, whilst C is a nonlocal operator that involves dierentiation in the µ variable. The notion of dierentiation in the measure variable follows Lions' denition (see [4]).

Since its introduction in Lions' lectures, there have been only a few papers on the master equation. In the notes he wrote following Lions' lectures (see [4]), Cardaliaguet discusses the particular case when players have deterministic trajectories, and where the solutions to the master equation is understood in the viscosity sense. In this framework, the existence of classical solutions has just been investigated for short time horizons by Gangbo and Swiech in the preprint [START_REF] Gangbo | Existence of a solution to an equation arising from the theory of Mean Field Games[END_REF]. Recently, in the independent works [START_REF] Bensoussan | The Master Equation in Mean Field Theory[END_REF][START_REF] Carmona | The master equation for large population equilibriums[END_REF][START_REF] Gomes | Mean eld games models -a brief survey[END_REF][START_REF] Gomes | Extended mean eld games -formulation, existence, uniqueness and examples[END_REF] and with dierent approaches, several authors revisited, mostly heuristically, the master equation in the case when the dynamics of the players are stochastic. A few months ago, in a lecture at the Collège de France [START_REF] Lions | Cours au collège de France[END_REF], Lions gave an outline of a proof, based on PDE arguments, for investigating the master equation rigorously in the latter case. In [START_REF] Bensoussan | The Master Equation in Mean Field Theory[END_REF][START_REF] Carmona | The master equation for large population equilibriums[END_REF],

the notion of master equation is extended to other types of stochastic control problem with players that obey a common policy controlled by a single center of decision.

The goal of this paper is to develop a probabilistic analysis of the class of equations (1.1). We seek classical solutions for a class of PDEs that incorporates the master equations for both types of policies (individual or collective) and for players with dynamics that can be either deterministic or stochastic. Beyond their purely theoretical interest, classical solutions (as opposed to viscosity solutions) are expected to be of use when handling approximated equilibria in a variety of situations: For instance, they help in proving the convergence of the equilibria, when computed over nite systems of players, toward the equilibria of the asymptotic game. This is indeed a challenging question that remains partially open. 2 Similarly, the analysis of numerical schemes for computing the equilibria certainly benets from robust regularity estimates for the solution of the master equation.

One of the reason for using a probabilistic approach is that there has been an expanding literature in probability theory on forward-backward systems, which have been widely used in stochastic control. Although mostly limited to the nite dimension, the existing theory gives a helpful insight into the general mechanism for deriving the master equation. One of the most noticeable results is that a forward-backward system may be 1 The master equation is introduced here in its forward form. However in its application to mean eld games it is used in its backward form, see equation (2.12). 2 See however the recent advances in [START_REF] Fischer | On the connection between symmetric N -player games and mean eld games[END_REF][START_REF] Lacker | A general characterization of the mean eld limit for stochastic dierential games[END_REF].

decoupled by means of a decoupling eld provided the system is uniquely solvable, see e.g. [START_REF] Ma | On wellposedness of forward-backward SDEs[END_REF][START_REF] Ma | On non-Markovian forward-backward SDEs and backward stochastic PDEs[END_REF]. More precisely, the decoupling eld allows one to express the backward component of the solution as a function of the forward one. When the coecients of the forward-backward system are deterministic, the decoupling eld satises (in a suitable sense) a quasilinear PDE. In the case of mean-eld games, the forward-backward system consists of two coupled PDEs, one of Fokker-Planck type and another one of Hamilton-Jacobi-Bellman type, and the corresponding quasilinear PDE is nothing but the master equation.

Another reason for analysing the master equation by means of probabilistic arguments is that equilibria in large population stochastic control problems driven by either individual or collective policies may be characterized as solutions of nite-dimensional forward-backward systems of the McKean-Vlasov type, see [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF][START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF]. The reformulation is based either on the connection between Hamilton-Jacobi-Bellman equations and backward SDEs or on the stochastic Pontryagin principle, see [START_REF] Fleming | Controlled Markov Processes and Viscosity Solutions[END_REF][START_REF] Yong | Stochastic Controls[END_REF] for the basic mechanisms in the non McKean-Vlasov framework. This reformulation has a crucial role as it allows one to reduce the innite-dimensional system made of the Fokker-Planck equation and of the Hamilton-Jacobi-Bellman equation to a nite dimensional system. The price to pay is that the coecients of the nite dimensional system may depend upon the law of the solution, in the spirit of McKean's theory of nonlinear SDEs. Inspired by Pardoux and Peng's work [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF] on the connection between backward SDEs and classical solutions to semilinear PDEs, we develop a systematic approach for analyzing the smoothness of the solution of the master equation by investigating the smoothness of the ow generated by the solution of the McKean-Vlasov forward-backward system with respect to the initial input. However, because of the McKean-Vlasov nonlinearity, the analysis is far from a straightforward adaptation of the classical result of Pardoux and Peng [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF]. The main issue is that the independent variable includes a probability measure, which requires a non-trivial extension of the notion of dierentiability with respect to a probability measure.

Several notions of derivatives with respect to a probability measure have been introduced in the literature. For example, the notion of Wasserstein derivative has been discussed within the context of optimal transport, see the monograph by Villani [START_REF] Villani | Optimal Transport. Old and New[END_REF].

An alternative, though connected, approach was suggested by Lions, see [4]. Generally speaking, Lions' approach consists in lifting (in a canonical manner) functions dened on the Wasserstein space P 2 pR d q (the space of probability measures on R d , with nite second-order moments endowed with the Wasserstein metric) into functions dened on L 2 pΩ, A, P; R d q, the space of square integrable d-dimensional random variables dened on the probability space pΩ, A, Pq. In this way, the operation of dierentiation with respect to a probability measure is dened as the Fréchet dierentiation in L 2 pΩ, A, P; R d q.

This approach is especially suited to the mean eld games framework. Indeed, the probabilistic representation we use yields a canonical lifted representation of the equilibria on L 2 pΩ, A, P; R d q that carries the underlying noise. The McKean-Vlasov forwardbackward system that models the equilibria consists of a forward component describing the dynamics of the population and a backward one describing the dynamics of the solution of the master equation along the state of the population. Any perturbation in L 2 pΩ, A, P; R d q of the initial condition of the forward component thus generates a perturbation in L 2 pΩ, A, P; R d q of the solution of the master equation. Using this strategy, the smoothness of the solution of the master equation is deduced by investigating the smoothness of the ow generated by the McKean-Vlasov forward-backward system with respect to an initial condition in L 2 pΩ, A, P; R d q.

In the sequel, we apply this strategy to general forward-backward systems of equations of McKean-Vlasov type. Under suitable assumption, we prove that existence and uniqueness of solutions holds for the system and that the corresponding decoupling eld is the unique classical solution of the time reversed version of the PDE (1.1). To do this we prove rst the smoothness of the decoupling eld by using the notion of dierentiation described above. Next, we apply a tailor-made chain rule on the Wasserstein space to identify the structure of the PDE from the coecients of the forward-backward system.

In general, the result holds for suciently small time intervals, as it is usually the case with forward-backward processes.

Inspired by [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF], we then show that, provided we have an a priori estimate for the gradient of the solution of the master equation, existence and uniqueness of a classical solution may be extended, via an inductive argument, to arbitrary large time intervals.

This requires the identication of a suitable space of solutions that is left invariant along the induction, which is one of the most technical issues of the paper. In the framework of large population stochastic control, we identify three classes of examples under which the a priori bound for the gradient is shown to hold. The rst two belong to the framework of mean-eld games. To bound the gradient in each of them, we combine either convexity (in the rst example) or ellipticity (in the second example) with the so-called Lasry-Lions condition, used for guaranteeing uniqueness of the equilibria, see [4]. To the best of our knowledge, except the aforementioned video by Lions [START_REF] Lions | Cours au collège de France[END_REF], the solvability of the master equation in the classical sense is, in both cases, a new result 3

. The third example concerns the situation when players obey a common center of decision, in which case the stochastic control problem may be reformulated as an optimization problem over controlled McKean-Vlasov diusion processes. In this last example, the proof mainly relies on convexity.

In a parallel work to ours made available recently, Buckdahn et al. [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF] adopted a similar approach to study forward ows, proving that the semigroup of a standard

McKean-Vlasov stochastic dierential equation with smooth coecients is the classical solution of a linear PDE dened on R d ˆP2 pR d q. The results in [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF] do not cover nonlinear PDEs of the type (1.1) that include master equations for large population equilibria.

It must be also noticed that a crucial assumption is made therein on the smoothness of the coecients, which restrict rather drastically the scope of applications. We avoid this, however, we do pay a heavy price for working under more tractable assumptions, see Remark 2.5 below.

We treat here systems of players driven by idiosyncratic (or independent) noises. Motivated by practical applications, see [START_REF] Carmona | The master equation for large population equilibriums[END_REF][START_REF] Guéant | Mean eld games and applications[END_REF], in subsequent work, the players will be driven by an additional common source of noise, in which case the McKean-Vlasov interaction in the forward-backward equations under consideration becomes random itself, as it then stands for the conditional distribution of the population given the common source of randomness.

The paper is organized as follows. The general set-up together with the main results are described in Section 2. The chain rule on the Wasserstein space is discussed in Section 3. The smoothness of the ow of a McKean-Vlasov forward-backward system is investigated in small time in Section 4. In Section 5, we provide some applications to large population stochastic control. The proofs of some technical results are given in Appendix.

2. General step-up and overview of the results

Let pΩ, A, Pq be a probability space supporting a d-dimensional Brownian motion pW t q tě0 and a square integrable random variable ξ, independent of pW t q tě0 . We denote by pF ξ,W t q tě0 the augmented ltration generated by ξ and pW t q tě0 . For a given terminal time T ą 0, we consider the following system of equations: "

X s " ξ `şs 0 bpX r , Y r , Z r , P pXr,Yrq qdr `şs 0 σpX r , Y r , P pXr,Yrq qdW r , Y s " gpX T , P X T q `şT s f pX r , Y r , Z r , P pXr,Yrq qdr ´şT s Z r dW r , s P rt, T s (2.1)

The processes X, Y and Z are d, m and m ˆd dimensional, respectively. The coecients b : R d ˆRm ˆRmˆd ˆP2 pR d ˆRm q Ñ R d , σ : R d ˆRm ˆP2 pR d ˆRm q Ñ R dˆd , f : R d ˆRm ˆRmˆd ˆP2 pR d ˆRm q Ñ R m and g : R d ˆP2 pR d q Ñ R m are measurable functions that satisfy conditions that will be imposed below. P pXr,Yrq denotes the law of pX r , Y r q. The system (2.1) is called a forward-backward system of McKean-Vlasov type. Notice that, for simplicity, the coecients b, σ and f are time homogeneous and X has same dimension as the noise W . These constraints can however be lifted and a similar analysis will apply.

In the following, we will show that, under convenient assumptions, there exists a unique solution of the forward-backward system (2.1) together with a decoupling eld U : r0, T s ˆRd ˆP2 pR d q Ñ R m to (2.1). Namely, U is a function such that Y s " U ps, X s , P Xs q , 0 ď s ď T .

(2.2)

Finally, we will show pt, x, µq P r0, T s ˆRd ˆP2 pR d q Ñ U pT ´t, x, µq is a classical solution of the equation (1.1).

2.1. Denition of U . The construction of the decoupling eld U is typically discussed under the assumption that the existence and uniqueness of the solution of the system (2.1) is holds. See, e.g. [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF][START_REF] Carmona | Mean-eld forward-backward stochastic dierential equations[END_REF][START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF], for conditions under which this holds for an arbitrary time horizon T . We adopt here a dierent approach: We rst focus on the case where T is suciently small so that the existence and uniqueness of the solution of the system (2.1) hold. This helps us construct the decoupling eld U for the same time horizon and, therefore deduce the existence of a unique local solution of PDE (1.1). Secondly we use results from [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF][START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF] to pass from a small time to an arbitrary time horizon and there justify the existence of a unique global solution to (1.1).

A common strategy to introduce the decoupling eld consists in letting the initial time in (2.1) vary. Without any loss of generality, we can assume that pΩ, A, Pq is equipped with a ltration pF t q tě0 (satisfying the usual condition) such that pΩ, F 0 , Pq is rich enough to carry R d -valued random variables with any arbitrary distribution in P 2 pR d q and pW t q tě0 is an pF t q tě0 -Brownian motion. In particular, ξ in (2.1) may be taken as an F 0 -measurable square-integrable random variable.

In the sequel, we often use the symbol µ to denote the law of ξ. We will use the notation rΘs :" P Θ to denote the law of the random variable Θ (so then µ " rξs).

Within this set-up, we consider the following version of (2.1) with the forward component starting at time t from ξ P L 2 pΩ, F t , P; R d q: # X with θ t,ξ " pX t,ξ , Y t,ξ , Z t,ξ q and θ t,ξ,p0q " pX t,ξ , Y t,ξ q.

A crucial remark for the subsequent analysis is to notice that the YamadaWatanabe theorem extends to equations of the same type as (2.3). More precisely, one can prove that, whenever pathwise uniqueness holds, solutions are also unique in law [START_REF] Kurtz | Weak and strong solutions of general stochastic models[END_REF]Example 2.14]. As a consequence, it follows that the law of pX t,ξ , Y t,ξ q only depends upon the law of ξ. In other words, rpX t,ξ r , Y t,ξ r qs is a function of rξs " µ. Given µ P P 2 pR d q, it thus makes sense to consider prpX t,ξ r , Y t,ξ r qsq tďrďT without specifying the choice of the lifted random variable ξ that has µ as distribution. We then introduce, for any x P R d , a stochastic ow associated to the system (2.3), dened as

# X t,
with θ t,x,µ " pX t,x,µ , Y t,x,µ , Z t,x,µ q and θ t,x,µ,p0q " pX t,x,µ , Y t,x,µ q.

We now have all the ingredients to give the denition of a decoupling eld to (2.3) on r0, T s ˆRd ˆP2 pR d q. For the following denition, assume for the moment that, for any pt, x, µq P r0, T s ˆRd ˆP2 pR d q and any random variable ξ P L 2 pΩ, F t , P; R d q with distribution µ, (2.3) has a unique (progressively-measurable) solution pX t,ξ s , Y t,ξ s , Z Denition 2.1 (The decoupling eld U ). The function U : r0, T s ˆRd ˆP2 pR d q Þ Ñ R m dened as U pt, x, µq " Y t,x,µ t , pt, x, µq P r0, T s ˆRd ˆP2 pR d q (2.5) is called the decoupling eld of the forward-backward system (2.3) (or, equivalently, of the corresponding stochastic ow (2.4)).

The decoupling property (2.2) of U is proved in Proposition 2.2 below, under assumptions that guarantee existence and uniqueness to (2.3) and (2.4).

Recall now that the 2-Wasserstein distance W 2 , dened on P 2 pR k q, k ě 1 is given by

W 2 pµ, νq " inf γ "ż pR k q 2
|u ´v| 2 γpdu, dvq; γp¨ˆR k q " µ, γpR k ˆ¨q " ν  1{2 .

As already mentioned, a very convenient way to prove strong existence and uniqueness to (2.3) and (2.4) consists in working rst with small time horizons. For T suciently small, there exists a unique solution to the systems (2.3) and (2.4) under the following assumption:

Assumption (pH 0q(i)). There exists a constant L ą 0 such that the mappings b, σ, f and g are L-Lipschitz continuous in all the variables, the distance on P 2 pR d ˆRm q, respectively P 2 pR d q being the 2-Wasserstein distance.

The existence of a local solution to the systems (2.3) and (2.4) under assumption pH0q(i) is not new (see for instance [6, Proof of Lemma 2]). The proof consists of a straightforward adaption of the results in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] for classical forward backward stochastic dierential equations (FBSDEs). To be precise, one shows that the systems (2.3) and

(2.4) are uniquely solvable under assumption pH0q(i) provided T ď c for a constant c :" cpLq ą 0. Examples where the result can be extended to long time horizons will be discussed in Section 5.

It is quite illuminating to observe that the system (2.4) can be rewritten as a classical coupled FBSDE with time dependent coecients, as follows # X t,x,µ s " x `şs t bt,µ pr, θ t,x,µ r qdr `şs t σt,µ pr, θ t,x,µ,p0q r qdW r , Y t,x,µ s " ĝt,µ pX t,x,µ T q `şt s ft,µ pr, θ t,x,µ r qdr ´şs t Z t,x,µ r dW r ,

with p bt,µ , ft,µ , σt,µ , ĝt,µ qpr, x, y, zq :" pb, f, σ, gqpx, y, z, rθ t,ξ,p0q r sq. Basically, for this new set of coecients, the dependence upon the measure is frozen since µ and rθ t,ξ,p0q s are xed and do not depend on x. In particular, when replacing x by ξ in (2.4) and (2.6), for some random variable ξ with µ as distribution, uniqueness of solutions to the classical (time-inhomogeous) FBSDE (2.6) implies that pX t,ξ,µ , Y t,ξ,µ , Z t,ξ,µ q " θ t,ξ . Then, the representation (2.6) allows us to characterize the decoupling eld of the system (2.3) as follows:

Under pH0q(i), we know from the classical theory of coupled FBSDEs [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] that, for T suciently small, for any t P r0, T s, there exists a continuous decoupling eld Ût,µ : rt, T s ˆRd Q ps, xq Þ Ñ Ût,µ ps, xq to (2.6) such that Y t,x,µ s " Ût,µ ps, X t,x,µ s q for s P rt, T s, the representation remaining true when x is replaced by an F t -measurable square-integrable random variable (see [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]Corollary 1.5]). In particular, choosing s " t, we get U pt, x, µq " Ût,µ pt, xq. We deduce that it as a function of the joint variable pv, µq. In the next section, we will present conditions under which a continuous version of B µ U pµqp¨q can be identied, such a version being uniquely dened on the support of µ. The next step is then to discuss the smoothness of the map R d ˆP2 pR d q Q pv, µq Þ Ñ B µ V pµqpvq. We say that V is partially C 2 if the mapping P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ V pµqpvq is continuous at any point pµ, vq such that v P Supppµq and if, for any µ P P 2 pR d q, the mapping R d Q v Þ Ñ B µ V pµqpvq is dierentiable, its derivative being jointly continuous with respect to µ and v at any point pµ, vq such that v P Supppµq. The gradient is then denoted by B v rB µ V pµqspvq P R dˆd .

Note that, B µ V pµqpvq is a d-dimensional row vector and B v rB µ V pµqspvq is a d ˆd matrix.

2.3. Solution of a Master PDE. In Section 3, we prove a chain rule for functions dened on the space P 2 pR d q which are partially C 2 in the above sense. Applying the chain rule to U pt, x, ¨q, we get:

U `t, x, rX t,ξ s s ˘´U `t, x, rξs " ż s t p E " B µ U `t, x, rX t,ξ r s ˘`xX t,ξ r y ˘b`x θ t,ξ r y, rθ t,ξ,p0q r s ˘ı dr (2.9) `1 2 ż s t p E " Tr " B v " B µ U `t, x, rX t,ξ r s ˘‰`x X t,ξ r y ˘`σσ : ˘`xθ t,ξ,p0q
r y, rθ t,ξ,p0q r s ˘‰ı dr.

The above identity relies on new notations. Indeed, in order to distinguish the original randomness in the dynamics of (2.3), which has a physical meaning, from the randomness used to represent the derivatives on the Wasserstein space, we will represent the derivatives on the Wasserstein space on another probability space, denoted by p Ω, Â, Pq. p Ω, Â, Pq is a copy of the original space pΩ, A, Pq. In particular, for a random variable ξ dened on pΩ, A, Pq, we denote by xξy its copy on Ω. All the expectations in the above expression may be translated into expectations under E. Nevertheless, we will refrain from doing this to avoid ambiguities between lifts and random variables constructed on the original space pΩ, A, Pq. We will state conditions under which the expectations in (2.9) are indeed well-dened.

Notice that, in (2.9), we used the same convention as in (2.8) for denoting gradients.

The term B µ U pt, x, rX t,ξ r sqpxX t,ξ r yq is thus seen as an m ˆd matrix and the trace term TrrB v rB µ U spt, x, rX t,ξ r sqpxX t,ξ r yqpσσ : qpxθ t,ξ,p0q r y, rθ t,ξ,p0q r sqs as a vector of dimension m. Combined with the analysis of the smoothness of U , we will then show that the function r0, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ U pt, x, µq solves, up to a time reversal, a PDE of the form (1.1). For the time being, we present a formal calculation to deduce this claim, the complete argument being given in Section 4. The basic observation is that, in the framework of Proposition 2.2, the time-increments of U may be expanded as U ps `h, x, rX t,ξ s sq ´U ps, x, rX t,ξ s sq " U ps `h, x, rX t,ξ s sq ´U ps `h, x, rX t,ξ s`h sq `Û t,µ ps `h, xq ´Û t,µ ps, xq, (2.10) for t ď s ď s `h ď T . Applying the chain rule to the dierence term U ps `h, x, rX t,ξ s sq Ú ps `h, x, rX t,ξ s`h sq on the right hand side of the previous equality and assuming that the derivatives in the chain rule are continuous in time so that we can let h tend to 0, we obtain " d dh Choosing s " t, we deduce that U is right dierentiable in time (it is then dierentiable in time provided that the right-hand side is continuous in time). Recalling (2.8) together with the notation µ " rξs and using the transfer theorem to express the expectations that appear in the chain rule as integrals over R d , we then get that U is a solution to the equation

‰ |h"0 U ps `h, x, rX t,ξ s sq " ´p E " b `xθ
B t U pt, x, µq `Bx U pt, x, µqb `x, U pt, x, µq, B σ x U pt, x, µq, νq `1 2 Tr " B 2 xx U pt, x, µq `σσ : ˘‰ `f `x, U pt, x, µq, B σ x U pt, x, µq, ν żR d B µ U pt, x, µqpvqb `v, U pt, v, µq, B σ x U pt, v, µq, ν ˘dµpvq `1 2 ż R d Tr " B v " B µ U pt, x, µq ‰ pvq `σσ : ˘`v, U pt, v, µq, ν ˘‰dµpvq " 0, (2.12)
for pt, x, µq P r0, T s ˆRd ˆP2 pR d q, with the terminal condition U pT, x, µq " gpx, µq, where ν is the law of pξ, U pt, ξ, µqq when rξs " µ and B σ x U pt, x, µq " B x U pt, x, µqσpx, U pt, x, µq, νq. In particular, upt, ¨, ¨q " U pT ´t, ¨, ¨q satises the equation (1.1), the operators A, B and C therein being dened as follows:

Aupt, x, µq " B x upt, x, µqb `x, upt, x, µq, B σ x upt, x, µq, ν

1 2 Tr " B 2
xx upt, x, µq `σσ : ˘`x, upt, x, µq, ν ˘‰ Bupt, x, µq " B x upt, x, µqσpx, upt, x, µq, νq Cupt, x, µqpvq " B µ upt, x, µqpvqb `v, upt, v, µq, B σ

x upt, v, µq, ν

1 2 Tr " B v " B µ upt, x, µq ‰ pvq `σσ : ˘`v, upt, v, µq, ν ˘‰, v P R d ,
with the initial condition up0, x, µq " gpx, µq, and with the same convention as above for the meaning of ν and of B σ x u.

Our rst main result is that, for small time horizons, all the partial derivatives that appear above make sense as continuous functions whenever the coecients are suciently smooth. In this sense, U is a classical solution of (2.12), see Theorem 2.7 right below.

We can actually prove that it is the unique one to satisfy suitable growth conditions, see Theorem 2.8. Our second main result is the extension to arbitrarily large time horizons for three classes of population equilibria. We refer the reader to Subsection 2.5 for a short account of the second result and to Section 5 for complete statements.

2.4. Assumptions. For an L 2 space, we use the notation } ¨}2 as a generic notation for the corresponding L 2 -norm. For a linear mapping Υ on an L 2 space, we let ~Υ~:" sup }υ} 2 "1 }Υpυq} 2 , and for a bilinear form on an L 2 space, we let in the same way ~Υ~:"

sup }υ 1 } 2 "1,}υ 2 } 2 "1 }Υpυ 1 , υ 2 q} 2 .
For a function h from a product space of the form R k ˆP2 pR l q into R, where k, l ě 1, we denote by B w hpw, µq the derivative (if it exists) of h with respect to the Euclidean variable w and by DHpw, χq the Fréchet derivative of the lifted mapping H : L 2 pΩ, A, P; R l q Q χ Þ Ñ hpw, rχsq. The Fréchet derivative is seen as a linear form on L 2 .

Concerning the rst order dierentiability of the coecients, we shall assume:

Assumption. pH1q In addition to pH0q(i), the mappings b, f , σ, g are dierentiable in pw " px, y, zq, µq 6 with jointly continuous derivatives in pw, µq 7 in the following sense: There exist a constant L (in addition to the constant L dened in pH0q(i)), a constant α ě 0 and a functional Φ α : rL 2 pΩ, A, P; R l qs 2 Q pχ, χ 1 q Þ Ñ Φ α pχ, χ 1 q P R `, continuous at any point pχ, χq of the diagonal, such that, for all χ, χ 1 P L 2 pΩ, A, P; R l q,

Φ α pχ, χ 1 q ď E ! `1 `|χ| 2α `|χ 1 | 2α `}χ} 2α 2 ˘|χ ´χ1 | 2 ) 1{2 when χ " χ 1 , (2.13)
and, for h matching any of the coordinates 8 of b, f , σ or g, for all w, w 1 P R k and χ, χ

1 P L 2 pΩ, A, P; R l q, ~DHpw, χq~ď L , ~DHpw, χq ´DHpw 1 , χ 1 q~ď L`| w ´w1 | `Φα pχ, χ 1 q ˘,
and

|B w hpw, rχsq| ď L , |B w hpw, rχsq ´Bw hpw 1 , rχ 1 sq| ď L`| w ´w1 | `Φα pχ, χ 1 q ˘.
Moreover, for any χ P L 2 pΩ, A, P; R l q, the family pDHpw, χqq wPR k , identied by Riesz' theorem with a collection of elements in L 2 pΩ, A, P; R l q, is uniformly square integrable.

Remark 2.3. (i) In particular we note that |DHpw, χq ¨χ1 | ď L}χ 1 } 2 (where `¨' denotes the action of the duality).

(ii) Notice that the right-hand side in (2.13) might not be nite. Actually, we shall make use of (2.13) when χ and χ 1 coincide outside a bounded subset R l , namely χpωq " χ 1 pωq whenever |χpωq| and |χ 1 pωq| are larger than some prescribed R ě 0, in which cases the right-hand side in (2.13) is nite. For instance, choosing χ " χ 1 , we get from (2.13) that Φ α is zero on the diagonal. Notice also that, when α " 0, we can directly choose Φ α pχ, χ 1 q " Er|χ ´χ1 | 2 s 1{2 . (iii) Proposition 3.8 below shows that, under pH1q, the function R l Q v Þ Ñ B µ hpw, µqpvq admits, for any w P R k and µ P P 2 pR l q, a continuous version. It allows to represent DHpw, χq, when identied with an element of L 2 pΩ, A, P; R l q by Riesz' theorem, in the form B µ hpw, rχsqpχq. We stress that such a continuous version of R l Q v Þ Ñ B µ hpw, µqpvq is uniquely dened on the support of µ. Reexpressing the bounds in pH1q, it satises

E "ˇˇB µ hpw, rχsqpχq ˇˇ2 ‰ 1{2 ď L, E "ˇˇB µ hpw, rχsqpχq ´Bµ hpw 1 , rχ 1 sqpχ 1 q ˇˇ2 ‰ 1{2 ď L |w ´w1 | `Φα pχ, χ 1 q ( , (2.14)
Moreover, the uniform square integrability property is equivalent to say that the family pB µ hpw, rχsqpχqq wPR k is uniformly square integrable for any χ P L 2 pΩ, A, P; R d q.

(iv) The uniform integrability assumption plays a major role in our analysis. Taking into account the fact that all the pDHpw, χqq wPR k have a norm less than L, this amounts 6

Here µ stands for the generic symbol to denote the measure argument.

7

Under the standing assumptions on the joint continuity of the derivatives, it is easily checked that the joint dierentiability is equivalent to partial dierentiability in each of the two directions w and µ.

8

For the presentation of the assumption, it is here easier to take h as a real-valued function, which explains why we identify h with a coordinate of b, f , σ or g; however, we will sometimes say rather abusively that h matches b, f , σ or g.

to require that lim PpAqÑ0,APA sup wPR k sup ΛPL 2 pΩ,A,P;R l q:}Λ} 2 ďL ˇˇDHpw, χq ¨`Λ1 A ˘ˇ" 0.

We stress the fact that it is automatically satised when α " 0 in (2.13). Indeed, we shall prove in (4.14) below that, whenever α " 0, there exists a constant C ě 0 such that, for all w P R k , |DHpw, χq| (identied with a random variable) is less than Cp1 `|χ| `}χ} 2 q.

Concerning the second order dierentiation of the coecients, we shall assume: Assumption (pH 2q). In addition to pH1q, all the mappings px, y, zq Þ Ñ bpx, y, z, µq, px, y, zq Þ Ñ f px, y, z, µq, px, yq Þ Ñ σpx, y, µq and x Þ Ñ gpx, µq are twice dierentiable for any µ P P 2 pR l q the second-order derivatives being jointly continuous in px, y, zq and µ. Moreover, for h equal to any of the coordinates of b, f , σ or g, for any w P R k and µ P P 2 pR l q, with the appropriate dimensions k and l, there exists a continuously dierentiable version of the mapping R l Q v Þ Ñ B µ hpw, µqpvq such that the mapping R k ˆRl Q pw, vq Þ Ñ B µ hpw, µqpvq is dierentiable (in both variables) at any point pw, vq such that v P Supppµq, the partial derivative R k ˆRl Q pw, vq Þ Ñ B v rB µ hpw, µqspvq being continuous at any pw, vq such that v P Supppµq and the partial derivative R k ˆSupppµq Q pw, vq Þ Ñ B w rB µ hpw, µqspvq being continuous in pw, vq. With the same constants L and α as in pH1q, for w P R k and χ P L 2 pΩ, A, P; R l q,

ˇˇB 2 ww hpw, rχsq ˇˇ`E "ˇˇB w " B µ hpw, rχsq ‰ pχq ˇˇ2 ‰ 1{2 `E"ˇˇB v " B µ hpw, rχsq ‰ pχq ˇˇ2 ‰ 1{2 ď L,
and, for w, w 1 P R k and χ, χ

1 P L 2 pΩ, A, P; R l q, ˇˇB 2 ww hpw, rχsq ´B2 ww hpw 1 , rχ 1 sqpχ 1 q ˇĚ "ˇˇB w " B µ hpw, rχsq ‰ pχq ´Bw " B µ hpw 1 , rχ 1 sq ‰ pχ 1 q ˇˇ2 ‰ 1{2 `E"ˇˇB v " B µ hpw, rχsq ‰ pχq ´Bv " B µ hpw 1 , rχ 1 sq ‰ pχ 1 q ˇˇ2 ‰ 1{2 ď L |w ´w1 | `Φα pχ, χ 1 q ( ,
In pH2q, we include the assumption:

Assumption (pH σq). The function σ is bounded by L.

Note that pH2q contains pH0q(ii) (and obviously pH0q(i) and pH1q).

Remark 2.4. The specic form of pH1q and pH2q is dictated by our desire to establish results for arbitrary large horizons. Generally speaking, such results are established by means of a recursive argument, which consists in using the current value U pt, ¨, ¨q of the decoupling eld at time t as a new boundary condition, or put it dierently in letting U pt, ¨, ¨q play at time t the role of g at time T when the FBSDEs (2.3) and (2.4) are considered on r0, ts instead of r0, T s. A delicate point in this construction is to choose a space of boundary conditions which is stable, namely in which U pt, ¨, ¨q remains along the recursion. We remark that we cannot prove that boundary conditions with globally Lipschitz derivatives in the measure argument are stable, even in small time. One of the contribution of the paper is thus to identify a space of terminal conditions which are indeed stable and which permits to apply the recursion method.

Remark 2.5. The reader may compare pH0q, pH1q and pH2q with the assumptions in [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF]. We rst point out that, in [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF], the rst L 2 bound in (2.14) is assumed to hold in L 8 . The example hprξsq " }ξ} 2 , for which the derivative has the form B µ hprξsqpvq " v{}ξ} 2 ,

shows that asking B µ h to be in L 8 is rather restrictive. We also observe that, dierently from [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF], we do not require the coecients to admit second-order derivatives of the type B 2 µµ . The reason is that we here establish the chain rule for functions from P 2 pR d q to R that may not have second-order derivatives of the type B 2 µµ , see Theorem 3.5.

2.5. Main results: from short to long time horizons and application to control.

Inspired by Assumptions pH0q(i), pH1q and pH2q, we let: Denition 2.6. Given non-negative real numbers β, a, b, with a ă b, we denote by D β pra, bsq the space of functions V : ra, bs ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ V pt, x, µq P R m for which we can nd a constant C ě 0 such that (i) For any t P ra, bs, the function V pt, ¨, ¨q : R d ˆP2 pR d q Q px, µq Þ Ñ V pt, x, µq satises the same assumption as g in pH0q(i), pH1q and pH2q, but with α replaced by β and with L and L replaced by C (and thus with w " x P R d , v P R d and χ P L 2 pΩ, A, P; R d q in the various inequalities where these letters appear);

(ii) For any x P R d and µ P P 2 pR d q, the function ra, bs Q t Þ Ñ V pt, x, µq is dierentiable, the derivative being continuous with respect to pt, x, µq on the set ra, bs ˆRd P2 pR d q. Moreover, the functions ra, bs ˆRd ˆL2 pΩ, A,

P; R d q Q pt, x, ξq Þ Ñ B x V pt, x, rξsq P R d , ra, bs ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B µ V pt, x, rξsqpξq P L 2 pΩ, A, P; R d q, ra, bs ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B 2 x V pt, x, rξsq P R d , ra, bs ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B x rB µ V pt, x, rξsqspξq P L 2 pΩ, A, P; R d q ra, bs ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B v rB µ V pt, x, rξsqspξq P L 2 pΩ, A, P; R d q are continuous.
For the reader's convenience, when ra, bs " r0, T s, we will simply use the notation D β for D β pr0, T sq. The extension to arbitrarily large time horizons will be discussed in Section 5. The principle for extending the result from small to long horizons has been already covered in several papers, including [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF][START_REF] Ma | On wellposedness of forward-backward SDEs[END_REF]. Basically, the principle is to prove that, following the recursive step, the decoupling eld remains in a space of admissible boundary conditions for which the length of the interval of solvability can be bounded from below. Generally speaking, this requires, rst, to identify a class of functions on R d ˆP2 pR d q left invariant by the recursive step and, second, to control the Lipschitz constant of the decoupling eld, uniformly along the recursion. In the current framework, the Lipschitz constant means the Lipschitz constant in both the space variable and the measure argument.

As suggested by Theorem 2.7, we are not able to prove that the space D β , for a xed β ě 0 is left invariant by the recursion step. In particular, for the case β " 0, this means that, even in short time, we cannot prove that the decoupling eld has Lipschitz derivatives in the direction of the measure when the terminal condition g has Lipschitz derivatives. This diculty motivates the specic form of the local Lipschitz assumption in pH1q and pH2q. Indeed, Theorem 2.7 shows that the set Ť βě0 D β is preserved by the dynamics of the master PDE under pH2q although none of the sets D β has been shown to be stable. More precisely, we allow the exponent α in pH1q and pH2q to increase by 1 at each step of the recursion, Theorem 2.7 guaranteeing that the set Ť βě0 D β is indeed stable along the induction.

In Section 5, we give three examples when the Lipschitz constant of the decoupling eld can be indeed controlled. First, we consider the forward-backward system deriving from the tailor-made version of the stochastic Pontryagin principle for mean-eld games.

Then, we establish a Lipschitz estimate of U , in the case when the extended Hamiltonian of the control problem is convex in both the state and control variables and when the Lasry-Lions monotonicity condition that guarantees uniqueness of the equilibrium is satised (see [4]). We then interpret U as the gradient in space of the solution of the master equation that arises in the theory of mean-eld games and, as a byproduct, we get that, in this framework, the master equation for mean-eld games is solvable. Second, we propose another approach to handle the master equation for mean-eld games when the extended Hamiltonian is not convex in x. We directly express the solution of the master equation as the decoupling eld of a forward-backward system of the McKean-Vlasov type. We then prove the required Lipschitz estimate of U when the cost functionals are bounded in x and are linear-quadratic in α, the volatility is non-degenerate and the Lasry-Lions condition is in force. Third, we consider the forward-backward system deriving from the stochastic Pontryagin principle, when applied to the control of McKean-Vlasov diusion processes. Then, we establish a similar estimate for the Lipschitz control of U , but under a stronger convexity assumption of the extended Hamiltonian namely, convexity must hold in the state and control variables and also in the direction of the measure (in which case there is no need of the Lasry-Lions condition). Again, this permits us to deduce that the master equation associated to the control problem has a global classical solution.

We may summarize with the following statement (again, we refer to Section 5 for a complete account):

Theorem 2.9. We can nd general examples taken from large population stochastic control such that, for a given T ą 0, (2.3) and (2.4) have a unique solution and the decoupling eld U belongs to Ť βě0 D β and satises the PDE (2.12). In particular, the corresponding forward equation (1.1) has a unique classical solution on r0, 8q.

2.6. Frequently used notations. For two random variables X and X 1 , the relationship X " X 1 means that X and X 1 have the same distribution. The conditional expectation given F t is denoted by E t . Let t P r0, T q. For a progressively-measurable process pX s q sPrt,T s with values in R l , for some integer l ď 1, we let

}X} H p ,t :" E t "ˆż T t |X s | 2 ds ˙p{2  1{p , }X} S p ,t :" E t " sup sPrt,T s |X s | p ı 1{p , }X} H p :" E "ˆż T t |X s | 2 ds ˙p{2  1{p , }X} S p :" E " sup sPrt,T s |X s | p ı 1{p .
(2.15)

In particular, we denote by S p prt, T s; R l q the space of continuous and adapted random processes from rt, T s to R l with a nite norm } ¨}S p and by H p prt, T s; R l q the space of progressively-measurable processes from rt, T s to R l with a nite norm } ¨}H p . In the sequel, the generic letter C is used for denoting constants the value of which may often vary from line to line. Constants whose precise values have a fundamental role in the analysis will be denoted by letters distinct from C.

3. Chain rule application to the proof of Theorem 2.8

In this section, we discuss the chain rule used in (2.9) and apply it to prove Theorem 2.8. Namely, we provide a chain rule for pU pµ t qq tě0 where U is an R-valued smooth functional dened on the space P 2 pR d q and pµ t q tě0 is the ow of marginal measures of an R d -valued Itô process pX t q tě0 . There are two strategies to expand pU pµ t qq tě0 . The rst one consists, for a given t ą 0, in dividing the interval r0, ts into sub-intervals of length h " t{N , for some integer N ě 1, and then in splitting the dierence U pµ t q ´U pµ 0 q accordingly:

U pµ t q ´U pµ 0 q " N ´1 ÿ i"0 " U pµ ih q ´U pµ pi´1qh q ‰ .
The dierences U pµ ih q ´U pµ pi´1qh q are expanded by applying Taylor's formula at order 2. Since the order of the remaining terms in the Taylor expansion are expected to be smaller than the step size h, we can derive the chain rule by letting h tend to 0. This strategy ts the original proof of Itô's dierential calculus and is presented in details in [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF]Section 6] and in [START_REF] Carmona | The master equation for large population equilibriums[END_REF]Section 6].

An alternative strategy consists in approximating the dynamics dierently. Instead of discretizing in time as in the previous strategy, it is conceivable to reduce the space dimension by approximating the ow pµ t q tě0 with the ow of empirical measures

ˆμ N t " 1 N N ÿ "1 δ X t ˙tě0 , N ě 1,
where pX 1 t q tě0 , . . . , pX N t q tě0 stand for N independent copies of pX t q tě0 . Letting @x 1 , . . . ,

x N P R d , u N px 1 , . . . , x N q " U ˆ1 N N ÿ "1 δ x ˙, (3.1)
we expand pu N pX 1 t , . . . , X N t qq tě0 by standard Itô's formula. Letting N tend to the innity, we then expect to recover the same chain rule as the one obtained by the rst method. Here u N is interpreted as a nite dimensional projection of U .

The rst strategy mimics the proof of the standard chain rule. The second one gives an insight into the signicance of the dierential calculus on the space of probability measures introduced by Lions in [4]. Both strategies require some smoothness conditions on U : Clearly, U must be twice dierentiable in some suitable sense. From this viewpoint, the strategy by particle approximation is advantageous: Taking benet of the nite dimensional framework, by using a standard mollication argument it works under weaker smoothness conditions required on the coecients. In particular, dierently from [START_REF] Buckdahn | Mean-eld stochastic dierential equations and associated PDEs[END_REF][START_REF] Carmona | The master equation for large population equilibriums[END_REF], we do not require the existence of B 2 µµ U to prove the chain rule, see Theorem 3.5.

3.1. Full C 2 regularity. We rst remind the reader of the notion of lifted version of U .

On L 2 pΩ, A, P; R d q (the σ-eld A being prescribed), we let

UpXq " U `rXs ˘, X P L 2 pΩ, A, P; R d q.

Instead of pΩ, A, Pq, we could use p Ω, Â, Pq, but since no confusion is possible with the physical random variables that appear in (2.3) and (2.4), we continue to work on pΩ, A, Pq. Following Lions' approach (see [4,Section 6]), the mapping U is said to be dierentiable on the Wasserstein space if the lift U is dierentiable in the sense of Fréchet on L 2 pΩ, A, Pq. By Riesz' theorem, the Fréchet derivative DUpXq, seen as an element of L 2 pΩ, A, P; R d q, can be represented as Section 6]. Recall that, as a gradient, B µ U prXsqpvq will be seen as a row vector.

DUpXq " B µ U `rXs ˘pXq, where B µ U prXsq : R d Q v Þ Ñ B µ U prXsqpvq P R d is in L 2 pR d , µ; R d q, see [4,
A natural question to investigate is the joint regularity of the function B µ U with respect to the variables µ and v. This requires a preliminary analysis for choosing a `canonical version' of the mapping B µ U pµq :

R d Q v Þ Ñ B µ U pµqpvq P R d ,
which is a priori dened just as an element of L 2 pR d , µ; R d q. In this perspective, a reasonable strategy consists in choosing a continuous version of the derivative if such a version exists. For instance, whenever DU is Lipschitz continuous, we know from [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF] that, for any µ P P 2 pR d q, there exists a Lipschitz continuous version of the function R d Q v Þ Ñ B µ U pµqpvq, with a Lipschitz constant independent of µ. This result is made precise in Proposition 3.8 below.

The choice of a continuous version is especially meaningful when the support of µ is the entire R d , in which case the continuous version is uniquely dened. Whenever the support of µ is strictly included in R d , some precaution is however needed, as the continuous version may be arbitrarily dened outside the support of µ. To circumvent this diculty, one might be tempted to look for a version of B µ U pµq : R d Q v Þ Ñ B µ U pµqpvq P R d , for each µ P P 2 pR d q, such that the global mapping

P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq P R d (3.2)
is continuous. Noticing, by means of a convolution argument, that the set tµ 1 P P 2 pR d q : supppµ 1 q " R d u is dense in P 2 pR d q, this would indeed permit to uniquely determine the value of B µ U pµqpvq for v outside the support of µ (when it is strictly included in R d ).

Unfortunately, in the practical cases handled below, the best we can do is to nd a version of B µ U pµq : R d Q v Þ Ñ B µ U pµqpvq P R d , for each µ P P 2 pR d q, such that the global mapping (3.2) is continuous at the points pµ, vq such that v P supppµq.

The fact that B µ U pµq : R d Q v Þ Ñ B µ U pµqpvq P R d is not uniquely determined outside the support of µ is not a problem for investigating the dierentiability of B µ U pµq in v. It is an issue only for investigating the dierentiability in µ. We thus say that the chosen

version of R d Q v Þ Ñ B µ U pµqpvq is dierentiable in v, for a given µ P P 2 pR d q, if the mapping R d Q v Þ Ñ B µ U pµqpvq is dierentiable in the standard sense, the derivative being denoted by R d Q v Þ Ñ B v rB µ U
pµqspvq (which belongs to R dˆd ). Note that there is no Schwarz' theorem for exchanging the derivatives as U does not depend on v.

Now, if we can nd a jointly continuous version of the global mapping (3.2), B µ U is said to be dierentiable in µ, at v P R d , if the lifted mapping L 2 pΩ, A, P; R d q Q X Þ Ñ B µ U prXsqpvq P R d is dierentiable in the Fréchet sense. Then, according to the previous discussion, the derivative can be interpreted as a mapping

R d Q v 1 Þ Ñ B µ rB µ U prXsqpvqspv 1 q P R dˆd in L 2 pR d , µ; R dˆd q, which we will denote by R d Q v 1 Þ Ñ B 2
µ U prXsqpv, v 1 q. In a rst step, we will prove Itô's formula when this additional assumption on the smoothness of B µ U in µ is in force. More precisely, we will say that U is fully C 2 if the global mapping B µ U in (3.2) is continuous and the mappings

P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq, P 2 pR d q ˆRd Q pµ, vq Þ Ñ B v rB µ U pµqspvq, P 2 pR d q ˆRd ˆRd Q pµ, v, v 1 q Þ Ñ B 2 µ U pµqpv, v 1 q,
are continuous for the product topologies, the space P 2 pR d q being endowed with the 2-Wasserstein distance.

Under suitable assumptions, it can be checked that full C 2 regularity implies twice Fréchet dierentiability of the lifting U. As we won't make use of such a result, we refrain from providing its proof in the paper. We will be much more interested in a possible converse: Can we expect to recover that U is C 2 regular (with respect to v and µ), given the fact that U has some Fréchet or Gâteaux dierentiability properties at the second-order? We answer this (more challenging) question in Subsection 3.4 below.

To clarify the signicance of the notion of full C 2 regularity, we now make the connection between the derivatives of u N and those of U : Proposition 3.1. Assume that U is C 1 . Then, for any N ě 1, the function u N is dierentiable on R N and, for all x 1 , . . . , x N P R d , the mapping

R d Q x i Þ Ñ B x i u N px 1 , . . . , x N q P R d reads B x i u N px 1 , . . . , x N q " 1 N B µ U ˆ1 N N ÿ "1 δ x ˙px i q.
If, moreover, U is fully C 2 , then, for any N ě 1, the function u N is C 2 on R N and, for all x 1 , . . . , x N P R d , the mapping

R d ˆRd Q px i , x j q Þ Ñ B 2 x i x j u N px 1 , . . . , x N q P R dˆd satises B 2 x i x j u N px 1 , . . . , x N q " 1 N B v " B µ U ˆ1 N N ÿ "1 δ x ˙px i qδ i,j `1 N 2 B 2 µ U ˆ1 N N ÿ "1 δ x ˙px i , x j q.
Proof. The formula for the rst order derivative has been already proved in [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF]. It remains to deduce the formula for the second order derivative. When i " j, it is a direct consequence of the rst order formula. When i " j, the computations require some precaution as dierentiability is simultaneously investigated in the directions of µ and v in B µ U pµqpvq, but, by the joint continuity of the second-order derivatives B 2 µ U pµqpvq and B v B µ U pµqpv, v 1 q, they are easily handled. Remark 3.2. Assume that U is fully C 2 . Then, for any X P L 2 pΩ, A, P; R d q and Y, Z P L 8 pΩ, A, P; R d q, the mapping from which we deduce that B 2 µ U prXsqpX, Xq " pB 2 µ U prXsqp X, Xqq : . By the same argument as above, we nally deduce that B 2 µ U pµqpv, v 1 q " pB 2 µ U pµqpv 1 , vqq : , for any v, v 1 P R d and any µ P P 2 pR d q.

ϑ : R 2 Q ph, kq Þ Ñ U `"X `hY `kZ ‰˘P R is of class C 2 on R 2 , with d dh " dϑ dk ‰ ph, kq " d dh E " B µ U `"X `hY `kZ ‰˘p X `hY `kZqZ ‰ " E " Tr `Bv B µ U `"X `hY `kZ ‰˘p X `hY `kZqZ b Y ˘‰ `EÊ " Tr `B2 µ U `"X `hY `kZ ‰˘p X `hY `kZ, X `h Ŷ `k ẐqZ b Ŷ ˘‰,
3.2. The chain rule for U fully C 2 . We consider an R d -valued Itô process dX t " b t dt `σt dW t , X 0 P L 2 pΩ, A, Pq, where pb t q tě0 and pσ t q tě0 are progressively-measurable processes with values in R d and, respectively, R dˆd respectively with respect to the (augmented) ltration generated by W , such that

@T ą 0, E "ż T 0 `|b t | 2 `|σ t | 4 ˘dt  ă `8. (3.5)
The following is the main result of this section Theorem 3.3. Assume that U is fully C 2 and that, for any compact subset K Ă P 2 pR d q,

sup µPK "ż R d ˇˇB µ U pµqpvq ˇˇ2 dµpvq `żR d ˇˇB v " B µ U pµq ‰ pvq ˇˇ2dµpvq  ă `8, (3.6)
Then, letting µ t :" rX t s and a t :" σ t pσ t q : , for any t ě 0,

U pµ t q " U pµ 0 q `ż t 0 E " B µ U pµ s qpX s qb s ‰ ds `1 2 ż t 0 E "
Tr `Bv `Bµ U pµ s q ˘pX s qa s ˘‰ds. (3.7) The proof relies on a mollication argument captured in the proof of the following result:

ϕ : L 2 pΩ, A, P; R d q Q X Þ Ñ ϕpXq.
It is then quite standard to check that:

B µ " U ‹ ϕ ‰ pµqpvq " ´d ÿ k"1 " B µ U `ϕ7µ ˘`ϕpvq ˘ık Bϕ k Bx i pvq ¯i"1,...,d , B 2 µ " U ‹ ϕ ‰ pµqpv, v 1 q " ´d ÿ k, "1 " B 2 µ U `ϕ7µ ˘`ϕpvq, ϕpv 1 q ˘ık, Bϕ k Bx i pvq Bϕ Bx j pv 1 q ¯i,j"1,...,d , B v " B µ " U ‹ ϕ ‰ pµqpvq ı " ´d ÿ k"1 " B µ U `ϕ7µ ˘`ϕpvq ˘ık B 2 ϕ k Bx i Bx j pvq `d ÿ k, "1 " B v " B µ U `ϕ7µ ˘‰`ϕ pvq ˘ık, Bϕ k Bx i pvq Bϕ
Bx j pvq ¯i,j"1,...,d .

(3.8)

Recall from Remark 3.2 that the second-order derivatives that appear in (3.8) have some symmetric structure. Now, since ϕ is compactly supported, the mapping P 2 pR d q Q µ Þ Ñ ϕ7µ has a relatively compact range 9 (in P 2 pR d q). By the continuity of U and its derivatives, we deduce that U ‹ ϕ and its rst and second order derivatives are bounded and uniformly continuous on the whole space.

Assume now that the chain rule has been proved for any bounded and uniformly continuous U with bounded and uniformly continuous derivatives of order 1 and 2. Then, for some U just satisfying the assumption of Theorem 3.3, we can apply the chain rule to U ‹ ϕ, for any ϕ as above. In particular, we can apply the chain rule to U ‹ ϕ n for any n ě 1, where pϕ n q ně1 is a sequence of compactly supported smooth functions such that pϕ n , B x ϕ n , B 2 xx rϕ n s 1 , . . . , B 2 xx rϕ n s d qpvq Ñ pv, I d , 0, . . . , 0q uniformly on compact sets as n Ñ 8, I d denoting the identity matrix of size d. In order to pass to the limit in the chain rule (3.7), the only thing is to verify some almost sure (or pointwise) convergence in the underlying expectations and to check the corresponding uniform integrability argument. 9 Tightness is obvious. By boundedness of ϕ, any subsequence converging in the weak sense is also convergent with respect to W2.

U ‹ ϕ n pµq Ñ U pµq, B µ " U ‹ ϕ n ‰ pµqpXq Ñ B µ U pµqpXq, B v " B µ `U ‹ ϕ n ˘‰pµqpXq Ñ B v " B µ U pµq ‰ pXq.
(3.10)

Moreover, we notice that Recall that the above is true for any µ P P 2 pR d q and any X P L 2 pΩ, A, P; R d q with µ as distribution. In particular, we can choose µ " µ s and X " X s in the above limits.

sup ně1 E " ˇˇB µ " U ‹ ϕ n ‰ pµqpXq ˇˇ2 `ˇB v " B µ `U ‹ ϕ n ˘pµq ‰ pXq ˇˇ2 ı ă 8. ( 3
As the bound Er|b s | 2 `|σ s | 4 s ă 8 is satised for almost every s P r0, ts, this permits to pass to the limit inside the integrals appearing in the chain rule applied to each of the pU ‹ ϕ n q ně1 . In order to pass to the limit in the chain rule itself, we must exchange the pathwise limit that holds for almost every s P r0, ts and the integral with respect to the time variable s. The argument is the same as in (3.11). Indeed, since the ow of measures prX s sq 0ďsďt is continuous for the 2-Wasserstein distance, the family of measures pprϕ n pX s qsq 0ďsďt q ně1 is relatively compact and thus

sup ně1 sup sPr0,ts E " ˇˇB µ U `rϕ n pX s qs ˘`ϕ n pX s q ˘ˇ2 `ˇB v " B µ U `rϕ n pX s qs ˘‰`ϕ n pX s q ˘ˇ2 ı ă 8,
which is enough to prove that the functions

´r0, ts Q s Þ Ñ E " B µ pU ‹ ϕ n qprX s sqpX s qb s ‰ `E Tr " B v `Bµ pU ‹ ϕ n qprX s sq ˘pX s qa s ‰( ¯ně1
are uniformly integrable on r0, ts.

We now turn to the proof of Theorem 3.3. We give just a sketch of the proof, as a rened version of Theorem 3.3 is given later, see Theorem 3.5 in the next subsection.

Proof. [Proof of Theorem 3.3.] By Proposition 3.4, we can replace U by U ‹ ϕ, for some compactly supported smooth function ϕ. Equivalently, we can replace pX t q tě0 by pϕpX t qq tě0 . In other words, we can assume that U and its rst and second order derivatives are bounded and uniformly continuous and that pX t q tě0 is a bounded Itô process.

Finally by the same argument as in the proof of Proposition 3.4, we can also assume that pb t q tě0 and pσ t q tě0 are bounded. Indeed, it suces to prove the chain rule when pX t q tě0 is driven by truncated processes and then to pass to the limit along a sequence of truncations that converges to pX t q tě0 . Let ppX t q tě0 q ě1 a sequence of i.i.d. copies of pX t q tě0 . That is, for any ě 1,

dX t " b t dt `σ t dW t , t ě 0,
where ppb t , σ t , W t q tě0 , X 0 q ě1 are i.i.d copies of ppb t , σ t , W t q tě0 , X 0 q.

Recalling the denition of the ow of marginal empirical measures:

μN t " 1 N N ÿ "1 δ X t ,
the standard Itô's formula yields together with Proposition 3.1, P-a.s., for any t ě 0

u N `X1 t , . . . , X N t ˘" u N `X1 0 , . . . , X N 0 1 N N ÿ "1 ż t 0 B µ U `μ N s ˘pX s qb s ds `1 N N ÿ "1 ż t 0 B µ U `μ N s ˘pX s qσ s dW s (3.12) `1 2N N ÿ "1 ż t 0 Tr B v " B µ U `μ N s ˘‰pX s qa s ( ds `1 2N 2 N ÿ "1 ż t 0 Tr B 2 µ U `μ N s ˘pX s , X s qa s ( ds,
with a s :" σ s pσ s q : . We take expectation on both sides of the previous equality and obtain (the stochastic integral has zero expectation due to the boundedness of the coecients), recalling (3.1),

E " U `μ N t ˘‰ " E " U `μ N 0 ˘‰ `1 N N ÿ "1 E "ż t 0 B µ U `μ N s ˘pX s qb s ds  `1 2N N ÿ "1 E "ż t 0 Tr B v " B µ U `μ N s ˘‰pX s qa s ( ds  `1 2N 2 N ÿ "1 E "ż t 0 Tr B 2 µ U `μ N s ˘pX s , X s qa s ( ds  .
All the above expectations are nite, due to the boundedness of the coecients. Using the fact that the processes ppa s , b s , X s q sPr0,ts q Pt1,...,N u are i.i.d., we deduce that

E " U `μ N t ˘‰ " E " U `μ N 0 ˘‰ `E"ż t 0 B µ U `μ N s ˘pX 1 s qb 1 s ds  (3.13) `1 2 E "ż t 0 Tr B v " B µ U `μ N s ˘‰pX 1 s qa 1 s ( ds
 (3.14) `1 2N E "ż t 0 Tr B 2 µ U `μ N s ˘pX 1 s , X 1 s qa 1 s ( ds  , (3.15)
In particular, because of the additional 1{N , the term in (3.15) converges to 0. Moreover, the coecients pa s q sPr0,ts and pb s q sPr0,ts being bounded, we know from [32, Theorem 10.2.7]:

lim N Ñ`8 E " sup 0ďsďt W 2 2 pμ N s , µ s q ‰ " 0. (3.16)
This implies together with the uniform continuity of U with respect to the distance W 2 , that E " U `μ N t ˘‰ (resp. E " U `μ N 0 ˘‰) converges to U pµ t q (resp. U pµ 0 q). Combining the uniform continuity of B µ U on P 2 pR d q ˆRd with (3.16), the second term in the right- hand side of (3.13) converges. Similar arguments lead to the convergence of the term in (3.14).

The notion of dierentiation as dened by Lions plays an essential role in the chain rule formula. It is the right dierentiation procedure to give the natural extension from the chain rule for empirical distribution processes to the chain rule for measure valued processes.

3.3. The chain rule for U partially C 2 . We observe that, in the formula for chain rule (3.7), the second order derivative B 2 µ U does not appear. It is thus a quite natural question to study its validity when B 2 µ U does not exist. This is what we refer to as `partial C 2 regularity'. More precisely, we will say that U is partially C 2 (in v) if the lift U is Fréchet dierentiable and, for any µ P P 2 pR d q, we can nd a continuous version of the mapping

R d Q v Þ Ñ B µ U pµqpvq such that:
' the mapping P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq is jointly continuous at any pµ, vq such that v P Supppµq, ' for any µ P P 2 pR d q, the mapping R d Q v Þ Ñ B µ U pµqpvq P R d is continuously dierentiable and its derivative is jointly continuous with respect to µ and v at any point pµ, vq such that v P Supppµq, the derivative being denoted by

R d Q v Þ Ñ B v rB µ U pµqspvq P R dˆd .
Recall from the discussion in Subsection 3.1 that, for each µ P R d , the mapping B µ U pµq : v Þ Ñ B µ U pµqpvq is uniquely dened on the support of µ.

The following is the chain rule for is partially C 2 : Theorem 3.5. Assume that U is partially C 2 and that, for any compact subset K Ă P 2 pR d q, (3.6) holds true. Then, the chain rule holds for an Itô process satisfying (3.5).

Notice that, in the chain rule, the mapping B µ U : P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq is always evaluated at points pµ, vq such that v belongs to the support of µ and thus for which B µ U pµqpvq is uniquely dened.

Proof. First step. We start with the same mollication procedure as in the proof of Theorem 3.3, see (3.8).

Repeating the computations, U ‹ ϕ and its rst and partial second order derivatives are bounded. Nevertheless, contrary to the argument in the proof of Theorem 3.3, we cannot claim here that B µ pU ‹ ϕq and B v rB µ pU ‹ ϕqs are continuous on the whole space since B µ U and B v rB µ U s are only continuous at points pµ, vq such that v is in the support of µ. In order to circumvent this diculty, we rst notice, from (3.8), that B µ pU ‹ ϕq and B v rB µ pU ‹ ϕqs are also continuous at points pµ, vq such that v is in the support of µ, the reason being that v P Supppµq implies ϕpvq P Supppϕ7µq. We then change P 2 pR d q Q µ Þ Ñ pU ‹ ϕqpµq into P 2 pR d q Q µ Þ Ñ pU ‹ ϕqpµ ‹ ρq where ρ is a smooth convolution kernel, with the entire R d as support and with exponential decay at innity, and µ ‹ ρ stands for the probability measure with density given by

R d Q x Þ Ñ ż R d ρpx ´yqdµpyq.
We then observe that

B µ "`U ‹ ϕ ˘pµ ‹ ρq ‰ pvq " ż R d B µ `U ‹ ϕ ˘pµ ‹ ρqpv ´v1 qρpv 1 qdv 1 , B v " B µ "`U ‹ ϕ ˘pµ ‹ ρq ‰‰ pvq " ż R d B v " B µ `U ‹ ϕ ˘pµ ‹ ρq ‰ pv ´v1 qρpv 1 qdv 1 .
Since the support of ρ is the whole R d , the measure µ ‹ ρ also has R d as support, so that, for any v P R d , pµ ‹ ρ, vq is a continuity point of both B µ pU ‹ ϕq and B v rB µ pU ‹ ϕqs. Since B µ pU ‹ ϕq and B v rB µ pU ‹ ϕqs are bounded, we deduce from Lebesgue's theorem that the maps pµ, vq Þ Ñ B µ pU ‹ ϕqpµ ‹ ρqpvq and pµ, vq Þ Ñ B v rB µ pU ‹ ϕqpµ ‹ ρqspvq are continuous on the whole P 2 pR d q ˆRd . Moreover, whenever ρ is chosen along a sequence that converges to the Dirac mass at 0 (for the W 2 distance), it is also easy to check that, for any µ P P 2 pR d q and any v P Supppµq, B µ pU ‹ ϕqpµ ‹ ρqpvq and B v rB µ pU ‹ ϕqspµ ‹ ρqpvq converge to B µ pU ‹ ϕqpµqpvq and B v rB µ pU ‹ ϕqspµqpvq. In particular, if Itô's formula holds true for functionals of the type P 2 pR d q Q µ Þ Ñ pU ‹ ϕqpµ ‹ ρq, it also holds true for functionals of the type P 2 pR d q Q µ Þ Ñ pU ‹ ϕqpµq and then for functionals of the type P 2 pR d q Q µ Þ Ñ U pµq by the same approximation argument as in the proof of Theorem 3.3.

Therefore, without any loss of generality, we can assume that U and its rst and partial second order derivatives are bounded and uniformly continuous on the whole space. As in the proof of Theorem 3.3, we can also assume that pX t q tě0 is a bounded Itô process.

Second step. The proof requires another mollication argument. Taking now ρ as a smooth compactly supported density on R d and using the same notations as above, we dene the convolution u N n of u N :

u N n px 1 , . . . , x N q " n N d ż pR d q N u N px 1 ´y1 , . . . , x N ´yN q N ź "1 ρ `ny ˘N ź "1 dy " E " U ˆ1 N N ÿ i"1 δ x i ´Y i {n ˙, (3.17)
where Y 1 , . . . , Y N are N i.i.d. random variables with density ρ. Recalling that

W 2 2 ˆ1 N N ÿ i"1 δ x i ´Y i {n , 1 N N ÿ i"1 δ x i ˙ď 1 N N ÿ i"1 `Y i n ˘2, we notice that E " W 2 2 ˆ1 N N ÿ i"1 δ x i ´Y i {n , 1 N N ÿ i"1 δ x i ˙ ď C n 2 , (3.18)
as ρ has compact support. Above and in the rest of the proof, the constant C is a general constant that is allowed to increase from line to line. Importantly, it does not depend on n nor N . Observe now that, for two random variables X, X 1 P L 2 pΩ, A, P; R d q, we can nd t P r0, 1s such that

|U prXsq ´U prX 1 sq| " ˇˇE " B µ U `"tX `p1 ´tqX 1 ‰˘`t X `p1 ´tqX 1 ˘pX ´X1 q ‰ˇď › › B µ U `"tX `p1 ´tqX 1 ‰˘`t X `p1 ´tqX 1 ˘› › 2 }X ´X1 } 2 ď C}X ´X1 } 2 ,
the last line following from the fact that the function P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq is bounded. Therefore, we deduce from (3.17) and (3.18) that ˇˇu N n px 1 , . . . , x N q ´uN px 1 , . . . , x N q ˇˇ" ˇˇˇE

" U ˆ1 N N ÿ i"1 δ x i ´Y i {n ˙´U ˆ1 N N ÿ i"1 δ x i ˙ˇˇˇď Cn ´1. (3.19)
Given a bounded random variable X with law µ, we know from [32, Theorem 10.2.1] that the quantity ErW 2 2 pµ, μN qs tends to 0 as N tends to the innity, μN denoting the empirical measure of a sample of size N of the same law as X. Moreover, the rate of convergence of pErW 2 2 pµ, μN qsq N ě1 towards 0 only depends upon the bounds for the moments of X. Together with (3.19), this says that we can nd a sequence pε q ě1 converging to 0 as tends to 8 such that, for any n, N ě 1 and for any t ě 0,

E "ˇˇu N n pX 1 t , . . . , X N t q ´U `µt ˘ˇ‰ ď E "ˇˇu N n pX 1 t , . . . , X N t q ´uN pX 1 t , . . . , X N t q ˇˇ‰ `E"ˇˇU `μ N t ˘´U `µt ˘ˇ‰ ď ε n `εN . (3.20)
(It is worth mentioning that the sequence pε q ě1 may be assumed to be independent of t.) By boundedness of U , we deduce that, for any p ě 1 and any t ě 0,

E "ˇˇu N n pX 1 t , . . . , X N t q ´U `µt ˘ˇp ‰ 1{p ď ε ppq n `εppq N , (3.21)
for a sequence pε ppq q ě1 that tends to 0 as tends to 8 (and the terms of which are allowed to increase from line to line). Now, by the rst part in Proposition 3.1, we compute

B x i u N n px 1 , . . . , x N q " n N d ż pR d q N B x i u N px 1 ´y1 , . . . , x N ´yN q N ź "1 ρpny q N ź "1 dy " n N d N ż pR d q N B µ U ˆ1 N N ÿ "1 δ x ´y ˙px i ´yi q N ź "1 ρpny q N ź "1 dy " 1 N E " B µ U ˆ1 N N ÿ "1 δ x ´Y {n ˙px i ´Y i {nq  .
Using the uniform continuity of B µ U on the whole space and following the proof of (3.20), we deduce that, for any t ě 0,

E "ˇˇN B x i u N n pX 1 t , . . . , X N t q ´Bµ U pµ t qpX i t q ˇˇ‰ ď ε n `εN . (3.22)
Again, by boundedness of B µ U , we deduce that, for any p ě 1 and any t ě 0,

E "ˇˇN B x i u N n pX 1 t , . . . , X N t q ´Bµ U pµ t qpX i t q ˇˇp ‰ 1{p ď ε ppq n `εppq N . (3.23)
Now, we dierentiate once more in x i :

B 2 x i x i u N n px 1 , . . . , x N q " n N d`1 N ż pR d q N " B µ U ˆ1 N N ÿ "1 δ x ´y ˙px i ´yi q * b ∇ρpny i q ź "i ρpny q N ź "1
dy , the tensorial product operating on elements of R d . We then split the derivative into two pieces:

N B 2 x i x i u N n px 1 , . . . , x N q " T 1,N n,i px 1 , . . . , x N q `T 2,N n,i px 1 , . . . , x N q, with T 1,N n,i px 1 , . . . , x N q " n N d`1 ż pR d q N " B µ U ˆ1 N ÿ "i δ x ´y `1 N δ x i ˙px i ´yi q * b ∇ρpny i q ź "i ρpny q N ź "1 dy T 2,N n,i px 1 , . . . , x N q " n N d`1 ż pR d q N ""ˆB µ U ˆ1 N N ÿ "1 δ x ´y Ḃµ U ˆ1 N ÿ "i δ x ´y `1 N δ x i ˙px i ´yi q * b ∇ρpny i q ź "i ρpny q N ź "1
dy .

By integration by parts (recall that

R d Q v Þ Ñ B µ U pµqpvq is dierentiable), we can split T 1,N n,i into T 1,N n,i px 1 , . . . , x N q " T 11,N n,i px 1 , . . . , x N q `T 12,N n,i px 1 , . . . , x N q, with T 11,N n,i px 1 , . . . , x N q " n N d ż pR d q N " B v " B µ U ˆ1 N N ÿ "1 δ x ´y ˙px i ´yi q * N ź "1 ρpny q N ź "1 dy T 12,N n,i px 1 , . . . , x N q " n N d ż pR d q N " B v " B µ U ˆ1 N ÿ "i δ x ´y `1 N δ x i Ḃµ U ˆ1 N N ÿ "1 δ x ´y ˙px i ´yi q * N ź "1 ρpny q N ź "1
dy .

The rst term is treated as per (3.20) and (3.22). Namely, we have, for any t ě 0,

E "ˇˇT 11,N n,i pX 1 t , . . . , X N t q ´Bv " B µ U pµ t q ‰ pX i t q ˇˇ‰ ď ε n `εN . (3.24)
Then, by boundedness of B v rB µ U s for any p ě 1 and any t ě 0,

E "ˇˇT 11,N n,i pX 1 t , . . . , X N t q ´Bv " B µ U pµ t q ‰ pX i t q ˇˇp ‰ 1{p ď ε ppq n `εppq N . (3.25)
To handle the second term, we use uniform continuity of B v rB µ U s. Indeed, we have

|T 12,N n,i px 1 , . . . , x N q| ď ε N as W 2 2 ˆ1 N ÿ "i δ x ´y `1 N δ x i , 1 N N ÿ "1 δ x ´y ˙ď 1 N |y i | 2 ď C N ,
since, in T 12,N n,i px 1 , . . . , x N q, ny i belongs to the (compact) support of ρ. This says that, for any t ě 0,

E "ˇˇT 12,N n,i pX 1 t , . . . , X N t q ˇˇ‰ ď ε N . (3.26)
And, then, for any p ě 1 and any t ě 0,

E "ˇˇT 12,N n,i pX 1 t , . . . , X N t q ˇˇp ‰ 1{p ď ε ppq N . (3.27)
We nally handle T 2,N n,i . Following the proof of (3.27), we have, for any p ě 1 and any t ě 0,

E "ˇˇT 2,N n,i pX 1 t , . . . , X N t q ˇˇp ‰ 1{p ď nε ppq N , (3.28)
the additional n coming from the dierentiation of the regularization kernel.

Third step. In order to complete the proof, we apply Itô's formula to pu N n pX 1 t , . . . , X N t qq tě0 for given values of n and N . We obtain

0 " u N n `X1 t , . . . , X N t ˘´u N n `X1 0 , . . . , X N 0 ˘´N ÿ "1 ż t 0 B x u N n `X1 s , . . . , X N s ˘b s ds ´N ÿ "1 ż t 0 B x u N n `X1 s , . . . , X N s ˘σ s dW s ´1 2 N ÿ "1 ż t 0 Tr B 2 x u N n `X1 s , . . . , X N s ˘a s ( ds,
with a s :" σ s pσ s q : . To compare with the expected result, we take the dierence with

∆ N t " U pµ t q ´U pµ 0 q ´1 N N ÿ "1 ż t 0 B µ U pµ s qpX s qb s ds ´1 N N ÿ "1 ż t 0 B µ U pµ s qpX s qσ s dW s ´1 2N N ÿ "1 ż t 0 Tr B v " B µ U pµ s q ‰ pX s qa s ( ds.
E " |∆ N t | ‰ ď ε n `p1 `nqε N ,
the sequence pε q ě1 now depending on T . Letting N tend to 8, we deduce from Fatou's lemma and the law of large numbers that

sup 0ďtďT |∆ t | ď ε n , (3.30)
where

∆ t " U pµ t q ´U pµ 0 q ´ż t 0 E " B µ U pµ s qpX s qb s ‰ ds ´1 2
ż t 0 E " Tr B v " B µ U pµ s q ‰ pX s qa s ( ı ds.
Letting n tend 8 in (3.30), we deduce that ∆ " 0, which completes the proof.

3.4.

A sucient condition for partial C 2 regularity. The following is a sucient criterion for partial C 2 regularity used in the next section:

Theorem 3.6. Let U : P 2 pR d q Ñ R be a function such that its lifted version U :

L 2 pΩ, A, P; R d q Q ξ Þ Ñ U prξsq P R is once continuously Fréchet dierentiable. Assume also that for any continuously dierentiable map R Q λ Þ Ñ X λ P L 2 pΩ, A, P; R d q,
with the property that all the pX λ q λPR have the same distribution and that

|rd{dλsX λ | ď 1 (in L 8 ), the mapping R Q λ Þ Ñ DUpX λ q ¨χ " E " B µ U prX λ sqpX λ qχ ‰ P R (3.31)
is continuously dierentiable for any χ P L 2 pΩ, A, P; R d q. Moreover assume that the derivative of the mapping R Q λ Þ Ñ DUpX λ q ¨χ at λ " 0 depends on the family pX λ q λPR only through the value of X 0 and of rd{dλs |λ"0 X λ (see footnote 10 below for more details), so that we can denote

B 2 ζ,χ UpXq :" d dλ |λ"0 " DUpX λ q ¨χ‰ ,
whenever X :" X 0 and ζ :" rd{dλs |λ"0 X λ . Finally, assume that there exist a constant C and an exponent α ě 0 such that, for any X, χ and ζ in L 2 pΩ, A, P; R d q, with |ζ| ď 1 (in L 8 ), it holds (with Φ α as in pH1q and in particular satisfying (2.13)):

piq |DUpXq ¨χ| `|B 2 ζ,χ U pXq| ď C}χ} 2 , piiq |DUpXq ¨χ ´DUpX 1 q ¨χ| `|B 2 ζ,χ UpXq ´B2 ζ,χ UpX 1 q| ď CΦ α pX, X 1 q}χ} 2 .
Then U is partially C 2 and satises for any compact subset K Ă P 2 pR d q:

sup µPK "ż R d ˇˇB µ U pµqpvq ˇˇ2 dµpvq `żR d ˇˇB v " B µ U pµq ‰ pvq ˇˇ2 dµpvq  ă 8,
so that the chain rule applies to an Itô process satisfying (3.5).

Remark 3.7. The thrust of Theorem 3.6 is to study the smoothness of the mapping v Þ Ñ B µ U pµqpvq independently of the smoothness in the direction µ by restricting the `test' random variables pX λ q λPR to an identically distributed family. One of the issue in the proof is precisely to construct such a family of test random variables.

Proof. In the proof, we use quite often the following result, which is a renement of

Q v Þ Ñ V pµqpvqq µ of Borel functions
from R d into R d indexed by elements µ P P 2 pR d q such that, for any µ P P 2 pR d q, the mapping

R d Q v Þ Ñ V pµqpvq P R d belongs to L 2 pµ, R d ; R d q.
Assume also that there exist a constant C and an exponent α such that, for any µ P P 2 pR d q and any ξ, ξ 1 P L 2 pΩ, A, P; R d q, such that ξ and ξ 1 have distribution µ, and

E " |V pµqpξq ´V pµqpξ 1 q| 2 ‰ 1{2 ď CE "`1 `|ξ| 2α `|ξ 1 | 2α `}ξ} 2α 2 ˘|ξ ´ξ1 | 2 ‰ 1{2 . (3.32)
Then, for any µ P P 2 pR d q, the mapping v Þ Ñ V pµqpvq admits a locally Lipschitz continuous version, that satises

|V pµqpvq ´V pµqpv 1 q| ď C " 1 `2 max `|v| 2α , |v 1 | 2α ˘`ˆż R d |x| 2 dµpxq ˙α 1{2 |v ´v1 |.
As a warm-up, we discuss what Proposition 3.8 says in the framework of Theorem 3.6. Representing DUpXq ¨χ as ErB µ U prXsqpXqχs, we can write (choosing X " ξ and X 1 " ξ 1 , with rξs " rξ 1 s " µ, in part (ii) of the statement of Theorem 3.6)

ˇˇE "`B µ U pµqpξ 1 q ´Bµ U pµqpξq ˘χ‰ˇď CE "`1 `|ξ| 2α `|ξ 1 | 2α `}ξ} α 2 ˘|ξ ´ξ1 | 2 ‰ 1{2 E " |χ| 2 ‰ 1{2 .
This says that, for any µ P P 2 pR d q, we can nd a locally Lipschitz continuous version of the mapping R d Q v Þ Ñ B µ U pµqpvq, the local Lipschitz constant being at most of αpolynomial growth, uniformly with respect to µ in W 2 -balls. In addition, Proposition 3.8 gives us a bit more. Consider a sequence pµ n q ně0 with values in P 2 pR d q such that µ n Ñ µ in the 2-Wasserstein distance. Then, the functions

pR d Q v Þ Ñ B µ U pµ n qpvqq ně0
are uniformly continuous on compact sets. Moreover, we notice, by Markov inequality that Pp|ξ n | ě 2}ξ n } 2 q ď 1{4, so that 3 4 inf

|v|ď2}ξn} 2 |B µ U pµ n qpvq| ď E " 1 t|ξn|ď2}ξn} 2 u |B µ U pµ n qpξ n q| 2 ‰ 1{2 ď E " |B µ U pµ n qpξ n q| 2 ‰ 1{2 ď C, (3.33)
where ξ n has distribution µ n , the last inequality following from (i) in the statement of Theorem 3.6. This says that the family pinf |v|ď2}ξn} 2 |B µ U pµ n qpvq|q ně0 is bounded. As the sequence p}ξ n } 2 q ně0 is bounded and the mappings pR d Q v Þ Ñ B µ U pµ n qpvqq ně0 are uniformly locally Lipschitz continuous, the sequence p|B µ U pµ n qp0q|q ně0 is also bounded. Therefore, the family pR d Q v Þ Ñ B µ U pµ n qpvqq ně0 is relatively compact for the topology of uniform convergence on compact subsets. Passing to the limit 11 (up to a subsequence) into the relationship

DUpξ n q ¨χ " E " B µ U pµ n qpξ n qχ ‰ ,
we deduce, by identication, that the limit of B µ U pµ n q must coincide with B µ U pµq on the support of µ. This says that the function P 2 pR d q ˆRd Q pµ, vq Þ Ñ B µ U pµqpvq is (jointly) continuous at any point pµ, vq such that v P Supppµq. Moreover, by point (i)

in the statement of Theorem 3.6, we have

ş R d |B µ U pµqpvq| 2 dµpvq ď C, for a constant C independent of
µ, which is the rst part in the condition (3.6) for applying the chain rule to partially C 2 functions.

To complete the proof we have two main steps. The rst one uses a new mollication argument. The second consists in a coupling lemma, which permits to choose relevant versions of the random variables along which the dierentiation is performed.

First step. Given a distribution µ and a random variable ξ with distribution µ, we introduce the convoluted version µ n of µ:

µ n " µ ‹ N d p0, 1
n I d q, n denoting an integer larger than 1 and N d p0, p1{nqI d q denoting the d-dimensional Gaussian distribution with covariance matrix p1{nqI d , where I d is the identity matrix of dimension d. Then, we can dene the mapping

V n pµ, vq " ż R d B µ U `µn ˘pv ´uqn d{2 ρ `n1{2 uqdu, (3.34)
where ρ stands for the standard d-dimensional Gaussian kernel. The mapping V n is the convolution of B µ U pµ n qp¨q with the measure N d p0, p1{nqI d q.

By the warm-up, the sequence pB µ U pµ n qp0qq ně1 is bounded and the functions pB µ U pµ n q : R d Q v Þ Ñ B µ U pµ n qpvq P R d q ně1 are locally Lipschitz, the Lipschitz constant being at most of α-polynomial growth, uniformly in n ě 1. In particular, the sequence of functions pV n pµ, ¨qq ně0 is relatively compact for the topology of uniform convergence on compact subsets. Any limit must coincide with B µ U pµq at points v in the support of µ or, put it dierently, any limit provides a version of B µ U pµq which is locally Lipschitz continuous, the Lipschitz constant being at most of α-polynomial growth, uniformly in µ in bounded subsets of P 2 pR d q. When µ has full support, the sequence pV n pµ, ¨qq ně0 converges to the unique continuous version of B µ U pµq, the convergence being uniform on compact subsets. Letting ξ n " ξ `n´1{2 G, where G is an N d p0, I d q Gaussian variable independent of ξ, we then observe that, for any R d -valued square integrable random variable χ such that the pair pξ, χq is independent of G,

DUpξ n q ¨χ " E " B µ U `µn ˘`ξ n ˘χ‰ " E "ˆż R d B µ U `µn qpξ ´uqn d{2 ρpn 1{2 uqdu ˙χ " E " V n pµ, ξqχ ‰ , (3.35)
where V n pµ, ξq is viewed as a row vector. We note that the mapping

R d Q v Þ Ñ V n pµ, vq
is dierentiable with respect to v (this was not the case for the original mapping

R d Q v Þ Ñ B µ U pµqpvq at this stage of the proof ).
Second step. We construct now, independently of the measure µ considered above, a family pY λ q λPR that is dierentiable with respect to λ in L 2 pΩ, A, P; Rq but which is, at the same time, invariant in law, all the Y λ , for λ P R, being uniformly distributed on r´π{2, π{2s. The strategy consists in starting with the uniform distribution:

Given two independent N p0, 1q random variables Z and Z 1 , we let, for any λ P R,

Z λ " cospλqZ `sinpλqZ 1 , Z 1,λ " ´sinpλqZ `cospλqZ 1 ,
so that pZ λ , Z 1,λ q has the same law as pZ, Z 1 q (because of the invariance of the Gaussian distribution by rotation). We then let

Y λ " arcsin `Zλ a pZ λ q 2 `pZ 1,λ q 2 ˘" arcsin `Zλ a Z 2 `pZ 1 q 2 ˘.
It is easy to check that Y λ has a uniform distribution on r´π{2, π{2s for any λ P R.

Pathwise, the mapping R Q λ Þ Ñ Y λ is dierentiable at any λ such that Z 1,λ " 0.
Noticing that rd{dλsZ λ " Z 1,λ (pathwise), we get:

d dλ Y λ " Z 1,λ a Z 2 `pZ 1 q 2 ´1 ´pZ λ q 2 pZ λ q 2 `pZ 1,λ q 2
¯´1{2 " sign `Z1,λ ˘.

On the event tZ 1,0 " 0u " tZ 1 " 0u, which is of probability 1, the set of λ's such that Z 1,λ " 0 is locally nite. The above derivative being bounded by 1, this says that, pathwise, the mapping

R Q λ Þ Ñ Y λ is 1-Lipschitz continuous.
Therefore, the random variables pY λ ´Y 0 q{λ, λ " 0, are bounded by 1. Moreover, still on the event tZ 1 " 0u, the above computation shows that

lim λÑ0 Y λ ´Y 0 λ " sign `Z1 ˘. (3.36)
Therefore, by Lebesgue's dominated convergence theorem, the mapping R Q λ Þ Ñ Y λ P L 2 pΩ, A, P; Rq is dierentiable at λ " 0 with signpZ 1 q as its derivative. In the sequel, we will denote Y 0 by Y . Actually, by a rotation argument, dierentiability holds at any λ P R,

with rd{dλsY λ " signpZ 1,λ q. It is then clear that R Q λ Þ Ñ signpZ 1,λ q P L 2 pΩ, A, P; R d q is continuous. In- deed, the path R Q λ Þ Ñ Z 1,λ is continuous.
Composition by the function sign preserves continuity since, for any λ P R, the set of zero points of Z 1,λ is of zero probability.

Third step. Assume now that µ denotes a given distribution as in the rst step. We then choose a random variable ξ with µ as distribution, ξ being independent of the pair pZ, Z 1 q. Given the same pY λ q λPR as above and some parameter δ ą 0, we let @λ P R, ξ λ " pδ ˆY λ qe `ξ, where e is an arbitrary deterministic unitary vector in R d . (We omit the dependence upon δ in the notation ξ λ .) Then, we know that the mapping R Q λ Þ Ñ ξ λ is continuously dierentiable in L 2 pΩ, A, P; R d q, with d dλ |λ"0 ξ λ " pδ ˆsignpZ 1 qqe.

Going back to (3.35), we get for another random variable χ P L 2 pΩ, A, P; R d q, with pξ, χ, Z, Z 1 q independent of G,

DU `ξλ `1 ? n G ˘¨χ " E " V n `rξ λ s, ξ λ ˘χ‰ ,
where V n pµ, vq is seen as a row vector. As the mapping R Q λ Þ Ñ ξ λ is continuously dierentiable in L 2 pΩ, A, P; R d q and since all the random variables pξ λ q λPR have the same distribution, we deduce that (for pξ, χ, Z, Z 1 q independent of G)

B 2 signpZ 1 qe,χ U `ξ `δY e `1 ? n G ˘" d dλ |λ"0 " DU `ξλ{δ `1 ? n G ˘¨χ ‰ " 1 δ d dλ |λ"0 " DU `ξλ `1 ? n G ˘¨χ ‰ " E " Tr B v V n `rξ `δY es, ξ `δY e ˘`psignpZ 1 qχq b e ˘(‰ .
Noticing that the random variable |signpZ 1 q| is equal to 1 almost surely, we can replace χ by signpZ 1 qχ (recall that |χ| must be less than 1) with pξ, χq independent of pZ, Z 1 q, so that

B 2 signpZ 1 qe,signpZ 1 qχ U `ξ `δY e `1 ? n G ˘" E " Tr B v V n `rξ `δY es, ξ `δY e ˘`χ b e ˘(‰ .
Finally, we let

W n,δ pµ, vq " ż R B v V n `µ ‹ p δ , v `δue ˘ppuqdu, (3.37)
where p is the uniform density on r´π{2, π{2s and p δ p¨q " pp¨{δq{δ is the uniform density on r´δπ{2, δπ{2s. Moreover µ‹p δ is an abbreviated notation for denoting the convolution of µ with the uniform distribution on the segment r´pδπ{2qe, pδπ{2qes. Since the pair pξ, χq is independent of pZ, Z 1 q, we end up with the duality formula:

B 2 signpZ 1 qe,signpZ 1 qχ U `ξ `δY e `1 ? n G ˘" E " Tr W n,δ pµ, ξq `χ b e ˘(‰ . (3.38)
By the smoothness assumption on B 2 ζ,χ U (see (ii) in the statement of Theorem 3.6), we deduce that, for another ξ 1 , with distribution µ as well, such that the triple pξ, ξ 1 , χq is independent of pZ, Z 1 q and the 5-tuple pξ,

ξ 1 , χ, Z, Z 1 q is independent of G, ˇˇE " Tr `Wn,δ pµ, ξq ´Wn,δ pµ, ξ 1 q ˘`χ b e ˘(‰ˇď CE " `1 `|ξ| 2α `|ξ 1 | 2α `|δY | 2α `| 1 ? n G| 2α `}ξ} 2α 2 ˘|ξ ´ξ1 | 2 ı 1{2 E " |χ| 2 ‰ 1{2 ď CE "`1 `|ξ| 2α `|ξ 1 | 2α `}ξ} 2α 2 ˘|ξ ´ξ1 | 2 ‰ 1{2 E " |χ| 2 ‰ 1{2 , (3.39)
where we used the independence of pξ, ξ 1 q and pZ, Z 1 q to pass from the second to the third line, the value of C varying from the second to the third line (but remaining independent of δ and n, when δ is taken in a bounded set). The above is true for any σpξ, ξ 1 q-measurable χ P L 2 pΩ, A, Pq. We deduce that, for any other e

1 P R d with |e 1 | " 1, E "ˇˇT r `Wn,δ pµ, ξq ´Wn,δ pµ, ξ 1 q ˘`e 1 b e ˘(ˇˇ2‰ ď CE "`1 `|ξ| 2α `|ξ 1 | 2α `}ξ} 2α 2 ˘|ξ ´ξ1 | 2 ‰ 1{2 . By Proposition 3.8, this says that R d Q v Þ Ñ TrtpW n,δ pµ, vqqpe 1 b equ has a locally
Lipschitz version, the local Lipschitz constant on a ball of center 0 and radius γ is less than Cp1 `γα q, the constant C being uniform with respect to ξ in L 2 balls.

Fourth step. From (3.34) and (3.37), we know that

W n,δ pµ, vq " ż R B v V n `µ ‹ p δ , v `δue ˘ppuqdu " n pd`1q{2 ż RˆR d B µ U `µ ‹ p δ ‹ N d p0, 1 n I d q, w `δue ˘ppuqρ 1 pn 1{2 pv ´wqqdudw.
Since µ ‹ N d p0, p1{nqI d has full support, we know from the warm-up that B µ U pµ ‹ p δ ‹ N d p0, p1{nqI d q, ¨q converges towards B µ U pµ ‹ N d p0, p1{nqI d q, ¨q as δ tends to 0, uniformly on compact subsets. We deduce that, as δ tends to 0, W n,δ pµ, vq converges to

W n pµ, vq " n pd`1q{2 ż R d B µ U `µ ‹ N p0, 1 n I d q, w ˘ρ1 `n1{2 pv ´wq ˘dw " B v ˆnd{2 ż R B µ U `µ ‹ N p0, 1 n I d q, w ˘ρpn 1{2 pv ´wqqdw ˙" B v V n pµ, vq.
Therefore, we deduce that the mappings

pR d Q v Þ Ñ TrtpB v V n pµ, vqqpe 1 b equq ně1 are locally Lipschitz continuous, uniformly in µ (the local Lipschitz constant being at most of α-polynomial growth). Since B v V n pµ, vq is independent of e, this says that the mappings pR d Q v Þ Ñ B v V n pµ
@n ě 1, inf |v|ď2|ξ| 2 |B v V n pµ, vq| ď C, (3.41)
where we used the same argument as in (3.33). This says that the sequence of mappings

pR d Q v Þ Ñ B v V n pµ
, vqq ně1 is relatively compact for the topology of uniform convergence. By the warm-up, the sequence of functions

pR d Q v Þ Ñ pV n pµ, vq, B v V n pµ, vqqq ně1 is
relatively compact for the topology of uniform convergence. As any limit of the sequence

pR d Q v Þ Ñ V n pµ, vqq ně1 provides a version of B µ U pµq, we deduce that there exists a version of B µ U pµq : R d Q v Þ Ñ B µ U
pµqpvq which is continuously dierentiable with respect to v. By (3.40), we deduce that, for any µ P P 2 pR d q and any ξ P L 2 pΩ, A, P; R d q with µ as distribution, Er|B v rB µ U pµqspξq| 2 s ď C, for a constant C independent of µ. Moreover, passing to the limit in (3.38) (rst on δ and then on n), we get

B 2 signpZ 1 qe,signpZ 1 qχ Upξq " E " Tr `Bv rB µ U pµqspξq ˘`χ b e ˘(‰ .
(3.42)

Combining the above identity and point (i) in the statement of Theorem 3.6, we recover

the fact that ş R d |B v rB µ U pµqspvq| 2 dµpvq ď C, for a constant C independent of µ, which is
a required condition for applying the chain rule.

Last step. We have just found, for any µ P P 2 pR d q, a version of the mapping B µ U pµq that is dierentiable in the variable v, with B v rB µ U pµqs denoting its derivative. In order to complete the proof, it remains to prove that the resulting mapping

P 2 pR d q ˆRd Q pµ, vq Þ Ñ B v rB µ U
pµqspvq is continuous in the joint variable pµ, vq at any point v P Supppµq. We already know that it is locally Lipschitz continuous with respect to v, the local Lipschitz constant being at most of α-polynomial growth, uniformly in µ in sets of probability measures with uniformly bounded second-order moments. For a sequence pµ n q ně1 in P 2 pR d q converging for the 2-Wasserstein distance to some µ, we deduce from the local Lipschitz property and by the same argument as in (3.41) that the sequence of functions pR Q v Þ Ñ B v rB µ U pµ n qspvqq ně1 is relatively compact for the topology of uniform convergence on compact subsets. By means of the bound sup ně1 Er|B v rB µ U pµ n qspξ n q| 2 s ď C, with ξ n " µ n , it is quite easy to pass to the limit in the right-hand side of (3.42). By (ii) in the statement of the theorem, we can also pass to the limit in the left-hand side. Equation (3.42) then permits to identify any limit with B v rB µ U pµqs on the support of µ. Since the mappings pB v rB µ U pµ n qsq ně1 are uniformly continuous on compact subsets, we deduce that, for an additional sequence pv n q ně1 , with values in R d , that converges to some v P Supppµq, the sequence pB v rB µ U pµ n qspv n qq ně1 converges, up to a subsequence, to B v rB µ U pµqspvq. Now, by relative compactness of the sequence pR Q v Þ Ñ B v rB µ U pµ n qspvqq ně1 , the sequence pB v rB µ U pµ n qspv n qq ně1 is bounded. By a standard compactness argument, the sequence pB v rB µ U pµ n qspv n qq ně1 must be convergent with B v rB µ U pµqspvq as limit.

3.5. Proof of Theorem 2.8. In order to prove Theorem 2.8, we rst need an extension of the chain rule to functions that depend on time, space and measure: Proposition 3.9. Consider an Itô process pX t q tPr0,T s driven by pb t , σ t q tPr0,T s satisfying (3.5) and a function V : r0, T s ˆRd ˆP2 pR d q Ñ R m belonging to Ť βě0 D β , see Denition 2.6. Then, P almost surely, for any t P r0, T s,

V `t, X t , rX t s ˘´V `0, X 0 , rX 0 s " ż t 0 ´Bt V pr, X r , rX r sq `Bx V pr, X r , rX r sqb r `p E " B µ V `r, X r , rX r s ˘`xX r y ˘xb r y ‰ ¯dr `1 2 ż t 0 ´Tr " B 2 xx V `r, X r , rX r s ˘`σ r σ : r ˘‰ `p E " Tr " B v " B µ V ‰`r , X r , rX r s ˘`xX r y ˘xσ r pσ r q : y ˘‰‰ ¯dr `ż t 0 B x V `r, X r , rX r s ˘σr dW r .
Remark 3.10. In comparison with Theorem 3.3, the formula is stated here in terms of the expectation Ê on the auxiliary probability space p Ω, Â, Pq. The goal is to distinguish the random variables X r , b r and σ r σ : r , observed on the physical space pΩ, A, Pq, from the random variables xX r y, xb r y and xσ r σ : r y that are used to express the derivatives in the direction µ.

Proof. The proof is similar to that outlined in Subsection 2.3. As in the proof of Theorem 3.3, we can assume that the processes pb t q tPr0,T s and pσ t q tPr0,T s are bounded.

Given s P r0, T s and h ą 0 such that s `h P r0, T s, we then expand

V `s `h, X s`h , rX s`h s ˘´V `s, X s , rX s s " V `s `h, X s`h , rX s`h s ˘´V `s `h, X s`h , rX s s V `s `h, X s`h , rX s s ˘´V `s, X s , rX s s ˘. (3.43)
Thanks to the regularity assumptions in pH1q and pH2q, we notice that, almost surely, the map P 2 pR d q Q µ Þ Ñ V ps `h, X s`h , µq satises the assumption of Theorem 3.6.

Therefore, we can write

V `s `h, X s`h , rX s`h s ˘´V `s `h, X s`h , rX s s " ż s`h s Ê" B µ V ps `h, X s`h , rX r sqpxX r yqxb r y ‰ dr `1 2 ż s`h s Ê" Tr `Bv " B µ V ps `h, X s`h , rX r sq ‰ pxX r yqxσ r σ : r y ˘‰dr. Recall that any versions of R d Q v Þ Ñ B µ V ps `h, X s`h , µqpvq and R d Q v Þ Ñ B v rB µ V ps h, X s`h ,
µqspvq may be used in the writing of the above formula. In particular, we can choose the versions of B µ V and B v rB µ V s that satisfy pH1q and pH2q. By the assumption we have on the regularity of B µ V and B v rB µ V s in the variable x, see pH1q and pH2q, and in the variable t, see (ii) in Denition (2.6), we deduce that there exists a sequence of non-negative random variables pε h q hą0 that tends to 0 in probability with h, such that

ˇˇˇV `s `h, X s`h , rX s`h s ˘´V `s `h, X s`h , rX s s ż s`h s Ê" B µ V pr, X r , rX r sqpxX r yqxb r y ‰ dr ´1 2 ż s`h s Ê" Tr `Bv " B µ V pr, X r , rX r sq ‰ pxX r yqxσ r σ : r y ˘‰dr ˇˇˇď hε h .
It must be noticed that the family pε h q hą0 may be chosen independently of s P r0, T s. The reason is that, thanks to pH1q and pH2q, for any continuous R d -valued path px t q tPr0,T s ,

lim δÑ0 sup r,sPr0,T s:|r´s|ďδ Ê" ˇˇB µ V `s, x s , rX r s ˘`xX r y ˘´B µ V `r, x r , rX r s ˘`xX r y ˘ˇı " 0,
with a similar result when B µ V is replaced by B v rB µ V s. By means of the standard Itô formula, the second dierence on the right hand side of (3.43) can be handled in a similar way, yielding a similar bound (for the relevant expansion) on an interval of length h. We then easily complete the proof by dividing any interval r0, ts Ă r0, T s into pieces of length less than h, applying the above bound on each piece of the subdivision and then by letting h tend to 0.

We now turn to

Proof. [Proof of Theorem 2.8.] The proof is a variant of the four-step-scheme used in [START_REF] Ma | Solving forward-backward stochastic dierential equations explicitly a four step scheme[END_REF]. We divide it into two steps.

First step. Given a solution U to (2.12) in the class Ť βě0 D β and given t P r0, T s and ξ P L 2 pΩ, F t , P; R d q, we build a solution to (2.3).

Letting

B σ

x U pt, x, µq " B x U pt, x, µqσ `x, U pt, x, µq, rpξ, U pt, ξ, µqqs

˘, with ξ " µ, we indeed claim that the McKean-Vlasov SDE dX s " b `Xs , U ps, X s , rX s sq, B σ x U ps, X s , rX s sq, " X s , U ps, X s , rX s sq ‰˘d s `σ`X s , U ps, X s , rX s sq, " X s , U ps, X s , rX s sq ‰˘d W s , X t " ξ, (3.44)
has a solution (the idea that we shall exploit in the proof being that the triplet process pX s , U ps, X s , rX s sq, B σ x U ps, X s , rX s sqq sPrt,T s solves the system (2.3)). The proof is not completely straightforward as B σ

x U is not Lipschitz continuous in the direction of the measure (see pH1q). In particular, we cannot apply Sznitman's result in [33], which relies on a contraction argument. Instead, we make use of Schauder's theorem, applying the same strategy as in [START_REF] Carmona | Mean-eld forward-backward stochastic dierential equations[END_REF].

The argument is as follows. Let Cprt, T s, P 2 pR d qq be the family of marginal measures pµ r q rPrt,T s with nite second-order moments such that the mapping rt, T s Q r Þ Ñ µ r P P 2 pR d q is continuous. For pµ r q rPrt,T s P Cprt, T s, P 2 pR d qq, we may solve

dX s " b `Xs , U ps, X s , µ s q, B σ x U ps, X s , µ s q, " X s , U ps, X s , µ s q ‰˘d s `σ`X s , U ps, X s , µ s q, " X s , U ps, X s , µ s q ‰˘d W s , X t " ξ.
By Sznitman's result, the above equation admits a unique solution, which we will denote by pX pµrq rPrt,T s s q sPrt,T s . We then consider the mapping

Φ : `µr ˘rPrt,T s Þ Ñ `"X pµrq rPrt,T s s ‰˘s Prt,T s ,
which maps Cprt, T s, P 2 pR d qq into itself. By standard stability arguments, it is quite clear that the mapping Φ is continuous, Cprt, T s, P 2 pR d qq being endowed with the supremum distance dppµ r q rPrt,T s , pμ r q rPrt,T s q :" sup rPrt,T s W 2 pµ r , μr q. Moreover, by boundedness of B x U and σ and by the Lipschitz property of U , we can nd a constant C (independent of the input pµ r q rPrt,T s ) such that, for any S P rt, T s

E t " sup sPrt,Ss |X pµrq rPrt,T s s | 4 ‰ 1{2 ď C ˆ1 `|ξ| 2 `ż S t ż R d |x| 2 dµ s pxqds ˙.
This proves that, when

@s P rt, T s, ż R d |x| 2 dµ s pxq ď Cp1 `}ξ} 2 2 q exp `Cps ´tq ˘, (3.45)
the same holds for Er|X pµrq rPrt,T s s | 2 s for all s P rt, T s. In such a case, we also have

E t " sup sPrt,Ss |X pµrq rPrt,T s s | 4 ‰ 1{2 ď C `1 `|ξ| 2 ˘`C `1 `}ξ} 2 2 ˘exppCT q.
so that, for any event A P A and any real R ą 0, Cauchy-Schwarz inequality yields

E " 1 A sup sPrt,Ss |X pµrq rPrt,T s s | 2 ‰ ď CE " E t r1 A s 1{2 ´`1 `|ξ| 2 ˘``1 `}ξ} 2 2 ˘exppCT q ¯ı, ď C ´`1 `R2 ˘``1 `}ξ} 2 2 ˘exppCT q ¯PpAq 1{2 `CE " 1 t|ξ|ěRu ´`1 `|ξ| 2 ˘``1 `}ξ} 2 2 ˘exppCT q ¯ı.
| 2 ‰ ď C 3{2 R ´4´`1 `R2 ˘``1 `}ξ} 2 2 ˘exppCT q ¯´`1 `}ξ} 2 2 ˘exppCT q ¯1{2 `CE " 1 t|ξ|ěRu ´`1 `|ξ| 2 ˘``1 `}ξ} 2 2 ˘exppCT q ¯ı. (3.46)
For a given s P rt, T s, we now denote by K s the subset of P 2 pR d q made of probability measures such that ş R d |x| 2 dµpxq is less than the right-hand side in (3.45) and ş t|x|ąR 4 u |x| 2 dµpxq is less than the right-hand side in (3.46) for any R ą 0. It is easy to checked that K s is a compact subset of P 2 pR d q. Indeed, any sequence in K s is tight and admits a subsequence that converges in the weak sense. Using (3.46), the subsequence is square-uniformly integrable. Using Skorohod's representation theorem, we deduce that the sequence converges in the W 2 -Wasserstein sense. By Fatou's lemma, K s is closed. Below, we let K " tpµ r q rPrt,T s P Cprt, T s, P 2 pR d qq : @r P rt, T s, µ r P K r u.

Notice now that, under (3.45), we have, for all s, s 1 P rt, T s,

E " |X pµrq rPrt,T s s 1 ´Xpµrq rPrt,T s s | 2 ‰ ď C 1 |s 1 ´s|,
for a constant C 1 depending upon C, }ξ} 2 and T . This says that the map is rt,

T s Q s Þ Ñ X
pµrq rPrt,T s s P L 2 pΩ, A, P; R d q, is continuous, uniformly in pµ r q rPrt,T s P K. Using the Arzelà-Ascoli theorem, we deduce that the restriction of Φ to K has a relatively compact range. Since K is closed and convex, Schauder's theorem applies and (3.44) has a solution.

Second step. We consider another solution U 1 to (2.12) in the class Ť βě0 D β . With X a solution of (3.44), we can apply the chain rule to both pY s " U ps, X s , rX s sqq sPrt,T s and pY 1 s " U 1 ps, X s , rX s sqq sPrt,T s , the drift of X being square-integrable and σ being bounded.

Letting pZ s " B σ x U ps, X s , rX s sqq sPrt,T s , pZ 1 s " B x U 1 ps, X s , rX s sqσpX s , Y 1 s , rX s , Y 1 s sqq sPrt,T s , pθ s " pX s , Y s , Z s qq sPrt,T s , pθ 1 s " pX s , Y 1 s , Z
Y s ´Y 1 s " ż T s `f pθ r , rθ 0q r sq ´f pθ r , rθ p0q1 r sq ˘dr `ż T s ! B x U 1 pr, X r , rX r sq ´b`θ r , rθ p0q r s ˘´b `θ1 r , rθ p0q1 r s ˘Ē " B µ U 1 pr, X r , rX r sqpxX r yq ´b`x θ r y, rθ p0q r s ˘´b `xθ 1 r y, rθ p0q1 r s ˘¯ı) dr `1 2 ż T s ! Tr " B 2 xx U 1 `r, X r , rX r s ˘´`σ σ : ˘`θ r , rθ p0q r s ˘´`σ σ : ˘`θ 1 r , rθ p0q1 r s ˘¯ı `Ê " Tr " B v " B µ U 1 ‰`r , X r , rX r s ˘`xX r y σσ : ˘`xθ p0q r y, rθ p0q r s ˘´`σ σ : ˘`xθ p0q1 r y, rθ p0q1 r s ˘¯ıı) dr ´ż T s `Zr ´Bx U 1 pr, X r , rX r sqσpθ p0q1 r , rθ p0q1 r sq ˘dW r .
By using Assumptions pH0q(i) and pHσq on the coecients and Assumptions pH1q and pH2q that enter in the denition of D β , we deduce from stability estimates for BSDEs, in the spirit of [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF], that

E " |Y s ´Y 1 s | 2 ‰ `E ż T s |Z r ´Bx U 1 pr, X r , rX r sqσpθ p0q1 r , rθ p0q1 r sq| 2 dr ď CE ż T s |Y r ´Y 1 r | 2 dr `1 2 E ż T s |Z r ´Z1 r | 2 dr,
from which we get, by the boundedness of

B x U 1 , that E " |Y s ´Y 1 s | 2 ‰ `E ż T s |Z r ´Z1 r | 2 dr ď CE ż T s |Y r ´Y 1 r | 2 dr `1 2 E ż T s |Z r ´Z1 r | 2 dr.
We deduce that Y s " Y 1 s for any s P rt, T s, that is U pt, ξ, rξsq " U 1 pt, ξ, rξsq almost surely. When rξs has full support over R d , continuity of U and U ' yield U pt, x, rξsq " U 1 pt, x, rξsq for all x P R d . When the support of rξs is strictly included in R d , we can approximate ξ by a sequence pξ n q ně1 that converges to ξ in L 2 such that, for each n ě 1, ξ n has full support over R d . Passing to the limit in the relationship U pt, x, rξ n sq " U 1 pt, x, rξ n sq, we complete the proof.

Smoothness for small time horizons proof of Theorem 2.7

The purpose of this section is to prove that the mapping U given in Denition 2.1 satises the required smoothness property for applying the chain rule. Generally speaking, this is proved by showing the smoothness of the corresponding stochastic ows dened in (2.3) and (2.4). More precisely, we prove that the stochastic ows dened in (2.3) and (2.4) are dierentiable with respect to ξ, x and µ in the sense discussed in Section 3. This is not a straightforward generalization of the method used by Pardoux and Peng in [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF] in order to prove the smoothness of the ow generated by the solution of a classical backward stochastic dierential equation as we are facing here two additional diculties: First, the initial conditions live in non-Euclidean spaces, which requires some special care; second, the backward equation is fully coupled to the forward equation. In order to handle the full coupling between the forward and backward components, we shall assume that T is small. In particular, throughout the whole section, T is less than 1. In the following section, we shall give sucient conditions for extending the results from small to arbitrary large time horizons.

Below, Assumption pH2q is in force. We shall use quite intensively the following lemma, which is an adaptation of the stability estimates in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]: Lemma 4.1. For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p ě 0 such that, for T ď c p , for any t P r0, T s,

x P R d and ξ P L 2 pΩ, F t , P; R d q, }X t,ξ } S p ,t `}Y t,ξ } S p ,t `}Z t,ξ } H p ,t ď C p `1 `|ξ| `}ξ 1 } 2 ˘, }X t,x,rξs } S p `}Y t,x,rξs } S p `}Z t,x,rξs } H p ď C p `1 `|x| `}ξ} 2 ˘, (4.1)
and, for any

x 1 P R d and ξ 1 P L 2 pΩ, F t , P; R d q, }X t,ξ ´Xt,ξ 1 } S p ,t `}Y t,ξ ´Y t,ξ 1 } S p ,t `}Z t,ξ ´Zt,ξ 1 } H p ,t ď C p " |ξ ´ξ1 | `W2 `rξs, rξ 1 s ˘‰, }X t,x,rξs ´Xt,x 1 ,rξ 1 s } S p `}Y t,x,rξs ´Y t,x 1 ,rξ 1 s } S p `}Z t,x,ξ ´Zt,x 1 ,rξ 1 s } H p ď C p " |x ´x1 | `W2 `rξs, rξ 1 s ˘‰. (4.2)
In the statement above, the notation c p :" c p pLq emphasizes the fact that c p only depends on the Lipschitz constant L introduced in pH0q ´pH1q. The constant C p is allowed to depend on the other parameters appearing in pH0q ´pH2q, but there is no need to keep track of them for our purpose.

X s " η `ż s t B `r, θ r , x θr yq `ϑr , x θp0q r y ˘dr `ż s t Σ `r, θ p0q r , x θ0q r y ˘`ϑ p0q r , x θ0q r y ˘dW r , Y s " G `XT , x XT yq `XT , x XT y ˘`ż T s F `r, θ r , x θp0q r y ˘`ϑ r , x θp0q r y ˘dr ´ż T s Z r dW r , (4.3)
where η is an initial condition in L 2 pΩ, F t , P; R d q, θ " pX, Y, Zq and θ " p X, Ŷ , Ẑq are solutions of (2.3) or (2.4), ϑ " pX , Y, Zq denotes the unknowns in the above equation and θ " p X , Ŷ, Ẑq is an auxiliary process, which may be ϑ itself (in which case it is unknown).

The exponent p0q denotes the restriction of the processes to the two rst coordinates, as in (2.3) and (2.4). The processes X, X, X and X have the same dimension, the same being true for the processes Y , Ŷ , Y and Ŷ and for the processes Z, Ẑ, Z and Ẑ. In particular, the mappings B, Σ, F and G take values in Euclidean spaces of according dimensions. The symbol x¨y is used to denote the copy of the underlying random variable onto the probability space p Ω, Â, Pq. Although the role of the copy is rather vague at this stage of the paper, it indicates that the coecients may depend in a non-Markovian way of the various stochastic processes involved. Here is an example:

Example 4.2. As a typical example for the coecients B, Σ, F and G, we may think of the derivatives, with respect to some parameter λ, of the original coecients b, f , σ and g when computed along some triplet pθ λ " pX λ , Y λ , Z λ qq solving (2.1). As a typical example for the parametrization by λ, we may think of the parametrization with respect to the initial condition which is applied to the entire system.

The shape of the coecients B, Σ, F and G can then be derived by replacing b, f , σ and g by a generic continuously dierentiable Lipschitz function h : pR d ˆRm ˆRmˆd q P2 pR d ˆRm q Ñ R. Given such a generic h, we can indeed consider a process of the form ´h`θ λ r , rθ λ,p0q r s ˘¯rPrt,T s where R Q λ Þ Ñ pθ λ r q rPrt,T s P S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q is dierentiable with respect to λ, with (derivatives being taken in the aforementioned space) θ λ r |λ"0 " θ r , d dλ |λ"0 θ λ r " ϑ r , r P rt, T s, the process pϑ r q rPrt,T s taking its values in R d ˆRm ˆRmˆd (and, for the moment, having nothing to do with the solution of (4.3)). Then, it is easy to check that the mapping going back to the general case, we apply the same procedure: In order to specify the shape of B, Σ, F and G (together with the assumptions they satisfy), it suces to make explicit the generic form of a function H that may be B, Σ, F or G and to detail the assumptions it satises. Given square-integrable processes pV r q rPrt,T s and p Vr q rPrt,T s , pV r q rPrt,T s possibly matching pX r q rPrt,T s , pθ p0q r q rPrt,T s or pθ r q rPrt,T s , and similarly for p Vr q rPrt,T s , together with other square-integrable processes pV r q rPrt,T s and p Vr q rPrt,T s , pV r q rPrt,T s possibly matching pX r q rPrt,T s , pϑ p0q r q rPrt,T s or pϑ r q rPrt,T s , and similarly for p Vr q rPrt,T s , we thus assume that Hpr, V r , x V p0q r yq acts on pV r , Vp0q r q in the following way:

R Q λ Þ Ñ phpθ λ r ,
Hpr, V r , x V p0q r yqpV r , Vp0q r q " H pV r , x V p0q r yqpV r , Vp0q r q `Ha prq, (4.5)
where H a prq is square-integrable and H pV r , x V p0q r yq acts linearly on pV r , Vp0q r q in the following sense

H pV r , x V p0q r yqpV r , Vp0q r q " h pV r , x V p0q r yqV r `Ê " Ĥ pV r , x V p0q r yqx Vp0q r y ‰ . (4.6)
Here h and Ĥ are maps from R k ˆL2 p Ω, Â, P; R l q into R l 1 and from R k ˆL2 p Ω, Â, P; R l q into L 2 p Ω, Â, P; R l 2 q respectively, for suitable k, l, l 1 and l 2 . Moreover, there exist three constants C, K, α ě 0 and a function Φ α : rL 2 pΩ, A, P; R l qs 2 Ñ R `, continuous at any point of the diagonal, such that, for w, w 1 P R k and V p0q ,

V p0q1 P L 2 pΩ, A, P; R d q, ˇˇh pw, x V p0q yq ˇˇ`Ê " | Ĥ pw, x V p0q yq| 2 ‰ 1{2 ď K, (4.7)
| Ĥ pw, x V p0q yq| ď C `1 `|x V p0q y| α`1 `} V p0q } α`1 2 ˘, (4.8) ˇˇh pw, x V p0q yq ´h pw 1 , x V p0q1 yq ˇˇ2 `Ê " | Ĥ pw, x V p0q yq ´Ĥ pw 1 , x V p0q1 yq| 2 ‰ ď C ! |w ´w1 | 2 `Φ2 α p V p0q , V p0q1 q ) , (4.9)
with the condition that, when V p0q " V p0q1 ,

Φ α p V p0q , V p0q1 q ď CE "`1 `| V p0q | 2α `| V p0q1 | 2α `} V p0q } 2α 2 ˘| V p0q ´V p0q1 | 2 ‰ 1{2 . (4.10)
We shall also require the additional assumption:

For any V p0q , the family `| Ĥ pw, x V p0q yq| 2 ˘wPR k is uniformly integrable. pw, µq, so that the derivatives are bounded, in L 8 in the direction w and in L 2 in the direction µ. Importantly (and as already suggested), the constant K corresponds to L in pH1q.

(2) Equation (4.8) expresses the fact that, for any pw, µq, v Þ Ñ B µ hpw, µqpvq admits a version that is at most of polynomial growth (in v) under pH1q (see the proof right below).

(3) Equation (4.9) says that the derivatives in the direction w and in the direction of the measure are continuous (in a suitable sense). Except when α " 0, derivatives may not be Lipschitz continuous in the direction of the measure, which is a crucial relaxation for our purpose.

(4) Condition (4.11) expresses the fact that, for χ P L 2 , the family pB µ hpw, µqpχqq wPR k must be uniformly square-integrable.

The existence of a version of v Þ Ñ B µ hpw, µqpvq that is at most of polynomial growth can be proved as follows. When h is understood as one of the coecients b, σ, f or g, we know that, under pH1q, B µ h (which might be identied with a Fréchet derivative) satises, for two random variables χ and χ 1 , with the same distribution µ,

E "ˇˇB µ hpw, µqpχq ´Bµ hpw, µqpχ 1 q ˇˇ2 ‰ 1{2 ď CE "`1 `|χ| 2α `|χ 1 | 2α `}χ} 2α 2 ˘|χ ´χ1 | 2 ‰ 1{2 , (4.12)
which implies that the mapping v Þ Ñ B µ hpw, µqpvq is locally Lipschitz continuous, see Proposition 3.8. More precisely, for a random variable χ with µ as distribution, ˇˇB µ hpw, µqpvq ´Bµ hpw, µqpv 1 q ˇˇď C `1 `|v| α `|v

w P R k , ˇˇB µ hpw, µqpvq ˇˇď C `C`1 `|v| α `}χ} α 2 ˘`|v| `}χ} 2 ď C `1 `|v| α`1 `}χ} α`1 2 ˘, (4.14)
which completes the proof of the polynomial growth property.

Remark 4.4. The reader may wonder about the sharpness of the bound (4.14). Indeed, when specialized to the case α " 0 and h independent of w, (4.14) provides just a linear growth bound for the derivative Denition 4.5. Given triplets pθ r " pX r , Y r , Z r qq rPrt,T s and p θr " p Xr , Ŷr , Ẑr qq rPrt,T s of the same form as above, we say that a subset J of L 2 pΩ ˆΩ, A b Â, P b P; R `q is admissible for pθ, θq if (i) for any r P rt, T s, for H matching B, Σ, F or G and pV r , V p0q r q matching pX r , Xr q, pθ r , θp0q r q or pθ p0q r , θp0q r q, there exists Λ P J such that Êr| Ĥ pr, V r , x

R l Q v Þ Ñ B µ hpµqpvq of a Lipschitz-continuous function h : P 2 pR l q Q µ Þ Ñ hpµq (the
V p0q r yq| 2 s 1{2 ď Λ; (ii) any Λ in J satises Pp ÊpΛ 2 q 1{2 ď Kq " 1.
Notations. Throughout 4.1.2, J is an admissible class for both pθ, θq and p θ, θq. For a real γ ě T , an integer p ě 1, a real C ą 0, a triplet ϑ " pX s , Y s , Z s q sPrt,T s with values in S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q and a pair of random variables pX, χq with values in a Euclidean space, we let

M p M pϑq :" M " sup sPrt,T s `|X s | p `γ1{2 |Y s | p ˘`γ 1{2 ˆż T t |Z s | 2 ds ˙p{2  , N p,C M pX, χq (4.16)
:

" esssup ΛPJ M " Ê" ! Λ ^"C `1 `Ê t " |xXy| 2α`2 ‰ 1{2 `}X} α`1 2 ˘ı) Êt " |xχy| 2 ‰ 1{2  p  ,
with the convention that M can be E t or E (in the latter case esssup is just a sup). Note that M p M pϑq depends on γ, t and T . We shall omit this dependence in the notation M p M pϑq. With these notations, we shall write M p M pϑ p0q q for M p M pX , Y, 0q. Similarly, we shall not specify the dependence upon t in the notation N p,C M pX, χq. We deduce that Lemma 4.7. For any p ě 1, there exist two constants Γ p :" Γ p pKq ě 1 and C ą 0 (C independent of p), such that, for T ď γ ď 1{Γ p and for any solution ϑ to a system of the same type as (4.15), it holds

M 2p Et pϑq ď Γ p " |η| 2p `γ1{2 M 2p Et p θq `R2p a `γ1{2 ! N 2p,C Et `X T , XT ˘`sup sPrt,T s N 2p,C Et ´θ p0q s , `M2 Et p θp0q q ˘1{2 ¯) . (4.18)
In particular (redening the value of Γ p if necessary),

M 2p Et pϑq ď Γ p " `|η| `}η} 2 ˘2p `R2p a `E" R 2 a ‰ p `γ1{2 ´M2p Et p θq `"M 2 E p θp0q q ‰ p `"M 2 E p θp0q q ‰ p ¯ı. (4.19)
Proof. We make use of standard results for solutions of an FBSDE. We can indeed start with the trivial case when the coecients B , Σ and F are null and Ĝ is also null (see (4.5) and (4.6) for the notations). Then, (4.15) reads as a system driven by the linear part g that appears in the decomposition (4.6) of G plus a remainder involving B a , Σ a , F a and G a . Without any McKean-Vlasov interaction, (4.18) follows from stability estimates for standard linear FBSDEs. For instance, following Delarue [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF], we get that, for any p ě 1, we can nd Γ p :" Γ p pKq ą 0 (the value of which is allowed to increase from line to line), such that for γ ď 1{Γ p , (4.18) holds, but with a simpler right-hand side just consisting of Γ p r|η| 2p `R2p a s. In the case when B , Σ , F are non-zero, we view them, when taken along the values of p θ, θp0q , θ, θp0q q as parts of B a , Σ a and F a . Similarly, we can see Ĝ , when taken along the values of pX T , x XT yq, as a part of G a . We are thus led back to the previous case, but with a generalized version of the remainder term R a . In order to complete the proof, it suces to bound this remainder in L 2p . The analysis of the remainder may be split into three pieces: One rst term involves b , σ and f ; another one involves B , Σ , F and Ĝ ; the last one involves B a , Σ a , F a and G a and corresponds to the original R a . As a nal bound, we get

M 2p

Et pϑq ď Γ p |η| 2p (4.20)

` Of course, we can use the same kind of argument for the rst term in (4.22) and get N 2p,C Et p XT , XT q as resulting bound.

The second claim follows from Lemma 4.6.

In particular, we have the following useful result for systems of the form (4.3) obtained by considering ϑ " θ and θ " θ in (4. [START_REF] Guéant | Mean eld games and applications[END_REF]) and setting γ small enough.

Corollary 4.8. For any p ě 1, there exists a constant Γ p :" Γ p pKq ě 1 such that, for T ď γ ď 1{Γ p and for any solution ϑ to a system of the same type as (4.3), it holds

M 2p Et pϑq ď Γ p " `η `}η} 2 ˘2p `R2p a `E" R 2 a ‰ p `γ1{2 " M 2 E p θp0q q ‰ p ı . (4.23)
When ϑ " θ, we have (modifying the constant Γ p if necessary):

M 2p Et pϑq ď Γ p " `η `}η} 2 ˘2p `R2p a `E" R 2 a ‰ p ı . (4.24)
Proof. Inequality (4.23) directly follows from (4.19). To get (4.24), we choose p " 1 and then take the expectation. For γ small enough, we obtain M 2 E pϑq ď Γ 1 p}η} 2 2 `ErR 2 a sq (up to a new value for Γ 1 ). Plugging the bound into (4.23), we deduce that (4.24) holds. 4.1.3. Stability estimates. The next step is to compare two solutions of (4.15) ϑ and ϑ 1 driven by two dierent sets of inputs p θ, θ, θ, θ, θ, θq and p θ1 , θ1 , θ1 , θ1 , θ1 , θ1 q but with the same starting point η. Throughout 4.1.3, J is an admissible class for pθ, θq and p θ, θq.

Given an integer p ě 1, dene similar notations to (4.10) and (4.16) (but without γ 1{2 in front of the terms in Y):

Φ α `θ p0q , θp0q1 ˘:" sup sPrt,T s Φ α `θ p0q s , θp0q1 s ˘( M2p pϑ, θq :" sup sPrt,T s ! E t " |X s | 2p `|Y s | 2p ‰ `} Xs } 2p 2 `} Ŷs } 2p 2 ) `}Z} 2p H 2p ,t , M2p `pϑ, θq, pϑ 1 , θ1 q ˘:" M2p `ϑ ´ϑ1 , θ ´θ 1 ˘`Φ 2p α `θ p0q , θp0q1 ˘, M2p ϑ :" M2p `ϑ, ϑ ˘, M2p ϑ, ϑ 1 :" M2p `pϑ, ϑq, pϑ 1 , ϑ 1 q ˘, (4.25)
and denote by ∆R 2p a the quantity (recall (4.17) for the denition of R 2p a ):

∆R 2p a :" E t " γ 1{2 ˇˇG a pT q ´G1 a pT q ˇˇ2p `ˆż T t ˇˇpB a ´B1 a , F a ´F 1 a qpsq ˇˇds ˙2p `ˆż T t ˇˇpΣ a ´Σ1 a qpsq ˇˇ2 ds ˙p . (4.26) (The notations B 1 a , F 1 a , Σ 1 a and G 1
a refer to the fact, along the processes labelled with a `prime', the remainders in the decomposition of the coecients may be dierent.) Then, we have Lemma 4.9. For any p ě 1, there exist three constants C (independent of p), Γ p :"

Γ p pKq ě 1 and C p ą 0, such that, for T ď γ ď 1{Γ p , M 2p Et `ϑ ´ϑ1 ˘ď Γ p γ 1{2 ! M 2p Et `θ ´θ 1 ˘`N 2p,C Et `X T , XT ´X 1 T sup sPrt,T s N 2p,C Et ´θ p0q s , `M2 Et p θp0q ´θ p0q1 q ˘1{2 ¯) (4.27) `Cp "´M 4p pϑ 1 , θ1 q `M 4p p θ1 , θ1 q ¯1{2 ˆ!1 ^´M 4p `pθ, θq, pθ 1 , θ1 q ˘`M 4p `pθ , θq, p θ1 , θ1 q ˘¯) 1{2 `∆R 2p a ı .
In particular, choosing p " 1 and taking expectation, we have, for some constant Γ 1 :" Γ 1 pKq such that T ď γ ď 1{Γ 1 and for some C 1 ą 0,

M 2 E `ϑ ´ϑ1 ˘ď Γ 1 γ 1{2 ! M 2 E `θ ´θ 1 ˘`M 2 E `θ p0q ´θ p0q1 ˘`M 2 E `θ p0q ´θ p0q1 ˘) `C1 E "´M 4 pϑ 1 , θ1 q `M 4 p θ1 , θ1 q ¯1{2 (4.28)
ˆ!1 ^´M 4 `pθ, θq, pθ 1 , θ1 q ˘`M 4 `pθ , θq, p θ1 , θ1 q ˘¯) 1{2 `∆R 2 a ı .

Remark 4.10. Specialized to the case when θ " θ " θ 1 " θ1 , θ " θ " θ1 " θ1 , ϑ " θ " θ " θ, ϑ 1 " θ1 " θ1 " θ1 and ∆R 2 a " 0, Lemma 4.28 reads as a uniqueness result to (4.3) in short time when ϑ " θ therein.

Proof. We start with the proof of (4.27). We take benet of the linearity to make the dierence of the two systems of the form (4.15) satised by ϑ and ϑ 1 . The resulting system is linear in ∆ϑ :" ϑ ´ϑ1 , ∆ θ :" θ ´θ 1 , ∆ θ :" θ ´θ 1 and ∆ θ :" θ ´θ 1 , but contains some remainders. We denote these remainders by ∆B a , ∆F a , ∆Σ a and ∆G a .

Using the notations introduced in (4.5) and (4.6), they may be expanded as:

∆H a psq " `h p θs , x θp0q s yq ´h p θ1 s , x θp0q1 s yq ˘θ 1 s `Ê "`Ĥ p θs , x θp0q s yq ´Ĥ p θ1 s , x θp0q1 s yq ˘x θp0q1 s y ‰ `Ha psq ´H1 a psq, ∆G a pT q " `g pX T , x XT yq ´g pX 1 T , x X1 T yq ˘X 1 T `Ê "`Ĝ pX T , x XT yq ´Ĝ pX 1 T , x X1 T yq ˘x X 1 T y ‰ `Ga pT q ´G1 a pT q, (4.29)
where H may stand for B, F or Σ, with a corresponding meaning for h , Ĥ and H a : h may be b , f , σ ; Ĥ may be B , F , or Σ ; H a may be B a , F a or Σ a ; and H 1 a may be B 1 a , F 1 a or Σ 1 a . With these notations in hand, the terms ∆H a psq and ∆G a pT q come from (recall (4.5)):

H `r, θr , x θp0q r y ˘`θ r , x θp0q r y ˘´H `r, θ1 r , x θp0q1 r y ˘`θ 1 r , x θp0q1 r y "

H `θ r , x θp0q r y ˘`∆ θr , x∆ θp0q r y ˘`∆H a prq , G `XT , x XT y ˘`X T , x XT y ˘´G `X1 T , x X1 T y ˘`X 1 T , x X 1 T y " G `XT , x XT y ˘`∆X T , x∆ XT y ˘`∆G a pT q .
(4.30)

We will apply Lemma 4.7. In the statement of the Lemma, we see from (4.30) that ϑ must be understood as ∆ϑ, θ as ∆ θ and similarly for the processes labelled with `hat' and `check'. Moreover, the remainder pB a , F a , Σ a , G a q in the statement must be understood as p∆B a , ∆F a , ∆Σ a , ∆G a q.

We estimate the remainder terms in (4.18), recalling (4.17) for the meaning we give to the remainder in the stability estimate. By (4.29), the remainder can be split into three pieces according to h , Ĥ and H a .

First step. Upper bound for the terms involving pb , f q, σ and g . We make use of the assumption (4.9) and of the conditional Cauchy-Schwarz inequality. Getting rid of the constant γ 1{2 in front of |G a pT q| 2p in (4.17), we let

∆r 2p :" E t " ˇˇ`g pX T , x XT yq ´g pX 1 T , x X1 T yq ˘X 1 T ˇˇ2p `ˆż T t ˇˇ`p b , f
E t "ˆż T t ˇˇ`p b , f qp θs , x θp0q s yq ´pb , f qp θ1 s , x θp0q1 s yq ˘θ 1 s ˇˇds ˙2p  ď C 1 M4p `pθ , θq, p θ1 , θ1 q ˘(1{2 M4p `θ 1 , θ1 ˘(1{2 .
It is pretty clear that we can get a similar bound when replacing pb , f q by σ (using the supremum norm to handle the fact that there is already a square inside the integral).

Finally, the term involving g can be also handled in a similar way, paying attention that the `bar' process has to be replaced by the `non-bar' process and the `check' process by the `hat' process. We thus get

∆r 2p ď C 1 M4p `pθ, θq, pθ 1 , θ1 q ˘`M 4p `pθ , θq, p θ1 , θ1 q ˘(1{2 M 4p `ϑ1 , θ1 ˘`M 4p `θ 1 , θ1 ˘(1{2 .
Using (4.7), we get another bound for the same quantity, just by taking advantage of the fact that pb , f q, σ and g are bounded: which satises the same bound as ∆r 2p . Above the passage from the rst to the third line may be applied with H equal to F or B and the passage from the second to the third line may be applied with H equal to Σ. We have a similar bound for the term involving Ĝ : are `equivalent' provided γ is not too small. In the sequel, we often choose γ exactly equal to 1{Γ p , so that M4p pϑ, θq and M 4p Et pϑq `pM 2 E p θp0q qq 2p can be indeed compared.

∆r 2p ď C 1 " E t "ˇˇX 1 T | 2p ‰ `sup sPrt,T s E t "ˇˇθ p0q1 s ˇˇ2p ‰ `Et "ˆż T t | θ1 s | 2 ds ˙p* , so that ∆r 2p ď C 1 " 1 ^ M4p `pθ, θq, pθ 1 , θ1 q ˘`M 4p `pθ , θq, p θ1 , θ1 q ˘(1{2 ı ˆ M 4p `ϑ1 , θ1 ˘`M 4p `θ 1 ,
E t " ˇˇÊ "`Ĝ pX T , x XT yq ´Ĝ pX 1 T , x X1 T yq ˘x X 1 T y ‰ˇˇ2 p ı ď C 1 sup sPrt,T s } θp0q1 s } 2p 2 " 1 ^´sup sPrt,T s E t " |θ p0q s ´θp0q,1 s | 2p ‰ `Φ2p α `θ p0q , θp0q1 ˘¯ı . ( 4
Corollary 4.12. Consider a family of progressively-measurable random paths ppθ ξ , θξ q : rt, T s Q s Þ Ñ pθ ξ s , θξ s qq ξ parametrized by ξ P L 2 pΩ, F t , P; R d q. Assume that, for any p ě 1, there exists a constant C p such that, for all ξ and ξ 1 (with the same notation as in (4. [START_REF] Lasry | Mean eld games[END_REF] but with Φ α dened on rL 2 pΩ, F t , P; R d qs 2 instead of rL 2 pΩ, A, P; R d qs 2 ):

`M 2p pθ ξ , θξ q ˘1{2p ď C p " 1 `|ξ| `}ξ} 2 ‰ , `M 2p `pθ ξ , θξ q, pθ ξ 1 , θξ 1 q ˘˘1{2p ď C p " |ξ ´ξ1 | `Φα `ξ, ξ 1 ˘‰, (4.35)
Assume also that we can nd a Borel subset O of a Euclidean space, a continuous functional Ψ from O ˆL2 p Ω, Â, P; R d q into L 2 pΩ, A, P; R `q and, for any ξ P L 2 pΩ, F t , P; R d q, an admissible class J ξ for pθ ξ , θξ q such that, for any Λ in J ξ , there exists a random variable λ : pΩ, A, Pq Ñ O satisfying Λpω, ¨q ď Ψpλpωq, xξyq, where Λpω, ¨q denotes the random variable Ω Q ω Þ Ñ Λpω, ωq on p Ω, Â, Pq.

With C as in Lemma 4.9, we then let, for ς P O and ξ P L 2 pΩ, F t , P; R d q,

Ψpς, ξqpωq " `Ψpς, ξqpωq ˘^! C `1 `|ξpωq| α`1 `}ξ} α`1 2 ˘), ω P Ω, (4.36)
where Ψpς, ξq is an abuse of notation for denoting the copy of the variable Ψpς, xξyq on the space Ω instead of Ω. (We may indeed assume that L 2 p Ω, Â, P; R d q is a copy of L 2 pΩ, A, P; R d q, in which case we can transfer (canonically) Ψpς, ¨q from one space to another.)

Then, for any p ě 1, there exist two constants Γ p :" Γ p pKq ě 1 and C 1 p ą 0, such that, for T ď γ ď 1{Γ p , choosing p θ, θq " pθ, ϑq, p θ, θq " p θ, θq, p θ1 , θ1 q " pθ 1 , ϑ 1 q and p θ1 , θ1 q " p θ1 , θ1 q in Lemma 4.9, with pθ, θq :" pθ ξ , θξ q and pθ 1 , θ1 q :" pθ ξ 1 , θξ 1 q, it holds that:

" M 2p Et `ϑ ´ϑ1 ˘‰1{2p ď C 1 p " " 1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰ ˆ´|η| `}η} 2 ``R 4p a ˘1{4p `E`R 2 a ˘1{2 ``M 2 E p θp0q1 q ˘1{2 ∆R 2p a ˘1{2p `sup ςPO sup }Λ 0 } 2 ďK " E " `Λ0 ^Ψpς, ξq ˘`M 2 Et p θp0q ´θ p0q1 q ˘1{2 ı * . (4.37)
When ϑ " θ and ϑ 1 " θ1 , we have (modifying the value of Γ p if necessary):

" M 2p Et `ϑ ´ϑ1 ˘‰1{2p ď C 1 p " " 1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰´| η| `}η} 2 ``R 4p a ˘1{4p `E" R 2 a ‰ 1{2 ¯``∆ R 2p a ˘1{2p * `C1 p " sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘"1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰ (4.38) ˆ´|η| `}η} 2 ``R 4 a ˘1{4 `E`R 2 a ˘1{2 ¯ı* `C1 p " sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘`∆R
ΛPJ ξ E t " Ê" ! Λ ^"C `1 `Ê t " |xXy| 2α`2 ‰ 1{2 `}X} α`1 2 ˘ı) Êt " |xX y| 2 ‰ 1{2  p  .
Simplifying the notations, the term inside the conditional expectation may be rewritten as ÊrpΛ ^xW yqxWys, for some random variables xW y and xWy in L 2 p Ω, Â, P; R `q and for Λ P J ξ . Allowing the constant C p in the assumption to increase from line to line, the following bound is proved right below:

E t " Ê" `Λ ^xW y ˘xWy ı p ı 1{p ď sup ςPO sup }Λ 0 } 2 ďK ! E " `Λ0 ^Ψpς, ξq ^W ˘Wı) . (4.39)
where, in the above expectation, Λ 0 P L 2 pΩ, A, P; R `q, W and W are the copies of xW y and xWy on the space Ω instead of Ω 1 .

We rst prove the remark:

Proof. [Remark 4.13.] By assumption on the structure of J ξ , we can nd λ such that ) .

Transferring the expectation appearing in the supremum into an expectation on Ω, we get (4.39).

l

We now turn to:

Proof. [Corollary 4.12.] The strategy is to make use of Lemma 4.9 and to estimate the various terms in (4.27). We use two values for the parameter γ in the denition (4.16) of M p M . As suggested in Remark 4.11, we rst use γ " 1{Γ p . Since we consider the case p θ1 , θ1 q " pθ 1 , ϑ 1 q and p θ1 , θ1 q " p θ1 , θ1 q, we deduce from (4.23) in Corollary 4.8 that there exists a constant C 1

p such that `M2p Et pϑ 1 q ˘1{2p ď C 1 p " |η| `}η} 2 ``R 2p a ˘1{2p `E`R 2 a ˘1{2 ``M 2 E p θp0q1 q ˘1{2 ı . (4.41)
Recalling again Remark 4.11 to compare M 4p Et and M4p and using in addition (4.35), the last term in (4.27), when put to the power 1{2p, gives the contribution:

C 1 p " " 1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰´| η| `}η} 2 ``R 4p a ˘1{4p `E`R 2 a ˘1{2 ``M 2 E p θp0q1 q ˘1{2 ∆R 2p a ˘1{2p ı .
We now discuss the other terms in (4.27). In this perspective, we use another value for γ, namely γ 1 ď 1{Γ p . Note that there is no conict with the previous choice for γ, which just permitted to handle the terms of the form M in (4.27). We thus turn to the two terms N 2p,C Et in (4.27). Taking them to the power 1{2p and making use of the rst line in (4.35), this brings us with a term of the same form as in the left-hand side of (4.39), with W " Cp1 `|ξ| 2α`1 `}ξ} 2α`1 2 q and W " rM 2 Et p θp0q ´θ p0q1 qs 1{2 . By (4.39), we get the following contribution:

sup ςPO sup }Λ 0 } 2 ďK ! E " `Λ0 ^Ψpς, ξq ˘`M 2 Et p θp0q ´θ p0q1 q ˘1{2 ı)
.

We obtain (modifying the constant Γ p in (4.27) in order to take into account the additional exponent 1{2p):

" M 2p Et `ϑ ´ϑ1 ˘‰1{2p ď C 1 p " " 1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰ ˆ´|η| `}η} 2 ``R 4p a ˘1{4p `E`R 2 a ˘1{2 ``M 2 E p θp0q1 q ˘1{2 ¯``∆ R 2p a ˘1{2p * `Γp pγ 1 q 1{4p sup ςPO sup }Λ 0 } 2 ďK " E " `Λ0 ^Ψpς, ξq ˘`M 2 Et p θp0q ´θ p0q1 q ˘1{2 ı * , (4.42)
which gives (4.37).

We now prove (4.38) when ϑ " θ and ϑ 1 " θ1 . We go back to (4.41). Applying (4.24) in Corollary 4.8 with p " 1 and taking expectation, we get, for γ small enough,

`M2p

Et pϑ

1 q ˘1{2p ď C 1 p " |η| `}η} 2 ``R 2p a ˘1{2p `E`R 2 a ˘1{2 ı ,
which means that, in (4.42), we can get rid of the term M 2p Et p θp0q1 q in the right-hand side. Let now p " 1 in (4.42). Multiply both sides by Λ 0 ^Ψpς, ξq for an R `-valued random variable Λ 0 such that }Λ 0 } 2 ď K and take the expectation and then the supremum over Λ 0 and ς. For γ 1 small enough, we get that

sup ςPO sup }Λ 0 } 2 ďK ! E " `Λ0 ^Ψpς, ξq ˘`M 2 Et pϑ p0q ´ϑp0q1 q ˘1{2 ı) ď C 1 " sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘"1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰ ˆ´|η| `}η} 2 ``R 4 a ˘1{4 `E`R 2 a ˘1{2 ¯ı* `C1 " sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘`∆R 2 a ˘1{2 ı * .
Plugging the above estimate into (4.42), we complete the proof.

Here is a very useful condition to check (4.35):

Lemma 4.14. Consider a family of progressively-measurable random paths ppθ ξ , θξ q : rt, T s Q s Þ Ñ pθ ξ s , θξ s qq ξ parametrized by ξ P L 2 pΩ, F t , P; R d q, with the property that the paths p θξ,p0q : rt, T s Q s Þ Ñ θξ,p0q s q ξ are continuous, and that p θξ,p0q s q sPrt,T s and p θξ 1 ,p0q s q sPrt,T s have the same distribution when ξ " ξ 1 .

Assume that, for any p ě 1, there exists a constant C p such that, for all ξ and ξ 1 , }θ ξ,p0q } S p ,t `}θ ξ,p0q } S p ,t `}θ ξ } H p ,t ď C p `1 `|ξ| `}ξ} 2 ˘, }θ ξ,p0q ´θξ 1 ,p0q } S p ,t `}θ ξ,p0q ´θ ξ

`M 2p pθ ξ , θξ q ˘1{2p ď C 1 p " 1 `|ξ| `}ξ} 2 ‰ , `M 2p `pθ ξ , θξ q, pθ ξ 1 , θξ 1 q ˘˘1{2p ď C 1 p " |ξ ´ξ1 | `Φ α pξ, ξ 1 q ‰ , where Φα pξ, ξ 1 q " E " |ξ ´ξ1 | 2 ‰ 1{2 `sup sPrt,T s Φ α p θξ,p0q s , θξ 1 ,p0q s q, ξ, ξ 1 P L 2 pΩ, F t , P; R d q. (4.44)
The functional Φα is continuous at any point of the diagonal of rL 2 pΩ, F t , P; R d qs 2 and satises (4.10) (up to a modication of the constant C therein).

Proof. The bound for p M2p pθ ξ , θξ qq 1{2p is a straightforward consequence of the rst line in (4.43). The bound for p M2p ppθ ξ , θξ q, pθ ξ 1 , θξ 1 qq 1{2p follows from the second line in (4.43) and from the denition of M2p in (4. [START_REF] Lasry | Mean eld games[END_REF].

The main issue is to check that Φα satises the same condition as Φ α . By (4.43), the map L 2 pΩ, A, P; R d q Q ξ Þ Ñ p θξ,p0q s q sPrt,T s P S 2 prt, T s; R l q (with the appropriate l) is continuous and, for any ξ P L 2 pΩ, A, P; R d q, the map rt, T s Q s Þ Ñ θξ,p0q s P L 2 pΩ, A, P; R l q is also continuous, proving that, for any sequence pξ n q ně1 converging to ξ in L 2 , the family of random variables p θξn,p0q s q sPrt,T s,ně1 is relatively compact. Since, for any compact subset K Ă L 2 pΩ, A, P; R d q, suptΦ α pχ, χ 1 q, χ, χ 1 P K, }χ ´χ1 } 2 ď δu tends to 0 with δ, continuity of Φα at any point of the diagonal easily follows. Now, we check that Φα satises (4.10) when ξ and ξ 1 have the same distribution.

Since θξ,p0q

| 2α `|θ ξ 1 ,p0q s | 2α `}θ ξ,p0q s } 2α 2 ˘|θ ξ,p0q s ´θ ξ 1 ,p0q s | 2 ‰ 1{2 ď E " E t "`1 `|θ ξ,p0q s | 4α `|θ ξ 1 ,p0q s | 4α `}θ ξ,p0q s } 4α 2 ˘‰1{2 E t " | θξ,p0q s ´θ ξ 1 ,p0q s | 4 ‰ 1{2 ı 1{2 ď CE "`1 `|ξ| 2α `|ξ 1 | 2α `}ξ| 2α 2 ˘|ξ ´ξ1 | 2 ‰ 1{2 . l
Example 4.15. We illustrate the meaning of (4.38) in the simplest (but crucial) case when R a " ∆R a " 0. Clearly, the most challenging term is

sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘"1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰`| η| `}η} 2 ˘ı, which is less than sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘2" 1 ^`|ξ ´ξ1 | `Φα pξ, ξ 1 q ˘‰2 ı 1{2 }η} 2 ď Φpξ, ξ 1 q}η} 2 , with Φpξ, ξ 1 q " sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘2" 1 ^|ξ ´ξ1 | 2 ‰ ı 1{2 `KΦ α pξ, ξ 1 q. (4.45)
Recalling the bound 0 ď Ψpς, ξq ď Cp1 `|ξpωq| α`1 `}ξ} α`1 2 q, there exists a constant C 1 such that, whenever ξ and ξ 1 have the same distribution,

Φpξ, ξ 1 q ď C 1 E "`1 `|ξ| 2α`2 `|ξ 1 | 2α`2 ˘|ξ ´ξ1 | 2 ‰ 1{2 ,
which ts (4.10), with α `1 instead of α, up to another multiplicative constant. The functional Φ is thus a candidate for being a function of the same type as Φ α`1 , according to the notation used in the assumptions (4.7)(4.11). Still, in order to guarantee that Φ indeed satises the same assumptions as Φ α`1 , it is necessary to prove that it is continuous at any point of the diagonal. We claim that it is the case under the two additional conditions (the proof is given right below):

(i) for each ξ P L 2 pΩ, F t , P; R d q, the family pΨ 2 pς, ξqq ςPO is uniformly integrable, (ii) the mappings pL 2 pΩ, F t , P; R d q Q ξ Þ Ñ Ψpς, ξq P L 2 pΩ, A, P; R d qq ςPO are equicontinuous.

As an example of a family pθ ξ , θξ q ξ and a functional Ψ : O ˆL2 pΩ, F t , P; R d q Q pς, ξq Þ Ñ Ψpς, ξq that satisfy the prescription in Corollary 4.12 together with piq and piiq, we can consider (again, the proof is given right below) pθ ξ :" θ t,ξ , θξ :" θt,ξ q ξ or pθ ξ :" θ t,x,rξs , θξ :" θt,ξ q ξ and Ψ `ς " pw, sq, ξ ˘" sup

H"B,Σ,F,G E t " ˇˇĤ `w, θt,ξ,p0q s ˘ˇ2 ı 1{2 , (4.46)
for w P R d ˆRm ˆRmˆd and s P rt, T s. (The denition of Ψpς, ξq for a random variable ξ that is not F t -measurable is useless here, since ξ is exclusively thought as an initial condition of the system (2.3) at time t.)

Proof. First step. We rst check that, under (i) and (ii), Φ is continuous at any point of the diagonal. Given two sequences pξ n q ně0 and pξ 1 n q ně0 converging in L 2 pΩ, F t , P; R d q towards some ξ, we already know that pΦ α pξ n , ξ 1 n qq ně0 converges to 0. Therefore, it suces to focus on the rst term in the right-hand side of (4.45). We have

ˇˇsup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξ n q ˘2" 1 ^|ξ n ´ξ1 n | 2 ‰ ı ´sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘2" 1 ^|ξ ´ξ1 | 2 ‰ ıˇˇď sup ςPO sup }Λ 0 } 2 ďK E " `Λ0 ^Ψpς, ξq ˘2ˇˇ1 ^|ξ ´ξ1 | 2 ´1 ^|ξ n ´ξ1 n | 2 ˇˇı `sup ςPO sup }Λ 0 } 2 ďK E "ˇˇˇ`Λ 0 ^Ψpς, ξq ˘2 ´`Λ 0 ^Ψpς, ξ n q ˘2ˇˇˇı . (4.47)
Recalling the bound Ψpς, ξq ď Ψpς, ξq, the rst term in the right-hand side is less than

sup ςPO E " Ψ 2 pς, ξq ˇˇ1 ^|ξ ´ξ1 | 2 ´1 ^|ξ n ´ξ1 n | 2 ˇˇı ,
which tends to 0 by uniform integrability of the family pΨpς, ξqq ςPO .

Consider now the second term in the right-hand side of (4.47). We have

sup ςPO sup }Λ 0 } 2 ďK E "ˇˇˇ`Λ 0 ^Ψpς, ξq ˘2 ´`Λ 0 ^Ψpς, ξ n q ˘2ˇˇˇı ď 2K sup ςPO sup }Λ 0 } 2 ďK E "ˇˇˇΛ 0 ^Ψpς, ξq ´Λ0 ^Ψpς, ξ n q ˇˇ2 ı 1{2 .
Recalling from (4.36) that Ψpς, ξq " Ψpς, ξq ^ϕpξq, with ϕpξq " rCp1 `|ξ| α`1 `}ξ} α`1 2 s, writing |Λ 0 ^Ψpς, ξq ´Λ0 ^Ψpς, ξ n q| ď |Λ 0 ^Ψpς, ξq ^ϕpξq ´Λ0 ^Ψpς, ξq ^ϕpξ n q| |Λ 0 ^Ψpς, ξq ^ϕpξ n q ´Λ0 ^Ψpς, ξ n q ^ϕpξ n q| and using the Lipschitz property of the map R Q x Þ Ñ a ^x, for any a P R, we deduce that

sup ςPO sup }Λ 0 } 2 ďK E "ˇˇˇ`Λ 0 ^Ψpς, ξq ˘2 ´`Λ 0 ^Ψpς, ξ n q ˘2ˇˇˇı ď 2K " sup ςPO E "ˇˇˇΨ pς, ξq ´Ψpς, ξ n q ˇˇ2 ı 1{2 `sup ςPO E
"ˇˇˇΨ pς, ξq ^ϕpξq ´Ψpς, ξq ^ϕpξ n q ˇˇ2 ı  1{2 .

By uniform continuity of the mappings pΨpς, ¨qq ςPO , the rst term in the right-hand side tends to 0. By uniform integrability of the family pΨ 2 pς, ξqq ςPO , the second one also tends to 0.

Second step. We now check the example. By Lemmas 4.1 and 4.14, (4.35) is satised with pθ ξ , θξ q :" pθ t,ξ , θt,ξ q or pθ ξ , θξ q :" pθ t,x,rξs , θt,ξ q. We prove that Ψ in (4.46) satises (i) and (ii). We check rst the uniform integrability property (i). It suces to check it for H equal to B, Σ, F or G (if uniform integrability holds for H equal to B, Σ, F or G, then the supremum over H equal to B, Σ, F or G also satises (i)). Given ξ P L 2 pΩ, F t , P; R d q, it suces to prove that the family psup rPrt,T s | Ĥ pw, θt,ξ,p0q r q| 2 q wPR k (for the appropriate k) is uniformly integrable. Consider a positive constant ε ą 0. Since the path rt, Therefore, denoting by pt " s 0 ă s 1 ă ¨¨¨ă s N " T q a subdivision of rt, T s with stepsize less than δ, we deduce that, for any event A P A,

T s Q r Þ Ñ θt,ξ,p0q
sup wPR k sup tďrďT E " ˇˇĤ `w, θt,ξ,p0q r ˘ˇ2 1 A ı ď sup wPR k sup i"0,...,N E " ˇˇĤ `w, θt,ξ,p0q s i ˘ˇ2 1 A ı `2Kε 1{2 .
By the uniform integrability of each of the family p| Ĥ pw, θt,ξ,p0q s i q| 2 q wPR k , for i " 0, . . . , N , see pH1q, we deduce that the left-hand side is indeed less than 4Kε 1{2 for δ small enough.

We check uniform continuity of the mappings pξ Þ Ñ Ĥ pw, θt,ξ,p0q s qq wPR k ,sPrt,T s :

sup wPR k sup sPrt,T s E " ˇˇĤ `w, θt,ξ,p0q s ˘´Ĥ `w, θt,ξ 1 ,p0q s ˘ˇ2 ı 1{2 ď C sup sPrt,T s Φ α `θ t,ξ,p0q s , θt,ξ 1 ,p0q s ˘,
which tends to 0 as ξ 1 ´ξ tends to 0, by the same argument as in Lemma 4.14. l

We complete the subsection with a very important observation: Remark 4.16. Example 4.15 ensures that Corollary 4.12 may be applied with pθ ξ , θξ q :" pθ t,ξ , θ t,ξ q or pθ ξ , θξ q :" pθ t,x,rξs , θ t,ξ q, in which case (4.35) holds for a suitable function Φ α (dened on rL 2 pA, F t , P; R d qs 2 ) and the the second term in the right-hand side of (4.38) may be bounded by a function of the type Φ α`1 (also dened on rL 2 pA, F t , P; R d qs 2 ).

It is worth mentioning that, with the construction that is suggested, both Φ α and Φ α`1 may depend on t, which is clear from (4.44) and (4.46).

Below, we want to use versions of both that are independent of t. This requires rst to restrict the domain of denition of both functionals to rL 2 pA, F 0 , P; R d qs 2 . Second, this requires a suitable adaptation of (4.44) and (4.46). When ξ P L 2 pΩ, F 0 , P; R d q, we may extend pθ t,ξ s q sPrt,T s to the interval r0, T s by letting X t,ξ s " ξ, Y t,ξ s " Y t,ξ t and Z t,ξ s " 0 for s P r0, ts. Then, for ξ, ξ 1 P L 2 pΩ, F 0 , P; R d q, instead of (4.44), we may let Φα pξ,

ξ 1 q " E " |ξ ´ξ1 | 2 ‰ 1{2 `sup tPr0,T s sup sPr0,T s Φ α p θt,ξ,p0q s , θt,ξ 1 ,p0q s q,
(that is we also take the supremum in t), and, instead of (4.46), we may let Ψ `ς " pw, t, sq, ξ ˘" sup

H"B,Σ,F,G E t " ˇˇĤ `w, θt,ξ,p0q s ˘ˇ2 ı 1{2 ,
(that is we include t in the variable ς).

Then, the resulting new functionals Φ α and Φ α`1 are independent of t, are continuous at any point of the diagonal of rL 2 pΩ, F 0 , P; R d qs 2 and satisfy (4.10) with respect to α and α `1. The proof works exactly as in Lemma 4.14 and in Example 4.15, noticing that that the mapping L 2 pΩ, F 0 , P; R d q ˆr0, T s ˆr0, T s Q pξ, s, tq Þ Ñ θ t,ξ s P L 2 pΩ, A, P; R d q is continuous (which is the main ingredient to make the argument work).

4.2. Analysis of the rst-order derivatives. L in pH0q(i)-pH1q now playing the role of the constant K in the above statements. In order to stress the fact that this subsection is devoted to the application of the general results proved above to the specic question of the dierentiability of the ow, we shall use constants cpLq or c p pLq instead of 1{ΓpKq or 1{Γ p pKq for quantifying small time constraints of the type T ď cpLq or T ď c p pLq.

To make things clear, we also recall the identication of h , Ĥ and H a in (4.4):

h pw, x V p0q yq " B w hpw, r V p0q sq, Ĥ pw, x V p0q yq " B µ hpw, r V p0q sqpx V p0q yq, H a " 0. (4.49)
The next results state the rst order dierentiability of the McKean-Vlasov system.

Lemma 4.17. Given a continuously dierentiable path of initial conditions R Q λ Þ Ñ ξ λ P L 2 pΩ, F t , P; R d q, t standing for the initial time in r0, T s, we can nd a constant c :" cpLq ą 0 such that, for T ď c, the path R Q λ Þ Ñ θ λ " pX λ , Y λ , Z λ q :" θ t,ξ λ P S 2 prt, T s; R d q ˆS2 prt, T s, R m q ˆH2 prt, T s; R mˆd q is continuously dierentiable. Proof. Under pH0q(i), existence and uniqueness of a solution to (2.1) may be proved for a small time horizon T by a contraction argument. As in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF], for T small enough, we can approximate pX λ , Y λ , Z λ q as the limit of a Picard sequence θ n,λ :" pX n,λ , Y n,λ , Z n,λ q, dened by

X n`1,λ s " ξ λ `ż s t bpθ n,λ r , rθ n,λ,p0q r sqdr `ż s t σpθ n,λ,p0q r , rθ n,λ,p0q r sqdW r Y n`1,λ s " gpX n`1,λ T , rX n`1,λ T sq `ż T s f pθ n,λ r , rθ n,λ,p0q r sqdr ´ż T s Z n`1,λ r dW r ,
where we have used the notation θ n,λ,p0q s " pX n,λ s , Y n,λ s q, with the initialization θ 0,λ " 0.

By the standard theory of Itô processes and backward equations (see in particular [START_REF] Pardoux | Backward stochastic dierential equations and quasilinear parabolic partial dierential equations[END_REF]), we can prove by induction that, for any n ě 0, the mapping R Q λ Þ Ñ θ λ " pX n,λ , Y n,λ , Z n,λ q P S 2 prt, T s; R d q ˆS2 prt, T s, R m q ˆH2 prt, T s; R mˆd q is continuously differentiable. We give just a sketch of proof. For the forward component, this follows from the fact that given a continuously dierentiable path

R Q λ Þ Ñ h λ P H 2 prt, T s, Rq, the paths R Q λ Þ Ñ p ş s t h λ r drq sPrt,T s and R Q λ Þ Ñ p
ş s t h λ r dW r q sPrt,T s , with values in S 2 prt, T s, R l q for a suitable dimension l, are continuously dierentiable, which is obviously true. To handle the backward component, it suces to prove rst that the path

R Q λ Þ Ñ pE s rh λ
T sq sPrt,T s , with values in S 2 prt, T s, Rq, is continuously dierentiable, which is straightforward by means of Doob's inequality. This is enough to handle the terminal condition and also the driver since we can split the integral from s to T into an integral from t to s (to which we can apply the result used for the forward component) and an integral from t to T (which can be seen as a new h T ). In this way, we can prove that

R Q λ Þ Ñ Y n`1,λ is continuously dierentiable from R to S 2 prt, T s, R m q. This shows that R Q λ Þ Ñ p ş s t Z n`1,λ r dW r q sPrt,T s is also continuously dierentiable from R to S 2 prt, T s, R m q.
By Itô's isometry, this nally proves that R Q λ Þ Ñ pZ n`1,λ s q sPrt,T s is continuously dierentiable from R to H 2 prt, T s, R mˆd q, the derivative of Z n`1,λ writing as the martingale representation term of the derivative of ş T t Z n`1,λ r dW r . The derivatives, denoted by pX n,λ , Y n,λ , Z n,λ q, satisfy the system

Y n`1,λ s " G p1q pX n`1,λ T , xX n`1,λ T yq `X n`1,λ T , xX n`1,λ T y ż T s F p1q `r, θ n,λ r , xθ n,λ,p0q r y ˘`ϑ n,λ r , xϑ n,λ,p0q r y ˘dr ´ż T s Z n`1,λ r dW r , (4.50)
where we have used the notations χ λ " rd{dλsξ λ , ϑ n,λ " pX n,λ , Y n,λ , Z n,λ q and ϑ n,λ,p0q " pX n,λ , Y n,λ q and where B, Σ, F and G are dened according to (4.49) and are denoted by B p1q , Σ p1q , F p1q and G p1q as in (4.4), the superscript p1q stressing the fact that we are dealing with rst-order derivatives. We thus obtain a system of the form (4.15) with θ " θ " θ n`1,λ , θ " θ " θ n,λ , ϑ " θ " ϑ n`1,λ and θ " θ " ϑ n,λ and χ λ playing the role of η. We now apply Lemma 4.7, noticing that the remainder R a therein is zero, see (4.49).

First, we set p " 1 in (4. [START_REF] Guéant | Mean eld games and applications[END_REF]) and choose γ " 1{Γ 1 pLq in (4.16), in agreement with Remark 4.11. We then take expectation on both sides. We get that, for T small, the sequence pM 2 E pϑ n,λ qq ně1 is at most of arithmetico-geometric type, with a geometric rate strictly less than 1. By induction, we deduce that there exist two constants c :" cpLq ą 0 and C ě 0 (the values of which are allowed to increase from one line to another), such that, for T ď c, sup ně0 M 2 E pϑ n,λ q ď C}χ λ } 2 2 . Inserting this estimate into (4.19) (with γ " 1{Γ 2 pLq therein), we can prove, in the same way, that, for possibly new values of c and C,

sup ně1 " M 4 Et `ϑn,λ ˘ı1{2 ď C " |χ λ | 2 `}χ λ } 2 2
‰ .

(4.51)

Exploiting Remark 4.11, we deduce that r M4 ϑ n,λ s 1{2 and r M4 ϑ n`1,λ s 1{2 in (4.25) are less than Cp|χ λ | 2 `}χ λ } 2 2 q. We now make use of (4.28) in Lemma 4.9, with p " 1, in order to compare ϑ n,λ and ϑ n`1,λ . Clearly, the remainder ∆R 2 a in (4.26) is zero since the R a terms are here equal to zero, recall (4.49). By the above argument, r M4 ϑ n,λ s 1{2 and r M4 ϑ n`1,λ s 1{2 in (4.25) are less than Cp|χ λ | 2 `}χ λ } 2 2 q. In order to apply Lemma 4.9, we also have to estimate r M4 θ n`1,λ , θ n,λ s 1{2 . Since T is small enough, the Picard scheme for solving (2.3) is geometrically convergent in L 2 and in any L p , p ě 2, conditional on F t , the geometric rate being independent of the initial conditions. To be precise, there exist ρ P p0, 1q and C 1 ě 0 such that, almost surely, r M4 θ n`1,λ ´θn,λ s 1{2 ď C 1 p1 `|ξ λ | 2 `}ξ λ } 2 2 qρ n . By continuity of the map R Q λ Þ Ñ ξ λ P L 2 pΩ, F t , P; R d q, this shows that Epr M4 θ n`1,λ ´θn,λ s 1{2 q converges to 0, uniformly in λ in compact subsets. Now, following the proof of Lemma 4.14 and using the fact that the map R Q λ Þ Ñ ξ λ P L 2 pΩ, F t , P; R d q is continuous, the family of random variables pθ n,p0q,λ s q ně0,sPrt,T s,λPK is relatively compact in L 2 pΩ, A, P; R d Rm q for any compact subset K Ă R. Therefore, Φ α pθ n`1,p0q,λ , θ n,p0q,λ q converges to 0, uniformly in λ in compact subsets. We deduce that Epr M4 θ n`1,λ , θ n,λ s 1{2 q converges to 0, uniformly in λ in compact subsets.

By (4.28) with γ 1{2 Γ 1 pLq " 1{4, we deduce that, for T ď c (allowing the value of c to decrease from line to line),

M 2 E `ϑn`1,λ ´ϑn,λ ˘ď 1 2 M 2 E `ϑn,λ ´ϑn´1,λ ˘`CE " p|χ λ | 2 `}χ λ } 2 2 qψ n pλq ‰˘ı , (4.52)
where pψ n pλqq ně0 is a sequence of random variables that are bounded by 1 and that converges in probability to 0 as n tends to 8, uniformly in λ in compact subsets. By a standard uniform integrability argument, we deduce from the bound ψ n pλq ď 1 and from the continuity property of the mapping R Q λ Þ Ñ χ λ P L 2 that Er `|χ λ | 2 `}χ λ } 2 2 ˘ψn pλqs tends to 0 as n tends to 8, uniformly in λ in compact subsets. Therefore, the left-hand side in (4.52) converges to 0, the convergence being geometric, uniformly in λ in compact subsets. By a Cauchy argument, the proof is completed.

l

We emphasize that the derivative process rd{dλs |λ"0 θ λ given by Lemma 4.17 satises (4.3) with η :" χ, with θ " θ :" θ 0 and ϑ " θ :" rd{dλs |λ"0 θ λ and with the coecients given in (4.49). In particular, for T small enough, the uniqueness of the solution to (4.3) (see Remark 4.10) ensures that the derivative process at λ " 0 depends only on the family pξ λ q λPR through ξ 0 and rd{dλs |λ"0 ξ λ . Thus, when ξ 0 :" ξ and rd{dλs |λ"0 ξ λ :" χ, we may denote by B χ θ t,ξ " pB χ X t,ξ , B χ Y t,ξ , B χ Z t,ξ q the tangent process at ξ in the direction χ. By linearity of (4.3), B χ θ t,ξ is linear in χ. By a direct application of Corollary 4.8 recall H a " 0 in the current case , we have Lemma 4.18. For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p , such that, for T ď c p and with γ " c p in (4.16),

" M 2p

Et `Bχ θ t,ξ ˘‰1{p2pq ď C p `|χ| `}χ} 2

˘.

Choosing p " 1 and taking the expectation, we get that the mapping L 2 pΩ, F t , P; R d q Q χ Þ Ñ B χ θ t,ξ P S 2 prt, T s; R d qˆS 2 prt, T s; R m qˆH 2 prt, T s; R mˆd q is continuous, which proves that L 2 pΩ, F t , P; R d q Q ξ Þ Ñ θ t,ξ P S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q is Gâteaux dierentiable. The next lemma shows that the Gâteaux derivative is continuous: Lemma 4.19. For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p , such that, for T ď c p and with γ " c p in (4.16),

" M 2p

Et `Bχ θ t,ξ ´Bχ θ t,ξ

}B χ X t,ξ ´Bχ X t,ξ 1 s } S 1 `}B χ Y t,ξ ´Bχ Y t,ξ 1 s } S 1 `}B χ Z t,ξ s ´Bχ Z t,ξ 1 s } H 1 ď C ´E"`1 ^|ξ ´ξ1 | 2 ˘‰1{2 `Φα`1 pξ, ξ 1 q ¯}χ} 2 .
By [2, Proposition A.3], the map L 2 pΩ, F t , P; R d q Q ξ Þ Ñ pX t,ξ , Y t,ξ , Z t,ξ q P S 1 prt, T s; R d qŜ 1 prt, T s; R m q ˆH1 prt, T s; R mˆd q is continuously Fréchet dierentiable.

4.2.2. First-order derivatives of the non McKean-Vlasov system with respect to the measure argument. We reproduce the same analysis as above, but with the process θ t,x,rξs instead of θ t,ξ by taking advantage of the fact that the dependence of the coecients of the system (2.4) upon the law is already known to be smooth. This permits to discuss the dierentiability of θ t,x,rξs in a straightforward manner.

We mimic the strategy of the previous subsection. Considering a continuously dierentiable mapping λ Þ Ñ ξ λ P L 2 pΩ, F t , Pq, we are to prove that λ Þ Ñ θ t,x,rξ λ s is continuously dierentiable.

The specic feature is that, for any λ, the coecients of the FBSDE (2.4) satised by θ t,x,rξ λ s depend in a smooth way upon the solution θ t,ξ λ of the FBSDE (2.3). Since we have already established the continuous dierentiability of the mapping λ Þ Ñ θ t,ξλ , it suces now to prove that the solution of a standard FBSDE depending on a parameter λ in a continuously dierentiable way is also continuously dierentiable with respect to λ. We shall not perform the proof, as it consists of a simple adaptation of the proof used to prove the dierentiability of the ow of a standard FBSDE, see [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF].

When ξ 0 " ξ and rd{dλs λ"0 ξ λ " χ, we shall denote the directional derivative at ξ along χ by `Bχ X t,x,rξs s , B χ Y t,x,rξs s , B χ Z t,x,rξs s ˘sPrt,T s , seen as an element of the space S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q. By the same argument as above, it only depends on the family pξ λ q λPR through the values of ξ and χ, pB χ X t,x,rξs , B χ Y t,x,rξs , B χ Z t,x,rξs q satisfying a `dierentiated' system, of the type (4.3), for which uniqueness holds. In (4.3), η " 0 (since rd{dλsX t,x,rξs " 0), θ " θ t,x,rξs , θ " θ t,rξs , ϑ " B χ θ t,x,rξs and θ " B χ θ t,ξ , the tangent process B χ θ t,ξ being given by Lemma 4.17. The coecients are of the general shape (4.5) and (4.6). When h stands for one of the functions b, f , σ or g and V for θ t,x,rξs , θ t,x,rξs,p0q or X t,x,rξs and V for θ t,ξ , θ t,ξ,p0q or X t,ξ , according to the cases, it holds, as in (4.49), h pV, x V p0q yq " B x hpV, r V p0q sq, Ĥ pV, x V p0q yq " B µ hpV, r V p0q sqpx V p0q yq, H a " 0. and

" M 2p E `Bχ θ t,x,rξs ´Bχ θ t,x,rξ 1 s ˘‰1{2p ď C p Φ α`1 pt, ξ, ξ 1 q}χ} 2 , (4.55)
where Φ α`1 pt, ¨, ¨q : rL 2 pΩ, F t , P; R d qs 2 Ñ R `is continuous at any point of the diagonal, does not depend on p and satises (4.10), with α replaced by α `1. The restriction of Φ α`1 pt, ¨, ¨q to rL 2 pΩ, F 0 , P; R d qs 2 may be assumed to be independent of t P r0, T s.

Remark 4.22. Note that there is no conditional expectation on F t in the above bounds as the initial condition of B χ X t,x,rξs is zero, which means that the ltration that is used for solving the linear equation can be assumed to be almost-surely trivial at time t. For that reason, the right hand side reduces to }χ} 2 . We stress the fact that it is not }χ} 2p but }χ} 2 , as the dependence upon χ comes through the McKean-Vlasov interaction terms, which is estimated in L 2 norm.

Proof. Equation (4.54) is a direct consequence of (4.23) in Corollary 4.8, with η " 0, R a " 0 and θ " B χ θ t,ξ , combined with Lemma 4.18 (to control the term θp0q " B χ θ t,ξ,p0q ). We now turn to (4.55). It follows from (4.37) in Corollary 4.12, with η " 0, R a " ∆R a " 0, θ ξ " θ t,x,rξs , ϑ ξ " θ t,x,rξs , θξ " θ t,ξ and θξ " ϑ t,ξ (and the same with ξ 1 instead of ξ).

L 2 pΩ, F t , P; R d q Q ξ Þ Ñ Upt, x, ξq " Y t,
x,rξs t is Fréchet continuously dierentiable. In particular, the function P 2 pR d q Q µ Þ Ñ U pt, x, µq is dierentiable in Lions' sense. Moreover, for all x P R d , for all ξ, ξ 1 P L 2 pΩ, A, P; R d q, we have, with µ " rξs and µ 1 " rξ 1 s, but with Φ α`1 dened on rL 2 pΩ, F 0 , P; R d qs 2 , which requires that ξ and ξ 1 belong to L 2 pΩ, F 0 , P; R d q. The main issue is to prove that Φ α`1 may be dened on the whole rL 2 pΩ, A, P; R d qs 2 . It is then worth mentioning that }B µ U pt, x, µqpξq ´Bµ U pt, x, µ 1 qpξ 1 q} 2 only depends on the law of pξ, ξ 1 q. Given pξ, ξ 1 q P rL 2 pΩ, A, P; R d qs 2 , we can always nd a pair p ξ, ξ1 q P rL 2 pΩ, F 0 , P; R d qs 2 with the same distribution (provided that pΩ, F 0 , Pq is rich enough). This says that, with the Φ α`1 given by Lemma 4.21, for all ξ, ξ 1 P L 2 pΩ, A, P; R d q, › › B µ U pt, x, µqpξq ´Bµ U pt, x, µ 1 qpξ 1 q › › 2 ď Φα`1 pξ, ξ 1 q, with Φα`1 pξ, ξ 1 q :" inf Φ α`1 p ξ, ξ1 q, p ξ, ξ1 q P L 2 pΩ, F 0 , P; R d q, p ξ, ξ1 q " pξ, ξ 1 q ( .

}B µ U pt, x, µqpξq} 2 ď C, }B µ U pt, x, µqpξq ´Bµ U pt, x, µ 1 qpξ 1 q} 2 ď CΦ α`1 pξ, ξ 1 q, ( 4
Clearly, Φα`1 is dened on the whole rL 2 pΩ, A, P; R d qs 2 . It satises (4.10). Continuity at any point of the diagonal may be proved as follows.

Given a sequence pξ n , ξ 1 n q ně1 that converges to some pξ, ξq in rL 2 pΩ, A, P; R d qs 2 , we may nd a pair p ξ, ξq P rL 2 pΩ, F 0 , P; R d qs 2 with the same law as pξ, ξq. Now, for any n ě 1, we can construct p ξn , ξ1 n q in rL 2 pΩ, F 0 , P; R d qs 2 such that the 4-tuple p ξn , ξ1 n , ξ, ξq has the same law as pξ n , ξ 1 n , ξ, ξq (it suces to use the conditional law of pξ n , ξ 1 n q given pξ, ξq). Then, p ξn , ξ1 n q ně1 converges to p ξ, ξq in L 2 . From the inequality Φα`1 pξ n , ξ 1 n q ď Φ α`1 p ξn , ξ1 n q, Φα`1 pξ n , ξ 1 n q tends to 0. l

We now discuss the Lipschitz property in x of B µ U pt, x, µq:

Lemma 4.24. For T ď c, with c :" cpLq ą 0, we can nd a constant C such that, for ξ with µ as distribution, @x,

x 1 P R d , }B µ U pt, x, µqpξq ´Bµ U pt, x 1 , µqpξq} 2 ď C|x ´x1 |.
Proof. Thanks to the relationship B χ Y t,x,rξs t " ErB µ U pt, x, rξsqpξqχs, it suces to discuss the Lipschitz property (in x) of the tangent process pB χ X t,x,ξ s , B χ Y t,x,ξ s , B χ Z t,x,ξ s q sPrt,T s , seen as an element of S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q, ξ and χ denoting elements of L 2 pΩ, F t , P; R d q.

Basically, the strategy is the same as in the proofs of Lemmas 4.19 and 4.21. It is based on a tailored-made version of Corollary 4.12, obtained by applying Lemma 4.1 and Lemma 4.9 with θ " θ :" θ t,x,rξs , θ 1 " θ1 :" θ t,x 1 ,rξs , θ " θ1 " θ " θ1 :" θ t,ξ and θ " θ1 " θ " θ1 :" B χ θ t,ξ . Informally, it consists in choosing η " 0 and in replacing |ξ ´ξ1 | by |x ´x1 | and Φ α pξ, ξ 1 q by 0 in the statement of Corollary 4.12. We end up with

|B χ Y t,x,rξs t ´Bχ Y t,x 1 ,rξs t | ď C|x 1 ´x|}χ} 2 .
4.2.3. Derivatives with respect to the space argument. We now discuss the derivatives of U with respect to the variable x. Since the process θ t,x,rξs " pX t,x,rξs , Y t,x,rξs , Z t,x,rξs q may be seen as the solution of a standard FBSDE parametrized by the law of ξ, we can apply the results in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] on the smoothness of the ow of a classical FBSDE in short time. Given t P r0, T s and ξ P L 2 pΩ, F t , P; R d q, we deduce that the function R d Q x Þ Ñ θ t,x,rξs " pX t,x,rξs , Y t,x,rξs , Z t,x,rξs q P S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q is continuously dierentiable, the derivative process at point x P R d being denoted by B x θ t,x,rξs " pB x X t,x,rξs , B x Y t,x,rξs , B x Z t,x,rξs q. To be self-contained, notice that the same result could be obtained by applying the results of Subsection 4.1, with the following version of Hpr, ¨q: Hpr, V t,x,rξs r qpV t,x,rξs r q " B x hpV t,x,rξs r , rV t,ξ r sqV t,x,rξs r .

(4.57)

As a consequence, we easily get, for T ď c p , c p :" c p pLq and with γ " c p in (4.16), rM 2p E pB x θ t,x,rξs qs 1{2p ď C p . Recalling the identity U pt, x, rξsq " θ t,x,rξs t

, we recover the

fact that R d Q x Þ Ñ U pt, x, rξsq is continuously dierentiable and that }B x U } 8 ď C,
see also [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]. On the same model (for instance by adapting Lemmas 4.19 or 4.21 to investigate the dierence B x θ t,x,rξs ´Bx θ t,x 1 ,rξs for two dierent x, x 1 P R d or by taking benet from the results proved in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]), it can be checked that, for any t P r0, T s, any ξ P L 2 pΩ, F t , P; R d q, the mapping

R d Q x Þ Ñ B x U pt, x, rξsq is C-Lipschitz continuous.
Intuitively, such a bound is much simpler to get than the bound for the continuity of B µ U because of the very simple structure of Hpr, ¨q in (4.57), the function B x h being Lipschitz-continuous with respect to the rst argument.

To get the smoothness of B x U in the direction µ, we may investigate the dierence B x θ t,x,rξs ´Bx θ t,x,rξ 1 s for two dierent ξ, ξ 1 P L 2 pΩ, F t , P; R d q. Reapplying Corollary 4.12, exactly in the same way as in the proof of Lemma 4.21, we deduce @x P R d , ξ, ξ 1 P L 2 pΩ, A, P; R d q, ˇˇB x U pt, x, rξsq ´Bx U pt, x, rξ 1 sq ˇˇď Φ α`1 pξ, ξ 1 q. (4.58) Actually, the above bound could be improved. Indeed, it also holds with Φ α`1 pξ, ξ 1 q replaced by Φ α pξ, ξ 1 q. The reason is that, in the analysis of B x θ t,x,rξs ´Bx θ t,x,rξ 1 s , there are no derivatives in the direction of the measure, whereas these are precisely these terms that make Φ α`1 pξ, ξ 1 q appear in the proof of Lemma 4.23 (or equivalently of Lemma 4.21). In order to keep some homogeneity between the various estimates we have on the derivatives of U , we feel it is more convenient to keep Φ α`1 pξ, ξ 1 q in the above right-hand side.

4.2.4. Final statement. The following is the complete statement about the rst-order dierentiability: Theorem 4.25. For T ď c, with c :" cpLq ą 0 and t P r0, T s, the function R d L2 pΩ, A, P; R d q Q ξ Þ Ñ U pt, x, rξsq " Upt, x, ξq is continuously dierentiable and there exists a constant C ě 0, such that for all x, x 1 P R d , for all ξ, ξ 1 P L 2 pΩ, A, P; R d q, |U pt, x, µq ´U pt,

x 1 , µ 1 q| ď C `|x ´x1 | `W2 pµ, µ 1 q |B x U pt, x, µq ´Bx U pt, x 1 , µ 1 q| `}B µ U pt, x, µqpξq ´Bµ U pt, x 1 , µ 1 qpξ 1 q} 2 ď C `|x ´x1 | `Φα`1 pξ, ξ 1 q ˘, (4.59)
where Φ α`1 : rL 2 pΩ, A, P; R d qs 2 Ñ R `is continuous at any point of the diagonal and satises (4.10), with α replaced by α`1. In particular, for any x P R d and µ P P 2 pR d q, we can nd a locally Lipschitz continuous version of the mapping

R d Q v Þ Ñ B µ U pt, x, µqpvq.
Moreover, the functions r0, T s ˆRd ˆL2 pΩ, A,

P; R d q Q pt, x, ξq Þ Ñ B x U pt, x, rξsq P R d and r0, T s ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B µ U pt, x, rξsqpξq P L 2 pΩ, A, P; R d q are continuous.
Finally, for any t P r0, T s, ξ P L 2 pΩ, F t , P; R d q and C 1 ě 0, lim

PpAqÑ0,APA sup pt,xqPr0,T sˆR d sup ΛPL 2 pΩ,A,P;R d q:}Λ} 2 ďC 1 ˇˇE " B µ U pt, x, rξsqpξqΛ1 A ‰ˇˇ" 0, (4.60)
which is the analogue of the uniform integrability property described in pH1q for the original coecients b, σ, f and g.

Proof. The Lipschitz property of U is a direct consequence of the bounds we have for B x U and B µ U (or equivalently of Lemma 4.1). The joint continuous dierentiability is a consequence of the partial continuous dierentiability and of the joint continuity properties of the derivatives. The extension of Φ α`1 to the whole rL 2 pΩ, A, P; R d qs 2 is achieved as in the proof of Lemma 4.23.

The local Lipschitz property

of R d Q v Þ Ñ B µ U pt,
x, µqpvq follows from Proposition 3.8. We now discuss the continuity of r0, T s Q t Þ Ñ B µ U pt, x, rξsqpξq P L 2 pΩ, A, P; R d q. Clearly, there is no loss of generality in assuming that ξ P L 2 pΩ, F 0 , P; R d q. Given ξ, χ P L 2 pΩ, F 0 , P; R d q and 0 ď t ď s ď T , it suces to bound the time increment Er `Bµ U pt, x, rξsqpξq ´Bµ U ps, x, rξsqpξq ˘χs by Cpt, sq}χ} 2 , the constant Cpt, sq being independent of χ and converging to 0 as s ´t tends to 0. We have

E "`B µ U pt, x, rξsqpξq ´Bµ U ps, x, rξsqpξq ˘χ‰ " Ê"`B µ U pt, x, rξsqpxξyq ´Bµ U ps, x, rξsqpxξyq ˘xχy ‰ " E Ê"`B µ U `s, X t,x,rξs s , " X t,ξ s ‰˘`@ X t,ξ s D˘´B µ U ps, x, rξsq `@ξ D˘˘x χy ‰ `EÊ "`B µ U pt, x, rξsq `xξy ˘´B µ U `s, X t,x,rξs s , " X t,ξ s ‰˘`x X t,
C " `1 `|x| `}ξ} 2 ˘ps ´tq 1{2 `sup ξ 1 :}ξ 1 ´ξ} 2 ďCp1`}ξ} 2 qps´tq 1{2 Φ α`1 pξ 1 , ξq ı }χ} 2 . (4.62)
Clearly, the term in brackets goes to 0 with s ´t.

We now handle the second term in the last line of (4.61). Dierentiating (with respect to ξ in the direction χ) the relationships U pt, x, rξsq " Y

r0, T s Q t Þ Ñ B µ U pt, x, rξsqpξq P L 2 pΩ, A, P; R d q easily follows. Continuity of r0, T s Q t Þ Ñ B x U pt, x, rξsq P R d may be
proved in the same way.

Together with the uniform continuity estimates (4.59), we deduce that the functions r0, T s ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B x U pt, x, rξsq P R d and r0, T s ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B µ U pt, x, rξsqpξq P L 2 pΩ, A, P; R d q are continuous.

We now prove (4.60). For A P A and Λ P L 2 pΩ, A, P; R d q with }Λ} 2 ď C We now apply (4.18) in Lemma 4.7 with θ " θ :" θ t,x,rξs , ϑ " θ :" B χ θ t,x,rξs , θ " θ :" θ t,ξ , θ " θ :" B χ θ t,ξ and η " 0. The coecients driving (4.3) are given by (4.53). By (4.18), we get that, for T ď γ with γ in (4.16) given by γ 1{2 Γ 2 pLq " minrp1{8L 2 q, 1{2s,

rB χ Y t,x,rξs t s 2 ď 1 4L 2 sup sPrt,T s N 2,C Et ´θt,ξ,p0q s , `M2
Et pB χ θ t,ξ,p0q q ˘1{2 ¯.

(4.64)

We use (4.39) (with Ψ given by (4.46)) to bound the above term. For any ε ą 0, sup

xPR d ,sPrt,T s N 2,C Et ´θt,ξ,p0q s , `M2 Et pB χ θ t,ξ,p0q q ˘1{2 ď sup pw,sqPR k ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘`M 2 Et pB χ θ t,ξ,p0q q ˘1{2 ı 2 ) (4.65) ď L 2 ε `sup pw,sqPR k ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘21 tM 2 E t pBχθ t,ξ,p0q qěεu ı 2 M 2 E pB χ θ t,ξ,p0q q ) ,
where we denoted R d ˆRm ˆRmˆd by R k and we used Cauchy-Schwarz' inequality in the last line. Recall that in the above suprema, Λ 0 takes values in R `. By Lemma 4.18, rM 2 E pB χ θ t,ξ,p0q qs 1{2 ď C}χ} 2 ď CC 1 . By uniform integrability of the family pΨ 2 ppw, sq, ξqq wPR k ,sPrt,T s , it thus suces to prove that lim PpAqÑ0

P `M2

Et pB χ θ t,ξ,p0q q ě ε ˘" 0 (4.66) in order to prove (4.60) (recall that the above probability depends on A through χ " Λ1 A ). We reapply (4.18) in Lemma 4.7, but with θ " θ " θ " θ :" θ t,ξ , ϑ " θ " θ " θ :" B χ θ t,ξ and η " Λ1 A . Following (4.64) and (4.65), we get

M 2
Et pB χ θ t,ξ,p0q q (4.67)

ď CΛ 2 1 A `1 4L 2 sup pw,sqPR k ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘`M 2 Et pB χ θ t,ξ,p0q q ˘1{2 ı 2 ) .
Multiplying by 1 A A and taking the expectation, we deduce that

E " 1 A A M 2 Et pB χ θ t,ξ,p0q q ‰ ď 1 2L 2 sup pw,sqPR k ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘`M 2 Et pB χ θ t,ξ,p0q q ˘1{2 1 A A ı 2 ) `1 2L 2 sup pw,sqPR k ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘`M 2 Et pB χ θ t,ξ,p0q q ˘1{2 1 A ı 2 ) ď 1 2 E " 1 A A M 2 Et pB χ θ t,ξ,p0q q ‰ `C sup pw,sqPR d ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppw, sq, ξq ˘21 A ı) ,
where we used Cauchy-Schwarz' inequality twice to get the last line. By uniform integrability of the family pΨ 2 ppw, sq, ξqq xPR k ,sPrt,T s , the second term in the last line tends to 0 with PpAq. Therefore, Er1 A A M 2 Et pB χ θ t,ξ,p0q qs also tends to 0 with PpAq.

Going back to (4.67), taking the root and then the expectation and splitting the expectation in the right-hand side according to the indicator functions of A A and A, we get in the same way

E "`M 2 Et pB χ θ t,ξ,p0q q ˘1{2 ‰ ď C `PpAq ˘1{2 `CE " 1 A A M 2 Et pB χ θ t,ξ,p0q q ‰ 1{2 `C sup px,sqPR d ˆrt,T s sup }Λ 0 } 2 ďL ! E " `Λ0 ^Ψppx, sq, ξq ˘21 A ı 1{2
) .

The right-hand side tends to 0 with PpAq, which proves (4.66). l 4.3. Study of the second-order dierentiability. The goal is now to discuss the second-order dierentiability of U .

4.3.1. Path property of Z t,x,rξs in S 2 prt, T s; R mˆd q. We start with the following remark.

In the previous subsection, we proved that the function B x U was Lipschitz continuous with respect to the variables x and µ. Recalling the standard representation formula 2.7), we may derive bounds for Z t,x,rξs in the space S 2 prt, T s, R mˆd q instead of H 2 prt, T s, R mˆd q (and similarly for Z t,ξ by replacing x by ξ in the above formula). Under assumption pH2q, which contains pHσq, σ is known to be bounded, so that Z t,x,rξs and Z t,ξ,rξs are indeed bounded (in L 8 ), independently of ξ. Moreover, for any p ě 1, for T ď c p with c p :" c p pLq, we can nd C p ě 0 such that, for ξ, Note that the term Φ α`1 pt, ξ, ξ 1 q comes from the fact that, when handling the dierence B x U ps, X

Z t,x,rξs s " B x U `s, X t,
ξ 1 P L 2 pΩ, F t , P; R d q, E t " sup rPrt,T s ˇˇZ t,x,rξs r ´Zt,x 1 ,rξ 1 s r ˇˇ2p ı 1{2p ď C p ´1 ^ |x ´x1 | `Φα`1 pt, ξ, ξ 1 q ( ¯, E t " sup rPrt,T s ˇˇZ t,ξ r ´Zt,ξ 1 r ˇˇ2p ı 1{2p ď C p ´1 ^ |ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 q ( ¯.
| Φα`1 pX t,ξ
s , X t,ξ 1 s qs as bound. We then apply (4.44) in Lemma 4.14 (with α `1 instead of α) to handle Φ α`1 pX t,ξ s , X t,ξ 1 s q. The part involving σ in the denition of Z t,x,rξs s can be treated by means of Lemma 4.1 using the fact that σ is Lipschitz continuous. Following Remark 4.16 and as in the statement of Lemma 4.19, the restriction of Φ α`1 pt, ¨, ¨q to the space rL 2 pΩ, F 0 , P; R d qs 2 may be assumed to be independent of t P r0, T s.

4.3.2. Partial smoothness of B µ U . Overview. By making use of (4.69), we rst discuss the existence and the smoothness of the second-order derivatives of U in the measure argument. The rst remark is that we only need to discuss partial C 2 dierentiability in order to prove the chain rule. This says that, when investigating the second-order derivatives, there is no need to prove that the function U has a twice Fréchet dierentiable lifted version. Roughly speaking, the only thing we need is the dierentiability of the mapping R d ˆRd Q px, vq Þ Ñ B µ U pt, x, µqpvq (at least when v is restricted to the support of µ), together with the continuity (in pt, x, µ, vq) of the derivatives (again, at least when v is restricted to the support of µ). In order to dierentiate in the direction v without dierentiating in the direction µ, we shall make use of Theorem 3.6, which has been specically designed for that purpose. Basically, we are to dierentiate the lifted version of B µ U pt, x, µq along trajectories pξ λ q λPR that are continuously dierentiable in L 2 pΩ, F t , P; R d q, with the constraint that all the pξ λ q λPR have the same distribution and the assumption that In this framework, we will make use of the following technical lemma: Lemma 4.26. Consider a function h : R k ˆP2 pR l q Ñ R as in pH2q, a continuously dierentiable mapping R Q λ Þ Ñ χ λ P L 2 pΩ, A, P; R l q with the property that all the χ λ have the same distribution, and a random variable P L 2 pΩ, A, P; R l q such that, for any bounded interval ra, bs Ă R, the family ´dχ λ dλ b ¯λPra,bs is uniformly square integrable (the tensorial product acting on R l ). Then, the function

@λ P R, › › d dλ ξ λ › › 8 ď 1,
R k ˆR Q pw, λq Þ Ñ E " B µ hpw, rχ λ sqpχ λ q ‰ " E " B µ hpw, rχ 0 sqpχ λ q ‰ is continuously dierentiable, with R k ˆR Q pw, λq Þ Ñ E " B w " B µ hpw, rχ 0 sq ‰ pχ λ q ı R k ˆR Q pw, λq Þ Ñ E " B v " B µ hpw, rχ 0 sq ‰ pχ λ q dχ λ dλ b ı
as respective partial derivatives in w and λ.

Proof. For w, w 1 P R k and λ, λ 1 P R, we write (thanks to pH2q):

B µ hpw 1 , rχ 0 sqpχ λ 1 q ´Bµ hpw, rχ 0 sqpχ λ q " B µ hpw 1 , rχ 0 sqpχ λ 1 q ´Bµ hpw 1 , rχ 0 sqpχ λ q `Bµ hpw 1 , rχ 0 sqpχ λ q ´Bµ hpw, rχ 0 sqpχ λ q "

ˆż 1 0 B v " B µ hpw 1 , rχ 0 sq ‰`s χ λ 1 `p1 ´sqχ λ ˘ds ˙`χ λ 1 ´χλ ˘(4.71) `ˆż 1 0 B w " B µ h `sw 1 `p1 ´sqw, rχ 0 s ˘‰`χ λ ˘ds ˙pw 1 ´wq.
Thanks to the L 2 bounds on B w rB µ hs and B v rB µ hs in pH2q, we deduce that, as pw 1 , λ 1 q Ñ pw, λq,

E "ˇˇˇˇż 1 0 B v " B µ hpw 1 , rχ 0 sq ‰`s χ λ 1 `p1 ´sqχ λ ˘ds ´Bv " B µ hpw, rχ 0 sq ‰`χ λ ˘ˇˇˇ2  Ñ 0, E "ˇˇˇˇż 1 0 B w " B µ h `sw 1 `p1 ´sqw, rχ 0 s ˘‰`χ λ ˘ds ´Bw " B µ hpw, rχ 0 sq ‰`χ λ ˘ˇˇˇ2  Ñ 0. (4.72)
Notice now from the uniform integrability property in the assumption that, as λ 1 Ñ λ (with λ 1 " λ), as for proving the rst-order continuous dierentiability and consists in discussing the continuous dierentiability of the derivative processes B χ θ t,ξ λ " pB χ X t,ξ λ , B χ Y t,ξ λ , B χ Z t,ξ λ q and B χ θ t,x,rξ λ s " pB χ X t,x,rξ λ s , B χ Y t,x,rξ λ s , B χ Z t,x,rξ λ s q with respect to λ when the family pξ λ q λPR satises the aforementioned prescriptions and χ is in L 2 pΩ, F t , P; R d q. Together with the relationship B χ Y t,x,rξ λ s t " DUpt, x, ξ λ q ¨χ, this will permit to apply Theorem 3.6 (compare in particular with (3.31)).

E " ˇˇ`χ λ 1 ´χλ λ 1 ´λ ´dχ λ dλ ˘b ˇˇ2 ı Ñ 0.
Intuitively, one has in mind to consider rst the partial second order tangent process of the McKean-Vlasov FBSDE (2.3) in the direction χ and ζ, which we shall denote by

B 2 ζ,χ θ t,ξ " pB 2 ζ,χ X t,ξ , B 2 ζ,χ Y t,ξ , B 2 ζ,χ Z t,ξ q :" rd{dλs |λ"0 B χ θ t,ξ λ
. Informally, this process should satisfy a system of the form (4.3), with coecients of the generic form (4.5).

Precisely, the coecients H should have the same decomposition as in the rst order case, see (4.49), V and V also standing for θ, θ p0q or X but V and V now standing for

B 2 ζ,χ θ, B 2 ζ,χ θ p0q or B 2
ζ,χ X (with the usual convention that the symbol p0q in V p0q and V p0q indicates the restriction to the two rst coordinates). Terms B a , Σ a , F a and G a in (4.5) should not be zero anymore and should be dened as follows for a generic coecient h that may be b, σ, f or g:

H a prq " B 2 ww h `θt,ξ r , rθ t,ξ,p0q r s ˘Bχ θ t,ξ r b B ζ θ t,ξ r `Ê " B w " B µ h `θt,ξ r , rθ t,ξ,p0q r s ˘‰`x θ t,ξ,p0q r y ˘xB χ θ t,ξ,p0q r y b B ζ θ t,ξ r ‰ `Ê " B v " B µ h `θt,ξ r , rθ t,ξ,p0q r s ˘‰`x θ t,ξ,p0q r y ˘xB χ θ t,ξ,p0q r y b xB ζ θ t,ξ,p0q r y ‰ ": H p2q a `θt,ξ r , xθ t,ξ,p0q r y, B χ θ t,ξ r , B ζ θ t,ξ r , xB χ θ t,ξ,p0q r y, xB ζ θ t, ξ,p0q r y ":
H ww a prq `Hwµ a prq `Hvµ a prq,

where H p2q a could be expressed (in an obvious way) as a function of general arguments θ r , x θp0q r y, ϑ Pay attention that there is no `second-order derivatives' in the direction of the measure (i.e. `B2 µµ h') in (4.74). Indeed, the fact that the initial conditions pξ λ q λ have the same distribution forces the solutions pθ λ q λ to be identically distributed as well. For the same reason, there is no crossed derivative of the form `Bµ rB w hs'. On the opposite, notice that pB χ θ λ q (resp. pB ζ θ λ q) are not identically distributed since the coupling between χ (resp. ζ) and ξ λ may vary. In particular, when dierentiating with respect to λ an expression of the form ÊrB µ hpθ λ , rθ λ,p0q sqpxθ λ,p0q yqxB χ θ λ,p0q ys for a function h as above, the input rθ λ,p0q s has a zero derivative as it is constant in λ, but the two last inputs, namely xθ λ,p0q y and xB χ θ λ,p0q y, may give a non-trivial contribution.

H p2q `r, θ r , x θp0q r y, ϑ 1 r , ϑ 2 r , x θ1,p0q r y, x θ2,p0q r y ˘`ϑ r , x θp0q r y :" B w hpθ r , r θp0q r sqϑ r `Ê " B µ hpθ r , r θp0q r sqpx θp0q r yqx θp0q r y ‰ `Hp2q a `θr , x θp0q r y, ϑ 1 r , ϑ 2 r , x θ1,p0q
On the model of (4.7), (4.8) and (4.9), we shall use the following assumptions on the coecients (compare also with pH2q):

|B 2 ww hpw, r V p0q sq| `p E " ˇˇB w " B µ hpw, r V p0q sq ‰ px V p0q yq ˇˇ2 ı 1{2 `p E " ˇˇB v " B µ hpw, r V p0q sq ‰ px V p0q yq ˇˇ2 ı 1{2 ď C, (4.76)
and

|B 2 ww hpw, r V p0q sq ´B2 ww hpw 1 , r V p0q1 sq| `p E " ˇˇB w " B µ hpw, r V p0q sq ‰ px V p0q yq ´Bw " B µ hpw 1 , r V p0q1 sq ‰ px V p0q1 yq ˇˇ2 ı 1 2 `p E " ˇˇB v " B µ hpw, r V p0q sq ‰ px V p0q yq ´Bv " B µ hpw 1 , r V p0q1 sq ‰ px V p0q1 yq ˇˇ2 ı 1 2 ď C `|w ´w1 | `Φα p V p0q , V p0q1 q ˘. ( 4
ď C p " ´M 4p `ϑ1 , θ1 ˘¯1{4p ´M 4p `ϑ2 , θ2 ˘¯1{4p `E" } θ1,p0q } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t ı 1{2  .
Proof. We start with the case H " B (resp. F ), or equivalently h " b (resp. f ). We use a decomposition of the same type as (4.74) (with the same notations). By conditional Cauchy-Schwartz inequality and (4.76), we can nd a constant C p such that

E t "ˆż T t |H ww a psq|ds ˙2p  ď C p }ϑ 1 } 2p H 4p ,t }ϑ 2 } 2p H 4p ,t . (4.79)
We now aim to obtain similar upper bound for the other terms in (4.74). We therefore observe, using (4.76), |H wµ a psq| ď C} θ1,p0q

s } 2 |ϑ 2 s |, so that E t "ˆż T t |H wµ a psq|ds ˙2p  ď C p }ϑ 2 } 2p H 2p ,t } θ1,p0q } 2p S 2 . ( 4
˙p 1{2p ď C p !´1 ^"E t `ε4p ˘1{4p `´M 4p `θ ´θ1 , θ ´θ 1 ˘¯1{4p `Φα `θ p0q , θp0q1 ˘ı" ´M 8p `ϑ1 , θ1 ˘¯1{8p ´M 8p `ϑ2 , θ2 ˘¯1{8p `E" } θ1,p0q } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t ı 1{2 ı) `Cp !´M 4p `ϑ1 ´ϑ11 , θ1 ´θ 11 ˘¯1{4p ´M 4p `ϑ2 , θ2 q ¯1{4p (4.82) `´M 4p `ϑ11 , θ11 ˘¯1{4p ´M 4p `ϑ2 ´ϑ21 , θ2 ´θ 21 ˘¯1{4p ) `Cp ! E " } θ1,p0q1 } 2 S 4 ,t } θ2,p0q ´θ 2,p0q1 } 2 S 4 ,t ı 1{2 `E" } θ1,p0q ´θ 1,p0q1 } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t ı 1{2 ) `Et "ˆż T t 1 t|θs´θ 1 s |ąεu |ϑ 1 s | |ϑ 2 s |ds ˙2p  1{2p .
Proof. We start with the case when H " B, F . As in the proof of Lemma 4.27, we make use of the decomposition (4.74). Denoting by H ww

E t "ˆż T t |H ww a psq ´Hww a 1 psq|ds ˙2p  1{2p ď E t "ˆż T t ˇˇ!B 2 ww h `θs , r θp0q s s ˘´B 2 ww h `θ1 s , r θp0q1 s s ˘) ϑ 1 s b ϑ 2 s ˇˇds ˙2p  1{2p `Et "ˆż T t ˇˇB 2 ww h `θ1 s , r θp0q1 s s ˘ ϑ 1 s ´ϑ11 s ( b ϑ 2 s ˇˇds ˙2p  1{2p `Et "ˆż T t ˇˇB 2 ww h `θ1 s , r θp0q1 s s ˘ϑ11 s b ϑ 2 s ´ϑ21 s ( ˇˇds ˙2p  1{2p :" A 1 `A2 `A3 . (4.83)
Bounding the dierence of the terms in B 2 ww h by a constant or by the increment of the underlying variables, we thus obtain, for any random variable ε with values in R `,

A 1 ď CE t " ! 1 ^´ε `sup sPrt,T s Φ α `θ p0q s , θp0q1 s ˘¯2p ) ˆż T t |ϑ 1 s | |ϑ 2 s |ds ˙2p  1{2p `Et "ˆż T t 1 t|θs´θ 1 s |ąεu |ϑ 1 s | |ϑ 2 s |ds ˙2p  1{2p ď C p ´1 ^"E t `ε4p ˘1{4p `sup sPrt,T s Φ α `θ p0q s , θp0q1 s ˘ı¯} ϑ 1 } H 8p ,t }ϑ 2 } H 8p ,t `Et "ˆż T t 1 t|θs´θ 1 s |ąεu |ϑ 1 s | |ϑ 2 s |ds ˙2p  .
We also have

A 2 `A3 ď C p ´}ϑ 1 ´ϑ11 } H 4p ,t }ϑ 2 } H 4p ,t `}ϑ 11 } H 4p ,t }ϑ 2 ´ϑ21 } H 4p ,t ¯.
Next, using a similar decomposition to (4.83), we compute

E "ˆż T t |H wµ a psq ´Hwµ a 1 psq|ds ˙2p  1{2p ď C p !´1 ^ }θ ´θ1 } H 4p ,t `sup sPrt,T s Φ α p θp0q s , θp0q1 s q ( ¯} θ1,p0q } S 2 }ϑ 2 } H 4p ,t `} θ1,p0q ´θ 1,p0q1 } S 2 }ϑ 2 } H 2p ,t `} θ1,p0q1 } S 2 }ϑ 2 ´ϑ21 } H 2p ,t
) .

We also get

E t "ˆż T t |H vµ a psq ´Hvµ a 1 psq|ds ˙2p  1{2p ď C p !´1 ^ }θ ´θ1 } H 2p ,t `sup sPrt,T s Φ α p θp0q s , θp0q1 s q ( ¯E" } θ1,p0q } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t ı 1{2 `E" } θ1,p0q1 } 2 S 4 ,t } θ2,p0q ´θ 2,p0q1 } 2 S 4 ,t ı 1{2 `E" } θ1,p0q ´θ 1,p0q1 } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t ı 1{2
) .

Collecting the various inequalities, we get (4.82).

The proof is quite similar when H " Σ or G, but there are two main dierences. The rst one is that, in the analysis of Σ s | ą εu. Then, E t rε 4p s 1{4p is exactly equal to }θ p0q ´θp0q1 } S 4p ,t , which is less than p M4p pθ ´θ1 , θ ´θ 1 qq 1{4p . l 4.3.5. Proof of the dierentiability of the McKean-Vlasov system. We claim:

Lemma 4.29. There exists c :" cpLq ą 0 such that, for T ď c, for χ and pξ λ q λ as in 4.3.2, the mapping

R Q λ Þ Ñ B χ θ t,ξ λ " `Bχ X t,ξ λ , B χ Y t,ξ λ , B χ Z t,ξ λ ˘,
with values in S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q, is continuously dierentiable. The derivative at λ " 0 only depends upon the family pξ λ q λPR through the value of ξ :" ξ 0 and ζ :" rd{dλs λ"0 ξ λ (see footnote 10 on page 26 for a precise meaning). It is denoted by B 2 ζ,χ θ t,ξ .

Proof.

We adapt the proof of Lemma 4.17.

To do so, we use the Picard sequence ppθ n,λ , B χ θ n,λ qq ně1 solving (4.50), with X n,λ t " ξ λ and χ λ " χ for any λ P R. The sequence converges in rS 2 prt, T s; R d q ˆS2 prt, T s, R m q ˆH2 prt, T s; R mˆd qs 2 towards pθ t,ξ λ , B χ θ t,ξ λ q, uniformly in λ in compact subsets. Pay attention that, in (4.50), the choice χ λ " χ, for all λ P R, ts the framework of Theorem 3.6 in which χ is kept frozen, independently of λ.

Similarly, we denote by ppθ n,λ , B ζ θ n,λ qq ně1 the Picard sequence solving (4.50), with X n,λ t " ξ λ and χ λ " rd{dλsξ λ for any λ P R. The sequence converges in rS 2 prt, T s; R d q Ŝ2 prt, T s, R m q ˆH2 prt, T s; R mˆd qs 2 towards pθ t,ξ λ , B ζ θ t,ξ λ q, uniformly in λ in compact subsets. In (4.50), the choice χ λ " rd{dλsξ λ , for any λ P R, ts the framework of Theorem 3.6 in which ζ " rd{dλs |λ"0 X λ , with X λ therein playing the role of ξ λ .

Notice that pθ n,λ q ně1 , which appears in each of the two Picard sequences, denotes the same process. The dierence between pB χ θ n,λ q ně1 and pB ζ θ n,λ q ně1 is that rd{dλsθ n,λ " B ζ θ n,λ but rd{dλsθ n,λ " B χ θ n,λ . The motivation for considering B χ θ n,λ is that B χ Y n,λ t converges to ErB µ U pt, ξ λ , rξ λ sqχs, which is precisely the quantity that we aim at dierentiating with respect to λ.

First step. The rst point is to prove that, for any n ě 0, the map λ Þ Ñ pB χ θ n,λ s q sPrt,T s is continuously dierentiable from R to S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q.

To do so, we recall the system (4.50):

B χ X n`1,λ s " χ `ż s t B p1q `r, θ n,λ r , xθ n,λ,p0q r y ˘`B χ θ n,λ r , xB χ θ n,λ,p0q r y ˘dr `ż s t Σ p1q pr, θ n,λ,p0q r , xθ n,λ,p0q r yq `Bχ θ n,λ,p0q r , xB χ θ n,λ,p0q r y ˘dW r B χ Y n`1,λ s " G p1q pX n`1,λ T , xX n`1,λ T yq `Bχ X n`1,λ T , xB χ X n`1,λ T y ż T s F p1q `r, θ n,λ r , xθ n,λ,p0q r y ˘`B χ θ n,λ r , xB χ θ n,λ,p0q r y ˘dr ´ż T s B χ Z n`1,λ r dW r , (4.84)
with B χ θ 0,λ " p0, 0, 0q as initialization, with a similar system for B ζ θ n,λ , replacing χ by rd{dλsX λ .

Generally speaking, the proof is the same as that of Lemma 4.17: We argue by induction, assuming at each step n ě 1 that λ Þ Ñ pB χ θ n,λ s q sPrt,T s is continuously dierentiable (the derivative being denoted by pB 2 ζ,χ θ n,λ s q sPrt,T s ); we prove rst the dierentiability of the forward component and then the dierentiability of the backward one in (4.84).

In comparison with the proof of Lemma 4.17, we must pay attention to the two following points. The rst question is to justify the dierentiability under the various expectation symbols that appear in the denitions of B p1q , Σ p1q , F p1q and G p1q . Thanks to (4.51) and from the fact that the sequence rd{dλspξ λ q is bounded in L 8 (see 4.70), we know that

sup ně1 " M 4 Et `Bχ θ n,λ ˘ı1{4 ď C " |χ| `}χ} 2 ‰ , sup ně1 " M 2p Et `Bζ θ n,λ ˘ı1{4 ď C, (4.85)
so that Lemma 4.26 applies with rd{dλsχ λ " B ζ θ n,λ and " B χ θ n,λ and permits to guarantee the dierentiability of the terms driven by an expectation.

Another problem is that the coecients now involve the product of two terms that are dierentiable in H 2 prt, T s; R k q (or S 2 prt, T s; R k q in some cases), for a suitable k ě 1, so that the product is dierentiable in H 1 prt, T s; R k q (or S 1 prt, T s; R k q) only (for another k).

We make this clear for p ş s t B p1q pr, θ n,λ r , xθ n,λ,p0q r yqpB χ θ n,λ r , xB χ θ n,λ,p0q r yqdrq tďsďT , the other terms being handled in a similar fashion. Repeating the analysis of Lemma 4.17, it is dierentiable from R to S 1 prt, T s; R d q, the derivative process writing

ż s t B p2q `r, Θ n,λ r ˘`B 2 ζ,χ θ n,λ r , xB 2 ζ,χ θ n,λ,p0q r y ˘dr, with Θ n,λ r " `θn,λ r , xθ n,λ,p0q r y, B χ θ n,λ r , B ζ θ n,λ r , xB χ θ n,λ,p0q r y, xB ζ θ n,λ,p0q r y ˘(4.86)
with s P rt, T s. As explained in (4.75) and on the model of Lemma 4.26, we here used the crucial assumption that all the pξ λ q λ are identically distributed to get the shape of B p2q . In order to prove dierentiability in S 2 prt, T s; R d q, a uniform integrability argument is needed. Assume indeed that a path R Q λ Þ Ñ ϑ λ " pϑ λ s q sPrt,T s P S 1 prt, T s, R k q, for some k ě 1, is continuously dierentiable and that, for any nite interval I, the family

psup sPrt,T s |rd{dλsϑ λ s | 2 q λPI is uniformly integrable. Then, R Q λ Þ Ñ ϑ λ P S 2 prt, T s; R k q is continuously dierentiable.
In our framework, the form of prd{dλsϑ λ s q sPrt,T s is explicitly given by (4.86). The coecient B p2q may be expanded by means of (4.75). Clearly, it involves a linear term in

pB 2 ζ,χ θ n,λ r , xB 2 ζ,χ θ n,λ,p0q r
yq and the remainder B p2q a pΘ n,λ r q. By pH1q and pH2q, we get that

sup sPrt,T s ˇˇd dλ ϑ λ s ˇˇď C "ˆż T t " |B 2 ζ,χ θ n,λ r | 2 `}B 2 ζ,χ θ n,λ r } 2 2 ‰ dr ˙1{2 `E ż T t |B p2q a pΘ n,λ r q|dr  . (4.87) By continuity of R Q λ Þ Ñ B 2
ζ,χ θ n,λ P H 2 prt, T s; R k q, the rst term in the right-hand side is uniformly square integrable. We thus discuss the term in B p2q a . Recalling Lemma 4.18 and the similar version (4.51) for the Picard scheme in Lemma 4.17 (which is given for p " 1 only but which could be generalized), we have the more general version of (4.85):

sup ně1 " M 2p Et `Bχ θ n,λ ˘ı1{2p ď C p " |χ| `}χ} 2 ‰ , sup ně1 " M 2p Et `Bζ θ n,λ ˘ı1{2p ď C p , (4.88)
so that, by Lemma 4.27 (with θ " θ :" θ n,λ , ϑ 1 " θ1 :" B χ θ n,λ and ϑ 2 " θ2 :" B ζ θ n,λ , and, as usual, for T ď c)

E t "ˆż T t |B p2q a pΘ n,λ r qdr ˙2p  1{2p ď C p `|χ| `}χ} 2 ˘. (4.89)
Now, choosing p " 2, we get that, for any event A P A,

E " 1 A ˆż T t |B p2q a pΘ n,λ r qdr ˙2 ď CE " E t r1 A s 1 2 `|χ| `}χ} 2 ˘2‰ " CE " E t " 1 A p|χ| `}χ} 2 q 2 ‰ 1 2 p|χ| `}χ} 2 q ı ď C}χ} 2 E " 1 A `|χ| `}χ} 2 ˘2‰ 1 2 ,
where we have used the fact that χ is F t measurable. The above bound permits to establish the required uniform integrability argument, a similar argument holding true for the terms driven by F p2q a , Σ p2q a and G p2q a . Inductively, this permits to prove that the map λ Þ Ñ B χ θ n,λ is continuously dierentiable from R to S 2 prt, T s, R d q ˆS2 prt, T s, R m q Ĥ2 prt, T s, R mˆd q. With the same notation as in (4.86), we have, for any n ě 0,

B 2 ζ,χ X n`1,λ s " ż s t B p2q `r, Θ n,λ r ˘`B 2 ζ,χ θ n,λ r , xB 2 ζ,χ θ n,λ,p0q r y ˘dr `ż s t Σ p2q `r, Θ n,λ,p0q r ˘`B 2 ζ,χ θ n,λ,p0q r , xB 2 ζ,χ θ n,λ,p0q r y ˘dW r , (4.90)
and

B 2 ζ,χ Y n`1,λ s " G p2q pΞ n`1,λ T q `B2 ζ,χ X n`1,λ T , xB 2 ζ,χ X n`1,λ T y ż T s F p2q `r, Θ n,λ r ˘`B 2 ζ,χ θ n,λ r , xB 2 ζ,χ θ n,λ,p0q r y ˘dr ´ż T s B 2 ζ,χ Z n`1,λ r dW r , (4.91)
where we have let:

Θ n,λ,p0q r " `θn,λ,p0q r , xθ n,λ,p0q r y, B χ θ n,λ,p0q r , B ζ θ n,λ,p0q r , xB χ θ n,λ,p0q r y, xB ζ θ n,λ,p0q r y ˘, Ξ n,λ T " `Xn,λ T , xX n,λ T y, B χ X n,λ T , B ζ X n,λ T , xB χ X n,λ T y, xB ζ X n,λ T y ˘.
Second step. Convergence of the sequence pB 2 ζ,χ θ n,λ q ně0 in the space S 2 prt, T s, R d q Ŝ2 prt, T s, R m q ˆH2 prt, T s, R mˆd q is then shown as in the proof of Lemma 4.17. Generally speaking, the point is to compare approximations at steps n and n `1 and then to prove that the norm of the dierence decays geometrically fast as n tends to 8. As in the rst step, some precaution is needed as the system diers from the one involved in the proof of Lemma 4.17, the dierence coming from the remainder term H p2q a in (4.74). Precisely, the proof of Lemma 4.17 relies on Lemmas 4.7 and 4.9, with 0 as remainder term R a , but, in the current framework, the remainder term is equal to pH p2q a pΘ n,λ s qq sPrt,T s when H " B, Σ or F and G p2q a pΞ n`1,λ T q when H " G, and is thus non-zero. The analysis thus imitates the proof of Lemma 4.17, but with a non-zero remainder term ∆R a in (4.28) that corresponds to the dierence of the remainders R a at steps n and n`1. In short, it is enough to prove that Er∆R 2 a s tends to 0 as n tends to 8 (to simplify, we omit to specify the index n in ∆R 2 a ). By convergence of pθ n,λ s , B χ θ n,λ s , B ζ θ n,λ s q sPrt,T s to pθ t,ξ λ s , B χ θ ξλ s , B ζ θ t,ξ λ s q sPrt,T s , we can deduce from Lemma 4.28 (with θ " θ :" θ n,λ , ϑ 1 " θ1 :" B χ θ n,λ , ϑ 2 " θ2 :" B ζ θ n,λ and θ 1 " θ1 :" θ n`1,λ , ϑ 1,1 " θ1,1 :" B χ θ n`1,λ , ϑ 2,1 " θ2,1 :" B ζ θ n`1,λ ) that ∆R 2 a tends to 0 in probability as n tends to 8, the convergence being uniform with respect to λ in compact subsets: In (4.82), we can check that all the terms not containing the variable ε tend 0; choosing ε as a small deterministic real, it is standard to prove that the expectation of the last term in (4.82) tends to 0. The latter property follows from the following fact: For any compact I Ă R, the sequence pB χ θ n,λ q ně1 and pB ζ θ n,λ q ně1 are convergent in the L 2 sense on Ω ˆrt, T s, so that the families pB χ θ n,λ q ně1,λPI and pB ζ θ n,λ q ně1,λPI are uniformly square integrable on Ω ˆrt, T s.

The convergence of ∆R 2 a to 0 actually holds in the L 1 sense on Ω, since the bound (4.89) (with similar bounds for F , Σ and G) allows to apply another argument of uniform integrability. The convergence is uniform with respect to λ in compact sets. This proves the continuous dierentiability of R Q λ Þ Ñ B χ θ t,ξ λ P S 2 prt, T s; R d q ˆS2 prt, T s; R m q Ĥ2 prt, T s; R mˆd q. The derivative at λ " 0 satises a system of the form (4.3) (obtained by an obvious adaptation of (4.90) and (4.91)), which is uniquely solvable in short time. This proves that the derivative at λ " 0 only depends on the family pX λ q λPR through X 0 and ζ.

We complete the analysis as in the proof of Lemma 4.17. 4.3.6. Estimates of the directional derivatives of the McKean-Vlasov system. We claim: Lemma 4.30. Recall the notations (4.16). For any p ě 1, there exist two constants c :" c p pLq ą 0 and C p , such that, for T ď c p (and with γ " c p in (4.16)),

" M 2p Et `B2 ζ,χ θ t,ξ ˘‰1{p2pq ď C p `|χ| `}χ} 2 ˘.
Proof. The result follows from Corollary 4.8 with η " 0, θ " θ :" θ t,ξ , ϑ " θ :" B 2

ζ,χ θ t,ξ , H given by (4.74), and, in particular, with remainders R 2p a and R 2 a coming from H p2q a in (4.74). Recalling Lemma 4.18 and the assumption }ζ} 8 ď 1, the remainders may be estimated by means of Lemma 4.27, with θ " θ :" θ t,ξ , ϑ 1 " θ1 :" B χ θ t,ξ and ϑ 2 " θ2 :" B ζ θ t,ξ . l

We now discuss the continuity with respect to ξ. We claim:

Lemma 4.31. For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p such that, for T ď c p (and with γ " c p in (4.16)),

" M 2p Et `B2 ζ,χ θ t,ξ ´B2 ζ,χ θ t,ξ 1 ˘ı1{2p ď C p ´1 ^ |ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 q ( ¯`|χ| `}χ} 2 ˘, (4.92)
where Φ α`1 pt, ¨q : rL 2 pΩ, F t , P; R d qs 2 Ñ R `is continuous at any point of the diagonal, does not depend on p and satises (4.10) with α replaced by α `1. The restriction of Φ α`1 pt, ¨, ¨q to rL 2 pΩ, F 0 , P; R d qs 2 may be assumed to be independent of t P r0, T s.

Proof. Generally speaking, the strategy is to apply Corollary 4.12, with η " 0, θ ξ " θξ :" θ t,ξ , ϑ " θ :" B 2 ζ,χ θ t,ξ (and the same for ξ 1 ), H given by (4.74) and, in particular, with remainders R 2p a and R 2 a coming from H p2q a in (4.74) (and the same for the remainders labelled with `prime'). As in the proof of the previous Lemma 4. [START_REF] Ma | On non-Markovian forward-backward SDEs and backward stochastic PDEs[END_REF], we can bound the remainders pR 2p a q 1{2p by C p p|χ| `}χ} 2 q. In order to estimate p∆R 2p a q 1{2p , we apply Lemma 4.28. A crucial fact is that we have (4.69). This says that, instead of working in conditional norm r M2p ¨ s 1{2p for estimating the distance between θ t,ξ and θ t,ξ 1 , we can directly work with the conditional norm } ¨}S 2p ,t `} ¨}S 2 . As a byproduct, we can choose ε " sup sPrt,T s |θ t,ξ s ´θt,ξ 1 s | in (4.82). By (4.69), we thus get C p p1 ^t|ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 quqp|χ| `}χ} 2 q as a bound for the terms containing the symbol ε in (4.82) (Φ α`1 being independent of t when ξ and ξ 1 are F 0 -measurable). By Lemmas 4. [START_REF] Gomes | Extended mean eld games -formulation, existence, uniqueness and examples[END_REF] for the process pθ t,x,rξs , B χ θ t,x,rξs q (instead of pθ t,ξ , B χ θ t,ξ q). Considering a continuously dierentiable path λ Þ Ñ ξ λ from R into L 2 pΩ, F t , P; R d q such that |rd{dλsξ λ | ď 1, we are rst to prove that the mapping R Q λ Þ Ñ B χ θ t,x,rξ λ s P S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆ H 2 prt, T s; R mˆd q is continuously dierentiable. Before we discuss the proof, we must say a word about the notation itself, which is slightly ambiguous. Since the law of ξ λ is independent of λ, we could be indeed tempted to say that B χ θ t,x,rξ λ s is independent of λ, which is obviously false. The reason is that, in the coecients driving the system satised by B χ θ t,x,rξ λ s , there are terms of the form ÊrB µ Hpθ t,x,rξ λ s , rθ t,ξ λ ,p0q sqpxθ t,ξ λ ,p0q yqxB χ θ t,ξ λ ,p0q ys, see (4.53), which explicitly depend upon the joint law of χ and ξ λ . Clearly, there is no reason for the joint law to be independent of λ.

Recalling (4.53), we know that B χ θ t,x,rξ λ s satises a standard linear FBSDE with ÊrB µ Hpθ t,x,rξ λ s , rθ t,ξ λ ,p0q sqpxθ t,ξ λ ,p0q yqxB χ θ t,ξ λ ,p0q ys as ane part. The coecients of the FBSDE read as coecients parametrized by λ through the values of pθ t,x,rξ λ s , θ t,ξ λ , B χ θ t,ξ λ q. Now that the continuous dierentiability of R Q λ Þ Ñ pθ t,x,rξ λ s , θ t,ξ λ , B χ θ t,ξ λ q has been proved, we can repeat the arguments used in the proof of Lemma 4.29 to show that

R Q λ Þ Ñ B χ θ t,
x,rξ λ s P S 2 prt, T s; R d q ˆS2 prt, T s; R m q ˆH2 prt, T s; R mˆd q is also continuously dierentiable. (The complete proof is left to the reader.)

With the notation ζ :" rd{dλs |λ"0 ξ λ , we denote the second-order tangent process by B 2

ζ,χ θ t,x,rξs :" rd{dλs |λ"0 B χ θ t,x,rξ λ s . It satises a system of the form (4.3) with θ " θ t,x,rξs , θ " θ t,ξ , ϑ " B

" M 2p E `B2 ζ,χ θ t,x,rξs ´B2 ζ,χ θ t,x 1 ,rξ 1 s ˘ı1{2p ď C p `|x ´x1 | `Φα`1 pt, ξ, ξ 1 q ˘}χ} 2 ,
where Φ α`1 pt, ¨q : rL 2 pΩ, F t , P; R d qs 2 Ñ R `is continuous at any point of the diagonal, does not depend on p and satises (4.10) with α replaced by α `1. The restriction of Φ α`1 pt, ¨, ¨q to rL 2 pΩ, F 0 , P; R d qs 2 may be assumed to be independent of t P r0, T s.

Proof. Loosely speaking, the result is similar to Lemmas 4.30 and 4.31, but with the realizations of ξ and ξ 1 therein replaced by x and x 1 . Actually, the main dierence with the computations made for the McKean-Vlasov system comes from the shape of the remainder R a that is implemented in the stability Corollary 4.12. In the proofs of Lemmas 4.30 and 4.31, the denition of the remainder R a is based on the formula (4.74).

In the current framework, it is based on the formula (4.93), which is slightly dierent. It can be estimated by means of Lemma 4.28. The proof is then completed as that one of Lemma 4.21.

l

' for any t P r0, T s and µ P P 2 pR d q, the function R d Q x Þ Ñ U pt, x, µq is C 2 and the functions r0, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ U pt, x, µq, r0, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ B x U pt, x, µq and r0, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ B 2 xx U pt, x, µq are continuous, ' for any pt, xq P r0, T s ˆRd , the function P 2 pRq Q µ Þ Ñ U pt, x, µq is partially C 2 ; for any pt, µq P r0, T s ˆRd ˆP2 pR d q, there exists a version of

R d Q v Þ Ñ B µ U pt, x, µqpvq P R d such that R d ˆRd Q px, vq Þ Ñ B µ U pt,
x, µqpvq P R d is dierentiable at any px, vq such that v P Supppµq, the partial derivative R d ˆRd Q px, vq Þ Ñ B v rB µ U pt, x, µqspvq being continuous at any pw, vq such that v P Supppµq and the partial derivative R d ˆSupppµq Q px, vq Þ Ñ B x rB µ U pt, x, , µqspvq being continuous in px, vq.

Moreover, we can nd a constant C such that, for all x P R d , for all ξ P L 2 pΩ, A, P; R d q,

ˇˇB 2 xx U pt, x, rξsq ˇˇ`E "ˇˇB x " B µ U pt, x, rξsq ‰ pξq ˇˇ2 ‰ 1{2 `E"ˇˇB v " B µ U pt, x, rξsq ‰ pξq ˇˇ2 ‰ 1{2 ď C,
and, for all x, x 1 P R d , for all ξ, ξ 1 P L 2 pΩ, A, P; R d q,

ˇˇB 2 xx U pt, x, rξsq ´B2 xx U pt, x 1 , rξ 1 sq ˇĚ "ˇˇB x " B µ U pt, x, rξsq ‰ pξq ´Bx " B µ U pt, x 1 , rξ 1 sq ‰ pξ 1 q ˇˇ2 ‰ 1{2 `E"ˇˇB v " B µ U pt, x, rξsq ‰ pξq ´Bv " B µ U pt, x 1 , rξ 1 sq ‰ pξ 1 q ˇˇ2 ‰ 1{2 ď C |x ´x1 | `Φα`1 pξ, ξ 1 q ( ,
where Φ α`1 : rL 2 pΩ, A, P; R d qs 2 Ñ R `satises (4.10), with α replaced by α `1. In particular, for any x P R d and µ P P 2 pR d q, we can nd a locally Lipschitz continuous version of the mappings

R d Q v Þ Ñ B x rB µ U pt, x, µqspvq and R d Q v Þ Ñ B v rB µ U pt, x, µqspvq.
The functions r0, T s ˆRd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B 2 xx U pt, x, rξsq P R d , r0, T s Rd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B x rB µ U pt, x, rξsqspξq P L 2 pΩ, A, P; R d q and r0, T s ˆRd L2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B v rB µ U pt, x, rξsqspξq P L 2 pΩ, A, P; R d q are continuous.

Proof. We rst apply Theorem 3.6 in order to prove the C 2 -partial property of µ Þ Ñ U pt, x, µq. By Theorem 4.25, we already know that the lifted version L 2 pΩ, A,

P; R d q Q ξ Þ Ñ Upt, x, ξq " U pt, x,
x P R d , a version of R d Q v Þ Ñ B µ U pt, x, µqpvq such that the derivative mapping R d Q v Þ Ñ B v rB µ U pt,
x, µqspvq is continuous on compact subsets of R d , uniformly in x P R d . Using the same trick as in (3.33), we deduce that the family pR d Q v Þ Ñ B v rB µ U pt, x, µqspvqq xPR d is relatively compact for the topology of uniform convergence on compact subsets. Considering a sequence px n q ně1 that converges to x, we already know that the sequence of functions

pR d Q v Þ Ñ B v rB µ U pt, x n , µqspvq P R dˆd q ně1 converges in L 2 pR d , µ; R dˆd q to R d Q v Þ Ñ B v rB µ U pt, x, µqspvq P R dˆd . Since B v rB µ U pt, x,
µqs is uniquely dened on the support of µ, the limit of any converging subsequence (for the topology of uniform convergence on compact subsets of R d ) of pB v rB µ U pt, x n , µqsp¨qq ně1 coincides with B v rB µ U pt, x, µqsp¨q on the support of µ. We deduce that the function R d ˆRd Q px, vq Þ Ñ B v rB µ U pt, x, µqspvq P R dˆd is continuous at any px, vq such that v P Supppµq.

Proving a similar version of Lemma 4.32, but for B 2 xx θ t,x,rξs , we can show in the same way that U is twice dierentiable in x and satises

ˇˇB 2 xx U pt, x, rξsq ˇˇ, ˇˇB 2 xx U pt, x, rξsq ´B2 xx U pt, x 1 , rξ 1 sq ˇˇď C " |x ´x1 | `Φα`1 pξ, ξ 1 q ‰ ,
We notice indeed that, for ξ " µ, B 2 xx Y t,x,rξs t coincides with B 2 xx U pt, x, µq. Similarly, we can investigate B x rB χ θ t,x,rξs s. By means of Lemma 6.1 in Appendix, we can prove that, once a continuous version of B µ U pt, x, µq has be chosen for any pt, x, µq P r0, T s ˆRd ˆP2 pR d q, the function R d Q x Þ Ñ B µ U pt, x, µqpvq is dierentiable at any point px, vq such that v P Supppµq, the derivative function R d ˆSupppµq Q px, vq Þ Ñ B x rB µ U pt, x, µqspvq being continuous. Combining with the continuous dierentiability property in v, we deduce that the mapping R d ˆRd Q px, vq Þ Ñ B µ U pt, x, µqpvq is dierentiable at any point px, vq such that v P Supppµq, with the aforementioned prescribed continuity properties of the partial derivatives.

Then B x rB χ Y t,x,rξs t s identies with ErB x rB µ U pt, x, µqspξqχs. Moreover,

E " |B x rB µ U pt, x, rξsqspξq| 2 ‰ 1{2 ď C, E " |B x rB µ U pt, x, rξsqspξq ´Bx rB µ U pt, x 1 , rξ 1 sqspξ 1 q| 2 ‰ 1{2 ď C " |x ´x1 | `Φα`1 pξ, ξ 1 q ‰ .
Generally speaking, time continuity of the derivatives can be proved as in Theorem 4.25. Anyhow, some precaution is needed since the drivers of the backward equations that represent all the second-order derivatives involve quadratic terms in B χ Z t,ξ and B χ Z t,x,rξs , see for instance (4.74). The a priori diculty is that, so far, we have exhibited bounds for B χ Z t,ξ and B χ Z t,x,rξs in H norm only, which might not suce for investigating the time regularity. The key point is then to notice that all these terms may be estimated in S instead of H norm. The trick is to invoke the representation formula (4.68) for the process Z t,x,rξs , to dierentiate it and then to make use of the bounds we just proved for B Recalling that θ t,ξ r " pX t,ξ r , Y t,ξ r , B x U pr, X t,ξ r , rX t,ξ r sqσpX t,ξ r , Y t,ξ r qq, we deduce from Theorem 4.25 (smoothness of B x U both in time and in space) that it converges (in L 2 ) to θ t,ξ t as r tends to t, proving that the supremum above tends to 0 as h tends to 0. Now, using the time continuity of the derivatives B µ U and B v rB µ U s (see Theorem 4.33), we deduce that there exists a function ε : R Q u Þ Ñ ε u P R `, with lim uÑ0 ε u " 0, such that ˇˇˇU `t `h, x, rX rU pt`h, x, rξsq´U pt, x, rξsqs{h has a limit as h tends to 0. As in Subsection 2.3, the right derivative in time satises (2.12) and is thus continuous in time. Since U is obviously continuous in time, we deduce that the mapping r0, T s Q t Þ Ñ U pt, x, rξsq is dierentiable and that the PDE (2.12) holds true.

5. Large population stochastic control proof of Theorem 2.9

In this section, we discuss two applications of our previous results to large population stochastic control. The rst application is related to mean-eld games, whilst the second one is related to the optimal control of McKean-Vlasov equations.

5.1. The global smoothness of the decoupling eld. So far, smoothness of the decoupling eld U has been discussed for small time intervals r0, T s; namely for T ď δ 0 where δ 0 ą 0 only depends upon the Lipschitz constants of the coecients b, f , σ and g, denoted by the common letter L in condition pH0q(i). A natural, though quite challenging, question concerns the possible extension of such a result to the case when T is arbitrarily large.

The principle for extending the result to an arbitrarily large time horizon is discussed in the earlier paper [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]. It consists of a backward recursion starting from the terminal time T . Thanks to the short time result proved in the previous section, the mapping rT ´δ0 , T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ U pt, x, µq P R m is rigorously dened as the initial value Y t,x,µ t of the backward component of the system (2.4), existence and uniqueness of the solution of the forward-backward system following from the condition T ´t ď δ 0 . By Lemma 4.1, U is Lipschitz continuous in px, µq, uniformly in t P rT ´δ0 , T s. Up to a modication of the choice of the constant δ 0 , δ 0 still depending on the Lipschitz constants of the coecients only, the results established in Section 4 show that, under the assumptions detailed in Subsection 2.4, U belongs to the class Ť βě0 D β prT ´δ0 , T sq. As in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF], we proceed by reapplying the short time existence, uniqueness and dierentiability result to a new interval of the form rT ´pδ 0 `δ1 q, T ´δ0 s, with the new terminal condition U pT ´δ0 , ¨, ¨q at time T ´δ0 replacing the terminal condition g at time T . A preliminary condition for iterating the short time solvability property is that U pT ´δ0 , ¨, ¨q is an admissible boundary condition. Under pH2q, Theorems 4.25 and 4.33 say that it is indeed the case, up to a deterioration of α into α `1, the exponent α driving the local Lipschitz regularity of the derivatives of the coecients in pH1q and pH2q. This makes possible to reapply the existence and uniqueness result for short time horizons with α be replaced by α `1. Fortunately, the length δ 1 of the new interval of existence and uniqueness only depends on the Lipschitz constant of b, f , σ and U pT ´δ0 , ¨, ¨q. In particular, it does not suer from the deterioration of the exponent α into α `1, which is a crucial fact. As a result we are able to extend the denition of U to rT ´pδ 0 `δ1 q, T δ0 sˆR d ˆP2 pR d q. Since the new terminal condition U pT ´δ0 , ¨, ¨q has the same properties as g (but possibly with a dierent Lipschitz constant and a dierent α), the extended version of U is in the class D α`1 prT ´pδ 0 `δ1 q, T sq Ă Ť βě0 D β prT ´pδ 0 `δ1 q, T sq.

The argument can be applied recursively on a sequence of small intervals of the form rT ´pδ 0 `¨¨¨`δ n`1 q, T ´pδ 0 `¨¨¨`δ n qs, n ě 0. Of course, the issue is that the lengths pδ n q ně0 may be smaller and smaller so that the sum ř ně0 δ n may not exceed T . This happens if the Lipschitz constant of U at times pT ´pδ 0 `¨¨¨`δ n qq ně1 blows up before that the sequence pδ 0 `¨¨¨`δ n q ně1 exceeds T . Put it dierently, the construction of the smooth decoupling eld U on r0, T s ˆRd ˆP2 pR d q can be achieved by means of a backward recursion provided that the Lipschitz constant of U pt, ¨, ¨q remain bounded as t runs backward along the induction.

The crux of the matter is thus to get such a Lipschitz estimate. In the following, we present two examples, derived from large population stochastic control, for which the following assumption holds true: Assumption(pH3q). For any t P r0, T s and any square integrable F t -measurable random variable ξ, the system (2.3) has a unique solution pX t,ξ s , Y t,ξ s , Z t,ξ s q sPrt,T s and it satises, for all ξ, ξ 1 P L 2 pΩ, F t , P; R d q,

E " |Y t,ξ t ´Y t,ξ 1 t | 2 ‰ 1{2 ď ΛE " |ξ ´ξ1 | 2 ‰ 1{2 , (5.1)
with Λ a positive constant that does not depend on ξ, ξ 1 nor on t.

We will show below that, under pH3q, the decoupling eld U constructed along the induction must satisfy at any time t at which it has been dened @ ξ, ξ 1 P L 2 pΩ, A, P; R d q, E " |U pt, ξ, rξsq ´U pt,

ξ 1 , rξ 1 sq| 2 ‰ 1{2 ď ΛE " |ξ ´ξ1 | 2 ‰ 1{2 . (5.2)
Although it is a rst step in the control of the Lipschitz constant for U , it remains insucient for our purposes. The reason is that the control is here stated along the diagonal only. Fortunately, the next Lemma permits to ll the gap and to bound the Lipschitz constant of U , in x and µ, on the entire domain:

Lemma 5.1. Under pH2q, assume that U has been constructed on some interval rT 0 , T s, for T 0 P r0, T s. Assume moreover that it satises (5.2) for any t P rT 0 , T s and that it is continuously dierentiable in the directions x and µ at any time t P rT 0 , T s. Then, we can nd a constant Λ, independent of T 0 , such that for t P rT 0 , T s, x, x 1 P R d and µ, µ 1 P P 2 pR d q:

|U pt, x, µq ´U pt,

x 1 , µ 1 q| ď Λ`| x ´x1 | `W2 pµ, µ 1 q ˘.
Proof.

Step 1. Applying Proposition 3.8 (with α " 0) we get that U is Λ-Lipschitz continuous in x, or equivalently that }B x U pt, ¨, ¨q} 8 ď Λ for t P rT 0 , T s.

Step 2a. Now, for t P rT 0 , T s, x P R d and ξ, ξ 1 P L 2 pΩ, A, P; R d q, we have |U pt, x, rξsq ´U pt, x, rξ 1 sq| " ˇˇˇż

1 0 E " B µ U pt, x, rp1 ´λqξ `λξ 1 sq `p1 ´λqξ `λξ 1 ˘`ξ ´ξ1 ˘‰dλ ˇˇď E " |ξ 1 ´ξ| 2 ‰ 1{2 ż 1 0 E "ˇˇB µ U pt, x, rp1 ´λqξ `λξ 1 sq `p1 ´λqξ `λξ 1 ˘ˇ2 ‰ 1{2 dλ.
In particular, in order to complete the proof, it suces to nd a constant C, independent of T 0 , such that, for all pt, x, µq P rT 0 , T s ˆRd ˆP2 pR d q, E " |B µ U pt, x, µqpξq| 2 ‰ 1{2 ď C.

(5.3)

Step 2b. Combining Step 1 and (5.2), we obtain

E " |U pt, ξ, rξsq ´U pt, ξ, rξ 1 sq| 2 ‰ 1{2 ď 2ΛE " |ξ ´ξ1 | 2 ‰ 1{2 ,
which at the level of the gradient says (choosing ξ 1 ´ξ " hχ, letting h tend to 0 and applying Fatou's lemma)

@χ P L 2 pΩ, A, P; R d q, E " Ê" B µ U pt, ξ, rξsqpxξyqxχy ‰ 2 ı 1{2 ď 2ΛEr|χ| 2 s 1{2 . (5.4)
This control is weaker than (5.3). In order to get (5.3), the strategy is to apply, on some small interval rt, Ss, the results proved in Section 4 on the rst-order dierentiability of U with respect to the measure. Assuming that ξ is F t measurable, we make use of Lemma 4.17 but on the interval rt, Ss and with g replaced by U pS, ¨, ¨q, the value of S being specied next. In the backward component of the system of the type (4.3) satised by the derivative process pB χ X t,ξ s , B χ Y t,ξ s , B χ Z t,ξ s q sPrt,Ss , the boundary condition reads as

B χ Y t,ξ S " B x U `S, X t,ξ S , rX t,ξ S s ˘Bχ X t,ξ S `Ê " B µ U `S, X t,ξ S , rX t,ξ S s ˘`xX t,ξ S y ˘xB χ X t,ξ S y ‰ .
Now, by the a priori bound (5.4) and Step 1, we get that

E " |B χ Y t,ξ S | 2 ‰ 1{2 ď CE " |B χ X t,ξ S | 2 ‰ 1{2 .
(5.5)

Above and in the computations below, the constant C may change from line to line, it depends on the parameters in assumptions and, importantly, is uniform with respect to 0 ď T 0 ď t ď S ď T . The bound (5.5) reads as a Lipschitz bound (in L 2 ), with a constant C.

We can make use of (4.24) in Corollary 4.8, with p " 1, γ ď 1{Γ 1 , g " 0, Ĝ " 0, G a pSq " B χ Y t,ξ S (which is to say, in rough terms, that we put the whole terminal condition in the remainder) and r0, T s replaced by rt, Ss.

;R d q Ê" |B µ U pt, x, rξsqpxξyq| 2 ‰ 1{2 ď C ´1 `sup xPR d ,ξPL 2 pΩ,A,P;R d q Ê" |B µ U `S, x, rξs ˘pxξyq| 2 ‰ 1{2 ¯.
Since the terms in the suprema only depend on the law of ξ, we can assume that the supremum in the left-hand side is taken over ξ P L 2 pΩ, A, P; R d q. Assuming without any loss of generality that C ě 1 and iterating the inequality, we get 1 `sup

xPR d ,ξPL 2 pΩ,A,P;R d q Ê" |B µ U pt, x, rξsqpxξyq| 2 ‰ 1{2 ď 2C ´1 `sup xPR d ,ξPL 2 pΩ,A,P;R d q Ê" |B µ U `S, x, rξs ˘pxξyq| 2 ‰ 1{2 ď p2Cq n ´1 `sup xPR d ,ξPL 2 pΩ,A,P;R d q Ê" |B µ g `x, rξs ˘pxξyq| 2 ‰ 1{2 ¯,
with n " rpT ´tq{cs. Recalling the notation L in pH0q(i), we deduce that sup

xPR d ,ξPL 2 pΩ,A,P;R d q Ê" |B µ U pt, x, rξsqpxξyq| 2 ‰ 1{2 ď LC T {c`1 ,
which proves (5.3) and thus completes the proof. l Proposition 5.2. Assume that b, f , σ and g satisfy pH2q and that the statement pH3q holds true. Then there exists a mapping U : r0, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ U pt, x, µq P R m , Lipschitz continuous in px, µq, uniformly in t P r0, T s, such that, for all t P r0, T s and ξ P L 2 pΩ, F t , P; R d q, Y t,ξ s " U `s, X t,ξ s , rX t,ξ s s ˘.

Moreover, U belongs to Ť βě0 D β and satises the master equation (2.12).

Proof. The proposition is proved by induction. Given a large integer N ě 1 (the value of which is xed below), let δ " T {N . The induction hypothesis reads, for n P t1, . . . , N u: pI n q : There exists a mapping U : rT ´nδ, T s ˆRd ˆP2 pR d q Q pt, x, µq Þ Ñ U pt, x, µq P R m that belongs to Ť βě0 D β prT ´nδ, T sq such that (i) for any t P rT ´nδ, T s, the function U pt, ¨, ¨q satises the same assumption as g in pH0q(i), pH1q pH2q, but with the constant L replaced by Λ coming from Lemma 5.1 ( L and α being replaced by some Ln and αn );

(ii) U satises the master PDE (2.12) on rT ´nδ, T s ˆRd ˆP2 pR d q (iii) for all t P rT ´nδ, T s and ξ P L 2 pΩ, F t , P; R d q, Y t,ξ t " U pt, ξ, rξsq.

Step 1. In this step, we rst specify the value of N and we prove that pI 1 q is satised. First, notice that Λ in Lemma 5.1 may be assumed to be larger than L in pH0q(i), pH1q and pH2q. We then choose N as the smallest integer such that δ :" T {N ď cp Λq, where c is given by Theorems 4.25 and 4.33 (or more precisely by the minimum of the c's in these two statements). For T ´t ď δ, we know that, for any x P R d and µ P P 2 pR d q, the system (2.4) has a unique solution pX t,x,µ s , Y t,x,µ s , Z t,x,µ s q sPrt,T s and, by Theorems 4.25 and 4.33, U belongs to Ť βě0 D β prT ´δ, T sq and satises the master equation on rT ´δ, T s ˆRd ˆP2 pR d q. Now, by Corollary 1.5 in [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] (which holds true for small time horizons), we can replace x by a square-integrable F t -measurable random initial condition ξ in (2.4). With obvious notations, it must satisfy Y t,ξ,µ t " U pt, ξ, µq. Choosing ξ with distribution µ, we deduce from uniqueness in small time to the system (2.3) that pX t,ξ,µ s , Y t,ξ,µ s , Z t,ξ,µ s q sPrt,T s coincides with pX t,ξ s , Y t,ξ s , Z t,ξ s q sPrt,T s . Indeed, pX t,ξ s , Y t,ξ s , Z t,ξ s q sPrt,T s solves (2.4) with x replaced by ξ and the system (2.4) has a unique solution. Therefore, we deduce that, with probability 1, Y t,ξ t " U pt, ξ, rξsq, for all t P rT ´δ, T s.

By pH3q, U satises (5.2) so that, by Lemma 5.1, pI 1 q is indeed satised.

Step 2 Assume that, for some n P t1, . . . , N ´1u, pI n q holds true. For any t P rT ´pn `1qδ, T s and ξ P L 2 pΩ, F t , P; R d q, we consider again the forwardbackward system (2.3). By pH3q, it admits a unique solution. In particular, by the uniqueness property guaranteed by pH3q, it must hold that Y t,ξ T ´nδ " Y T ´nδ,X t,ξ T ´nδ T ´nδ .

(5.7)

By the induction hypothesis, Y t,ξ T ´nδ must have the form Y t,ξ T ´nδ " U `T ´nδ, X t,ξ T ´nδ , rX t,ξ T ´nδ s ˘.

Therefore, we now consider (2.3) but on rT ´pn `1qδ, T ´nδs, with U pT ´nδ, ¨, ¨q as terminal boundary condition. By the induction hypothesis, we know that U pT ´nδ, ¨, ¨q is Λ-Lipschitz continuous, so that existence and uniqueness to (2.3) with U pT ´nδ, ¨, ¨q as terminal boundary condition hold true. This permits to extend the denition of U to rT ´pn `1qδ, T ´nδs. By Theorems 4.25 and 4.33, the extension of U belongs to Ť βě0 D β prT ´pn `1qδ, T ´nδsq and thus to Ť βě0 D β prT ´pn `1qδ, T sq. Moreover, it satises the master equation on rT ´pn `1qδ, T s ˆRd ˆP2 pR d q.

Consider now the restriction of the global solution pX t,ξ s , Y t,ξ s , Z t,ξ s q sPrt,T s to the small interval rT ´pn `1qδ, T ´nδs. By (5.7), it must coincide with the short time solution constructed on rt, T ´nδs with U pT ´nδ, ¨, ¨q as terminal boundary conditions. By the same arguments as in Step 1, we thus get that Y t,ξ t " U pt, ξ, rξsq with probability one. This shows that U satises (5.2) and applying Lemma 5.1, we get that pI n`1 q is satised. l 5.2. Mean-eld games.

5.2.1. General set-up. Mean-eld games were introduced simultaneously by Lasry and

Lions [START_REF] Lasry | Jeux à champ moyen I. Le cas stationnaire[END_REF][START_REF] Lasry | Jeux à champ moyen II. Horizon ni et contrôle optimal[END_REF][START_REF] Lasry | Mean eld games[END_REF] and by Huang, Caines and Malhamé [START_REF] Huang | Large population stochastic dynamic games: closedloop McKean-Vlasov systems and the Nash certainty equivalence principle[END_REF]. Their purpose is to describe asymptotic Nash equilibria within large population of controlled agents interacting with one another through the empirical distribution of the system. When players are driven by similar dynamics and subject to similar cost functionals, asymptotic equilibria are expected to obey some propagation of chaos, limiting the analysis of the whole population to the analysis of one single player and thus reducing the complexity in a drastic way.

The dynamics of one single player read as dX t " bpX t , µ t , α t qdt `σpX t , µ t qdW t , t P r0, T s,

for some possibly random initial condition X 0 , where pW t q tPr0,T s is an R d -valued Brownian motion and b : R d ˆP2 pR d q ˆRk Ñ R d and σ : R d ˆP2 pR d q are Lipschitz-continuous on the model of pH0q(i). Above, pα t q tPr0,T s denotes the control process. It takes values in R k and is assumed to be progressively-measurable and to satisfy:

E ż T 0 |α t | 2 dt ă `8.
The family pµ t q tPr0,T s denotes an arbitrary ow of probability measures in P 2 pR d q. It is intended to describe the statistical equilibrium of the game, the notion of equilibrium being dened according to some cost functional

J `pα t q tPr0,T s ˘" E " GpX T , µ T q `ż T 0 F pX t , µ t , α t qdt  ,
and being actually given by the solution of a xed point problem, the description of which is taken from [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF]:

(i) Given the family pµ t q tPr0,T s , solve the optimization problem inf pαtq tPr0,T s J `pα t q tPr0,T s ˘.

Assume that the optimal path is uniquely dened and denote it by p Xpµsq sPr0,T s t q tPr0,T s .

(ii) Find pµ s q sPr0,T s such that r Xpµsq sPr0,T s t s " µ t for all t P r0, T s.

Generally speaking, there are two ways to characterize the optimal paths in (i) by means of an FBSDE. The rst one is to represent the value function of the optimization problem (i) as the decoupling eld of a forward-backward system, in which case equilibria solving (ii) may be described through a McKean-Vlasov FBSDE along the lines of [START_REF] Carmona | A probabilistic weak formulation of mean eld games and applications[END_REF].

Another way is to make use of the stochastic Pontryagin principle to represent directly the optimal path in (i) as the forward component of the solution of a forward-backward system, in which case equilibria solving (ii) may be described through a McKean-Vlasov FBSDE along the lines of [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF]. When using the stochastic Pontryagin principle, the decoupling eld of the underlying forward-backward system is then understood as the gradient of the value function of the optimization problem (i).

Here we are willing to show that, in both cases, the decoupling eld of the McKean-Vlasov FBSDE used to characterize equilibria of the game is indeed a classical solution of a master PDE of the type (2.12) and, then, to make the connection with the so-called master equation presented in Lions' lectures at the Collège de France. In each case, we exhibit sucient conditions under which the master PDE is solvable for an arbitrary time horizon T . In short, the two types of representation apply under slightly dierent assumptions. The direct representation of the value function is well-tted to cases when σ is uniformly non-degenerate, since standard theory for uniformly parabolic semilinear PDEs then applies. The stochastic Pontryagin principle is more adapted to cases when the underlying Hamiltonian is convex in both the space and control variables, σ being possible degenerate. In both cases, we shall implement the Lasry-Lions monotonicity condition, see pH4q(iii) below, in order to investigate the Lipschitz property of the solution of the corresponding master PDE in the direction of the measure.

Use of the Stochastic Pontryagin

Principle. We rst explain how things work when using the stochastic Pontryagin principle in order to characterize the optimal paths in (i).

Then, following [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF], the matching problem (ii) is solved by forcing the forward component of the FBSDE derived from the Pontryagin principle to have pµ t q tPr0,T s as marginal laws.

The resulting system becomes (pY s q sPrt,T s being seen as a row vector process) Hpx, µ, y, αq " y : bpx, µ, αq `F px, µ, αq, x, y P R d , α P R k , µ P P 2 pR d q,

(5.10) and αpx, µ, yq denotes the minimizer: αpx, µ, yq " argmin α Hpx, µ, y, αq.

(5.11)

We shall specify below assumptions under which the minimizer is indeed well-dened.

For the moment, we concentrate on the regularity properties we need on the coecients.

As we aim at applying Proposition 5.2, we let: Assumption (pH 4q(i)). The running cost F may be decomposed as F px, µ, αq " F 0 px, µq `F1 px, αq, x P R d , µ P P 2 pR d q, α P R k ,

(5.12) the function F 1 being three times dierentiable, with bounded and Lipschitz-continuous derivatives of order 2 and 3. The functions F 0 and G are locally Lipschitz continuous in x and µ, the Lipschitz constant being at most of linear growth in |x| and in p ş R d |x 1 | 2 dµpx 1 qq 1{2 . Moreover, F 0 and G are dierentiable with respect to x and the coefcients f 0 " B x F 0 and g " B x G are Lipschitz in px, µq and satisfy pH1q and pH2q with h " f 0 , g and w " x.

In particular, there exists a constant C such that, for all x P R d , µ P P

2 pR d q, α P R k , |Gpx, µq| ď C " 1 `|x| 2 `żR d |x 1 | 2 dµpx 1 q ı , |F 0 px, µq| `|F 1 px, αq| ď C " 1 `|x| 2 `żR d |x 1 | 2 dµpx 1 q `|α| 2
ı .

(5.13)

Actually, the decomposition (5.12) is motivated by the uniqueness criterion we use below. We introduce it now and not later since makes the exposition of the regularity assumption much simpler. The growth conditions on the Lipschitz constant of the derivatives are motivated by the typical example when F and G have a quadratic structure in x and α (see [START_REF] Carmona | Control of McKean-Vlasov versus Mean Field Games[END_REF]).

The reader may notice that nothing is said about the smoothness of b and σ. The reason is the following. Generally speaking, the uniqueness of the minimizer in (5.11) is ensured under strict convexity of the Hamiltonian in the direction α, but, for our purpose, we will use more. We indeed require the full extended Hamiltonian H 1 px, µ, y, z, αq " Hpx, µ, y, αq `Trace `zσpx, µq ˘, for x P R d , µ P P 2 pR d q, y P R d , z P R dˆd , α P R k , to be convex in px, αq, namely H 1 px 1 , µ, y, z, α 1 q ´H1 px, µ, y, z, αq ´xx 1 ´x, B x H 1 px, µ, y, z, αqy ´xα 1 ´α, B α H 1 px, µ, y, z, αqy ě λ|α 1 ´α| 2 , (5.14)

for some λ ą 0. In order to guarantee the convexity of H, we must assume that bpx, µ, αq is a linear function in px, αq of the form b 0 pµq `b1 x `b2 α, for some matrices b 1 P R dˆd and b 2 P R dˆk and b 0 : P 2 pR d q Ñ R d . Moreover, because of the uniqueness criterion we use below, we shall restrict ourselves to the case b 0 " 0 so that the drift reduces to the linear combination bpx, αq " b 1 x `b2 α. Similarly, we must assume that σpx, µq is a linear function in x, which implies that σ is independent of x as we need it to be bounded (see pHσq). Again, because of the uniqueness criterion we use below, we restrict ourselves to the case when σ is also independent of µ, namely σpx, µq " σ for some constant matrix σ of dimension d ˆd. Then, the convexity property (5.14) holds provided F satises it.

In particular, H 1 px 1 , µ, y, z, α 1 q ´H1 px, µ, y, z, αq " Hpx 1 , µ, y, α 1 q ´Hpx, µ, y, αq so that the analysis of the full extended Hamiltonian H 1 may be reduced to the analysis of the extended Hamiltonian H. We thus require Assumption (pH 4q(ii)). There exist b 1 P R dˆd , b 2 P R dˆk and σ P R dˆd such that bpx, µ, αq " b 1 x `b2 α and σpx, µq " σ, for any x P R d , µ P P 2 pR d q and α P R k . Moreover, F satises (5.14) and the mapping R d Q x Þ Ñ Gpx, µq P R is convex in the x-variable for any µ P P 2 pR d q.

We then notice that αpx, y, µq solves the equation:

y : b 2 `Bα F `x, µ, αpx, µ, yq ˘" 0.

(5.15)

Since B α F " B α F 1 does not depend upon µ, we deduce that αpx, µ, yq reduces to αpx, yq.

It is then straightforward to prove from the implicit function theorem that the mapping px, yq Þ Ñ αpx, yq is twice dierentiable with respect to px, yq with bounded and Lipschitzcontinuous derivatives. This says in particular that, in (5.9), there is no McKean-Vlasov interaction in the forward equation. Moreover, we deduce, by composition, that Assumption pH2q is satised (and thus pH0q and pH1q as well).

Existence, uniqueness and dierentiability of the solution. In [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF], it is proved that (5.9) admits a unique solution provided the following assumption is in force (in addition to pH4q(i) and pH4q(ii)):

Assumption (pH 4q(iii)). There exists c ą 0 such that

(1) For all x P R d , |B α F 1 px, 0q| ď c, (2)
ż R d `F0 px, µq ´F0 px, µ 1 q ˘d`µ ´µ1 qpxq ě 0, ż R d
`Gpx, µq ´Gpx, µ 1 q ˘d`µ ´µ1 qpxq ě 0.

Actually, not only existence and uniqueness hold, but also the key Lipschitz estimate (5.1) is true, justifying pH3q. The argument is the same as the one given in [7, Proposition 3.7] for proving uniqueness. The only dierence is that initial conditions may be dierent. More precisely, given t P r0, T s and two square-integrable F t -measurable random variables ξ and ξ 1 , the same argument as in [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF], combined with (3.6) therein to take into account the fact that the initial conditions are dierent, shows that

2λE ż T t |αpX t,ξ s , Y t,ξ s q ´αpX t,ξ 1 s , Y t,ξ 1 s q| 2 ds ď E " xξ ´ξ1 , Y t,ξ t ´Y t,ξ 1 t y ‰ . (5.16)
(Here pX t,ξ , Y t,ξ , Z t,ξ q and pX t,ξ 1 , Y t,ξ 1 , Z t,ξ 1 q satisfy (5.9) with X t,ξ t " ξ and X t,ξ 1 t " ξ 1 .)

Now, it is quite straightforward to see that

E " |Y t,ξ t ´Y t,ξ 1 t | 2 ‰ ď C ´sup sPrt,T s E " |X t,ξ s ´Xt,ξ 1 s | 2 ‰ `E ż T t |αpX t,ξ s , Y t,ξ s q ´αpX t,ξ 1 s , Y t,ξ 1 s q| 2 ds ¯, (5.17)
and,

sup sPrt,T s E " |X t,ξ s ´Xt,ξ 1 s | 2 ‰ ď C ´E" |ξ ´ξ1 | 2 ‰ `E ż T t |αpX t,ξ s , Y t,ξ s q ´αpX t,ξ 1 s , Y t,ξ 1 s q| 2 ds
¯.

(5.18) Therefore, from (5.17) and (5.18),

E " |Y t,ξ t ´Y t,ξ 1 t | 2 ‰ ď C ´E" |ξ ´ξ1 | 2 ‰ `E ż T t |αpX t,ξ s , Y t,ξ s q ´αpX t,ξ 1 s , Y t,ξ 1 s q| 2 ds ¯,
Plugging (5.16) into the above equation, we get (5.1).

Master equation. The fact that pH3q holds permits us to apply Proposition 5.2. It follows that the decoupling eld U of the forward-backward equation (5.9) satises the corresponding master PDE (2.12).

We emphasize that the master PDE that we derive is not the standard master equation in mean-eld games theory. Loosely speaking, the master equation in mean-eld games is the equation satised by V , such that U is the gradient of V , which stands for the value function of the game, namely

V pt, x, µq

" E " G `Xt,x,µ T , rX t,ξ T s ˘`ż T t F `Xt,x,µ s , rX t,ξ s s, αpX t,x,µ s , Y t,x,µ s q ˘ds  , ξ " µ, (5.19)
in other words V pt, x, µq is the optimal cost when the private player is initialized at x and the equilibrium strategy for the population is initialized at µ. (Here pX t,x,µ , Y t,x,µ , Z t,x,µ q solves (2.4) with the coecients of (5.9).)

Now that U is known to belong to Ť βě0 D β , we can see X t,ξ and X t,x,µ as solutions of autonomous forward SDEs driven by smooth Lipschitz-continuous coecients (the drift being just obtained by a composition of b with αp¨, U p¨, ¨, ¨qq). In particular, X t,ξ and X t,x,µ must have the same smoothness properties as in the various results of Section 4, but for arbitrary time since the backward constraint has been removed. Another way to understand that claim is to prove regularity inductively, by means of a forward induction, applying successively the results obtained in Section 4 on rt, T ´nδs, rT ´nδ, T ´pn´1qδs, ..., rT ´δ, T s, for the same δ as in the proof of Proposition 5.2 and for n such that t P rT ´pn `1qδ, T ´nδq. The induction is then based on the ow property, which says that, for s P rT ´kδ, T ´pk ´1qδs,

X t,ξ s " X T ´kδ,X t,ξ T ´kδ s and X t,x,rξs s " X T ´kδ,X t,x,rξs T ´kδ ,rX t,ξ T ´kδ s s , (5.20)
and, thus, permits the transfer from one interval to another.

Basically, this permits us to prove that V is smooth in x and µ by dierentiating under the expectation, provided that G and F 0 are smooth enough in the direction of the measure. Motivated by the fact that the coecients are required to satisfy the convexity assumption pH4q(ii), assume for instance that Assumption (pH 4q(iv)). The functions

R d ˆP2 pR d q Q px, µq Þ Ñ F 0 px, µq b 1 `|x| 2 `şR d |v| 2 dµpvq , R d ˆP2 pR d q Q px, µq Þ Ñ Gpx, µq b 1 `|x| 2 `şR d |v| 2 dµpvq , (5.21)
satisfy pH0q(i)pH1qpH2q (for some values of the parameters therein). In particular, F 0 and G satisfy the same dierentiability property as in pH0q(i)pH1qpH2q but the derivatives are locally (instead of globally) controlled.

Then, we can dierentiate the representation formula for V as we dierentiated the backward components of (2.3) and (2.4) in Section 4, up to the slight dierence that the derivatives of G and F p¨, ¨, αp¨, ¨qq in x, y and µ may be of linear growth in all the arguments. The key point to circumvent it is to notice from pH4q(iv) that the random variable

GpX t,x,µ T , rX t,ξ T sq b 1 `|X t,x,µ T | 2 `}X t,ξ T } 2 2
satises the same rst-order and second-order dierentiability properties as θ Theorem 5.3. Under pH4q(iiv), the function V pt, ¨, ¨q satises the same assumption as F 0 and G in pH4q(iv), the parameters that appear in pH0q(i)pH1qpH2q being uniform in t P r0, T s. Moreover, for any x P R d and µ P P 2 pR d q, the function r0, T s Q t Þ Ñ V pt, x, µq is continuously dierentiable, the derivative being continuous in pt, x, µq. For any x P R d and ξ P L 2 pΩ, A, P; R d q, the functions r0, T s Rd ˆL2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B µ V pt, x, rξsqpξq P L 2 pΩ, A, P; R d q and r0, T s ˆRd L2 pΩ, A, P; R d q Q pt, x, ξq Þ Ñ B v rB µ V pt, x, rξsqspξq P L 2 pΩ, A, P; R d q are continuous. Finally, V satises the master equation

B t V pt, x, µq `Bx V pt, x, µqb `x, αpx, U pt, x, µqq ˘`F `x, µ, αpx, U pt, x, µqq żR d B µ V pt, x, µqpvqb `v, αpv, U pt, v, µq ˘dµpvq `1 2 Tr "ˆB 2 xx V pt, x, µq `żR d B v `Bµ V pt, x, µq ˘pvqdµpvq ˙σσ :  " 0, (5.22)
with V pT, x, µq " Gpx, µq as terminal condition and with U denoting the decoupling eld of (5.9).

q tďsďT , we get that the right-hand side is equal to Y t,x,µ t . We omit the details of the computation here. In particular, contrary to the last paragraph, we shall prove existence and uniqueness to (2.3) without relying on results in [START_REF] Carmona | Probabilistic analysis of mean eld games[END_REF]. Of course, as previously, we shall need to check that the processes that enter the representation of the value function satisfy pH3q, or equivalently, that the key estimate (5.2) holds true. We shall assume:

Assumption (pH 5q). The running cost F may be decomposed as F px, µ, αq " F 0 px, µq `F1 px, αq, x P R d , µ P P 2 pR d q, α P R d , where ' the functions F 0 and G are bounded and satisfy pH0q(i), pH1q, pH2q; ' the function F 1 is bounded in x and at most of quadratic growth in α, uniformly in x; it is is three times dierentiable in px, αq, the derivatives of order 2 and 3 being bounded and Lipschitz-continuous, the derivative of order 1 in x being bounded and the derivative of order 1 in α being at most of linear growth in α, uniformly in x; there exists λ ą 0 such that it satises the convexity assumption F 1 px, α 1 q ´F1 px, αq ´xα 1 ´α, B α F 1 px, αqy ě λ|α 1 ´α| 2 , admits a unique solution. It satises |Z t,x,pµuq uPrt,T s s | ď C R ds b dP almost everywhere, for a constant C R that may depend upon R (but not on pµ u q uPrt,T s ).

We now prove that C R may be chosen independently of R. The proof is as follows.

We write f px, ϕ R pzq, µq " f px, 0, µq `ˆż 1 0 B z f px, ϕ R pλzq, µq∇ϕ R pλzqdλ ˙z: . as drift in the forward component: The driver in the backward component is bounded and, by (5.23), the drift in the forward component is bounded in the variable x and at most of linear growth in the variable z. In particular, by [START_REF] Delarue | Estimates of the Solutions of a System of Quasi-Linear PDEs. A probabilistic Scheme[END_REF], there exists a constant Γ, independent of R and pµ u q uPrt,T s , such that, we indeed have |Z t,x,pµuq uPrt,T s s | ď Γ.

The coecients G and F 0 being bounded, we also have |Y t,x,pµuq uPrt,T s s | ď C, for C independent of R and of pµ u q uPrt,T s . Second step. We now construct δ ą 0 such that the master PDE (5.22) admits a solution in Ť βě0 D β prT ´δ, T sq on rT ´δs ˆRd ˆP2 pR d q. With the same Γ as in the previous step, we indeed apply Theorem 2.7 with pb R , σ, f R , gq instead of pb, σ, f, gq, for some R ą Γ}σ ´1} 8 . This says that, for some δ P p0, T s, there exists a function V : rT ´δ, T s ˆRd ˆP2 pR d q Ñ R in Ť βě0 D β prT ´δ, T sq that solves (5.22) with b replaced by b R and f replaced by f R . Now, for any pt, x, µq P rT ´δ, T s ˆRd ˆP2 pR d q, for any s P rt, T s, B x V ps, X t,x,µ s , rX t,ξ s sq " Z t,x,µ s σ ´1pX t,x,µ s q,

where ξ " µ and pX t,ξ , Y t,ξ , Z t,ξ q and pX t,x,µ , Y t,x,µ , Z t,x,µ q solve (2.3) and (2.4) with pb, f q replaced by pb R , f R q. In particular, |B x V pt, x, µq| ď Γ}σ ´1} 8 ă R. Therefore, V also solves (5.22). It also satises |V | ď C, for some C independent of R. Basically, this proves (ii) in pI 1 q.

Third step. In order to prove (i) in pI 1 q and more generally in pI n q for any n " 2, . . . , N , we must identify the constant Λ rst. We thus proceed as follows. We assume that there exists a time t P r0, T s such that, on rt, T s ˆRd ˆP2 pR d q, the master PDE (5.22) has a solution V in Ť βě0 D β prt, T sq. We are then willing to provide a bound for sup xPR d ,ξPL 2 pΩ,A,P;R d q }B µ V pt, x, rξsqpξq} 2 , independently of t P r0, T s.

Since V P Ť βě0 D β prt, T sq, we can nd some R ą 0 such that }B x V ps, ¨, µq} 8 }σ} 8 ă R for any s P rt, T s and µ P P 2 pR d q. In particular, V also solves the master PDE associated with pb R , σ, f R , gq instead of pb, σ, f, gq. Since pb R , σ, f R , gq satises pH0q(i) pH1qpH2q, we can imitate the proof of Theorem 2.8 and build a solution to (2.3) for any ξ P L 2 pΩ, F t , P; R d q. The forward process is dened as a solution of (3.44). We shall prove right below that it is uniquely dened, so that we can denote it by pX t,ξ s q sPrt,T s .

Uniqueness is a consequence of a more general result of stability, the proof of which is as follows. Given ξ, ξ 1 P L 2 pΩ, F t , P; R d q, we consider two solutions pX t,ξ s q sPrt,T s and pX t,ξ 1 s q sPrt,T s to the SDE (3.44), with ξ and ξ 1 as respective initial solutions. We then expand, by means of Itô's formula pV ps, X t,ξ s , rX t,ξ s sq ´V ps, X t,ξ 1 s , rX t,ξ s sqq sPrt,T s (observe that, in both terms, the measure argument is driven by ξ). By Proposition 3.9 (either by generalizing to the case when the process plugged in the spatial argument is not the same as the one plugged in the measure argument or by extending the dimension in order to see pX t,ξ s , X t,ξ 1 s q sPrt,T s as a single process), we get for s P rt, T s (using the fact that R ą }B x V } 8 }σ} 8 ) s , B x V ps, X t,ξ 1 s , rX t,ξ s sq ˘¯ı ds `Bx V ps, X t,ξ 1 s , rX t,ξ s sqσpX t,ξ 1 s qdW s .

d " V `s, X t,
(5.26)

Taking the dierence between (5.25) and (5.26) and using the same notation H for the Hamiltonian as in (5.10), we obtain d " V `s, X t,ξ s , rX t,ξ s s ˘´V `s, X t,ξ 1 s , rX t,ξ s s

˘‰

"

´"f `Xt,ξ s , rX t,ξ s s, α`X t,ξ s , B x V ps, X t,ξ s , rX t,ξ s sq ˘f `Xt,ξ 1 s , rX t,ξ s s, α`X t,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘˘ı ds

´"H ´Xt,ξ 1 s , rX t,ξ s s, B x V ps, X t,ξ 1 s , rX t,ξ s sq, α`X t,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘H ´Xt,ξ 1 s , rX t,ξ s s, B x V ps, X t,ξ 1 s , rX t,ξ s sq, α`X t,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ s sq ˘¯ı ds `"B x V ps, X t,ξ s , rX t,ξ s sqσpX t,ξ s q ´Bx V ps, X t,ξ 1 s , rX t,ξ s sqσpX t,ξ 1 s q ı dW s .

Therefore, taking the expectation and integrating in s from t to T , we get from the convexity of H in α (that follows from the convexity of F (5.27)

When ξ " ξ 1 , the left-hand side is zero. Denoting by X and X 1 two solutions to the SDE (3.44) with the same initial condition ξ, the above inequality (with the formal identication X " X t,ξ and X 1 " X t,ξ 1 ) says that αpX 1 s , B x V ps, X 1 s , rX 1 s sqq " αpX 1 s , B x V ps, X 1 s , rX s sqq. Then, uniqueness to (3.44) follows from the fact that, by assumption, B x V is Lipschitz continuous in x.

Fourth step. Given the ow of probability measures prX t,ξ s sq sPrt,T s we just constructed, we know from [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF] that, for any x P R d , the FBSDE (2.4), when driven by pb R , σ, f R , gq and by µ " rξs, is uniquely solvable. By the rst step, the solution must solve (2.4), when driven by pb, σ, f, gq. Moreover, it satises |Z t,x,µ Another way to make the connection with (2.4) is to see expansions (5.25) and (5.26) as standard verication arguments, as often used in stochastic control theory. Indeed, we are just using the fact that the mapping ps, xq Þ Ñ V ps, x, rX t,ξ s sq is a solution of a standard HJB equation, corresponding to the optimization problem (i) in the description of a mean-eld game on page 80. We can indeed dierentiate in time V ps, x, µ s q for a given x P R d , where µ s " rX t,ξ s s. Applying the chain rule proved in Section 3 and combining with the master PDE (5.22), we then recover the HJB equation:

B s " V ps, x, µ s q ‰ `Bx V ps, x, µ s q `b0 pxq `αpx, B x V ps, x, µ s qq 1 2 Tr " σσ : pxqB 2 xx V ps, x, µ s q ‰ `F `x, µ s , αpx, B x V ps, x, µ s qq ˘" 0,

(5.28)

for s P rt, T s and x P R d , with V pT, x, µ T q " Gpx, µ T q. We know that B x V is bounded by Γ}σ ´1} 8 . Therefore, (5.28) reads as a standard semilinear uniformly parabolic equation driven by smooth coecients in x. Since f is Lipschitz-continuous in the direction of the measure and rt, T s Q s Þ Ñ µ s is 1{2-Hölder continuous (the drift of the diusion X t,ξ being bounded), the coecients are 1{2-Hölder continuous in time. By Schauder's theory for semilinear parabolic equation (see [START_REF] Friedman | Partial dierential equations of parabolic type[END_REF]Chapter 7]), we can nd a bound Γ 1 for B 2 xx V that is independent of t P r0, T s. Now, going back to (3.44), we may use the bound for B 2 xx V as a Lipschitz bound for B x V in the direction x. It is then pretty standard to deduce, from Gronwall's lemma, that, for any ξ, ξ

the last line following from (5.27) (paying attention that the last term in the right-hand side is non-negative).

We now make use of Remark 5.4. By dierentiating pY t,x,µ s q sPrt,T s with respect to x (which is licit as it reads pY t,x,µ s " V ps, X t,x,µ s , rX t,ξ s sqq sPrt,T s and pX t,x,µ s q sPrt,T s solves a standard SDE with smooth coecients) and then, by applying Itô's formula, we can indeed check that pB x Y t,x,µ s pB x X t,x,µ s q ´1q sPrt,T s , solves the backward SDE in (5.9), so that (5.31)

B x V pt,
We now look at the backward equation in (2.3) (driven by pb R , σ, f R , gq). Now that we have proven a Lipschitz estimate for the forward component, it is standard to prove a similar estimate for the backward one. We deduce that (5.1) and thus (5.2) hold true.

Applying Lemma 5.1, we get the required Λ in (i) of the induction property pI n q.

Last step. From the second and fourth steps, it is clear that (i) in pI 1 q holds true, which completes the proof of pI 1 q.

We then apply Theorem 2.7 iteratively along the lines of the proof of Proposition 5.2.

Notice that here there is no need of the assumption (iii) in the induction scheme used in the proof of Proposition 5.2. Indeed, by the fourth step above, we have a direct way to establish (5.2), whereas, in the proof of Proposition 5.2, the bound (5. is the optimal control of McKean-Vlasov equations. We refer to [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF][START_REF] Carmona | Control of McKean-Vlasov versus Mean Field Games[END_REF] for a complete review. The idea here is to minimize the cost functional J `pα t q tPr0,T s ˘" E " GpX T , rX T sq `ż T 0 Assumption (pH 6q(ii)). The function R d ˆP2 pR d q ˆRk Q px, µ, αq Þ Ñ B α F px, µ, αq satisfy pH2q (and thus pH0q and pH1q as well) (with w " px, αq in the notations used in pH1q and pH2q).

F
Then, Lemma 5.6. Under pH6q(i) and pH6q(ii), the function R d ˆP2 pR d q ˆRk Q px, µ, αq Þ Ñ αpx, µ, yq satises pH0q, pH1q and pH2q (with w " px, yq in the notations used in pH1q and pH2q).

Proof. The starting point is (5.15). By (5.34) and by the Lipschitz property of B α F , we can reproduce the argument used in [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF] to prove that α is also Lipschitz continuous. More generally, the smoothness in x, y follows from a standard application of the implicit function theorem.

We now discuss the regularity of α in the direction µ. Given ξ, χ P L 2 pΩ, A, Pq, we deduce from (5.15) that, for any t P R, y : b 2 `Bα F `x, rξ `tχs, αpx, rξ `tχs, yq ˘" 0.

By the standard implicit function theorem, we deduce that the function R Q t Þ Ñ αpx, rξ tχs, yq P R k is dierentiable and that E B µ " B α F `x, rξs, αpx, rξs, yq ˘pξqχ ‰( `B2 αα F `x, rξs, αpx, rξs, yq ˘d dt |t"0 " αpx, rξ `tχs, yq ‰ " 0.

By strict convexity, the matrix B 2 αα F px, rξs, αpx, rξs, yqq is invertible. We easily deduce that the mapping L 2 pΩ, A, P; R d q Q ξ Þ Ñ αpx, rξs, yq P R k is Fréchet dierentiable. In particular, the mapping P 2 pR d q Q µ Þ Ñ αpx, µ, yq is dierentiable in Lions' sense and B µ αpx, µ, yqpvq " " B 2 αα F `x, µ, αpx, µ, yq ˘‰´1 B µ " B α F `x, µ, αpx, µ, yq ˘‰pvq.

The corresponding bounds in pH1q together with the uniform integrability property are easily checked. Now, the smoothness in v follows from that one of B µ rB α F s and the related bounds in pH2q hold true. The smoothness of B µ α in x, y is satised once we have the smoothness of α in x, y. l 5.3.2. Master equation. The point is now to apply Proposition 5.2 with b as above, σ constant and f px, y, νq " B x H `x, π 1 7ν, αpx, π 1 7ν, yq ˘`h px, νq, x, y P R d , ν P P 2 pR d ˆRd q, gpx, µq " B x Gpx, µq `gpx, µq, x P R d , µ P P 2 pR d q.

Notice also that pH3q is satised, see again [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF]. It thus remains to check that pH2q is satised.

We thus assume that Assumption (pH 6q(iii)). The functions R d ˆP2 pR d q ˆRk Q px, µ, αq Þ Ñ B x F px, µ, αq and R d ˆP2 pR d q Q px, µq Þ Ñ B x Gpx, µq satisfy pH0q(i), pH1q and pH2q (with w " px, αq and w " x respectively). For any px, µ, αq P R d ˆP2 pR d q ˆRk , there exist versions of B µ F px, µ, αqp¨q and of B µ Gpx, µqp¨q such that R d ˆP2 pR d q ˆRk ˆRd Q px, µ, α, vq Þ Ñ B µ F px, µ, αqpvq and R d ˆP2 pR d q ˆRd Q px, µ, vq Þ Ñ B µ Gpx, µqpvq that satisfy pH0q(i), pH1q and pH2q (with w " px, α, vq and w " px, vq respectively). where Y t,ξ t and U pt, ξ, rξsq are seen as row vectors. By Fubini's theorem, this identies U pt, ξ, rξsq with B x V pt, ξ, rξsq `ÊrB µ V pt, xξy, rξsqpξqs. This proves (5.35) when the law of ξ has R d as support. In the general case, we can approximate ξ by random variables with R d as support. Passing to the limit in (5.35), this completes the proof of the identication.

We refer to [START_REF] Carmona | The master equation for large population equilibriums[END_REF] for additional comments about the dierences between the shapes of the master equation in mean-eld games and in the control of McKean-Vlasov equations.

6. Appendix 6.1. Proof of Proposition 3.8. The proof is a straightforward adaptation of Lemma 3.3 in [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF]. Basically, it suces to prove the result when µ has a smooth positive density denoted by p, and p and its derivatives being at most of exponential decay at the innity. It is then possible to construct a quantile function U : p0, 1q d Q pz 1 , . . . , z d q Þ Ñ U pz 1 , . . . , z d q P R d (this is the notation used in [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF], but this has nothing to do with the generic notation U used in the paper for denoting a function of the measure) such that U pη 1 , . . . , η d q has law µ when η 1 , . . . , η d are i.i.d. random variables with uniform distribution on p0, 1q. Moreover, BU i {Bz i " 0 and BU j {Bz i " 0 if i ă j.

Going Then, for any x P R d and any µ P P 2 pR d q, we can nd a continuous version of V px, µ, ¨q, uniquely dened on Supppµq, such that the mapping R d ˆSupppµq Q px, vq Þ Ñ V px, µ, vq is dierentiable with respect to x. Moreover, we can nd a mapping R d ˆRd Q px, vq Þ Ñ BV px, µ, vq, continuous in v for any given x P R d , jointly continuous at any point px, vq with v P Supppµq, such that BV px, µ, vq identies with B x V px, µ, vq whenever v P Supppµq. In particular, B x V p¨, µ, ¨q is continuous on R d ˆSupppµq.

Proof. By Riesz' theorem, for any i P t1, . . . , du, for any x P R d and any ξ P L 2 pΩ, A, P; R d q, we can nd an element V i x,ξ P L 2 pΩ, A, P; R By assumption, we thus get that h ´1pV px `he i , rξs, ξq ´V px, rξs, ξqq ´V i x,ξ tends to 0 in L 2 pΩ, A, P; R d q. Therefore V i

x,ξ is a random variable in L 2 pΩ, σpξq, P; R d q and we can express it as B i V px, rξs, ξq where B i V px, rξs, ¨q is a function in L 2 pR d , rξs; R d q.

We have E " |BV px, rξs, ξq ´BV px 1 , rξ 1 s, ξ 1 q| 2 ‰ ď C `|x ´x1 | 2 `Φ2 α pξ, ξ 1 q ˘.

Choosing x " x 1 , we deduce from Proposition 3.8 that, for any x P R d and any ξ P L 2 pΩ, A, P; R d q, there exists a version of the mapping R d Q v Þ Ñ BV px, rξs, vq " pB 1 V px, rξs, vq, ..., B d V px, rξs, vqq P R dˆd that is continuous on compact subsets of R d , uniformly in x P R d , such a version being uniquely dened on the support of rξs. By the same method as in (3.33), we deduce that the family pR d Q v Þ Ñ BV px, rξs, vq P R dˆd q xPR d is relatively compact for the topology of uniform convergence on compact subsets. Considering a sequence px n q ně1 that converges to x P R d , we already know that the sequence of functions pR d Q v Þ Ñ BV px n , rξs, vq P R dˆd q ně1 converges in L 2 pR d , rξs; R dˆd q to R d Q v Þ Ñ BV px, rξs, vq P R dˆd . Since BV px, rξs, ¨q is uniquely dened on the support of rξs, the limit of any converging subsequence (for the topology of uniform convergence on compact subsets of R d ) of pBV px n , rξs, ¨qq ně1 coincides with BV px, rξs, ¨q on the support of rξs. We easily deduce that the function R d ˆRd Q px, vq Þ Ñ BV px, rξs, vq P R dˆd is continuous at any point px, vq such that v P Suppprξsq.

Similarly, we deduce from the identity E "A 1 h

´`V px `he i , rξs, ξq ´V px, rξs, ξq ˘´`V px 1 `he i , rξ 1 s, ξ 1 q ´V px 1 , rξ 1 s, ξ 1 q ˘¯, χ

Eı " ż 1 0 E !A `Bi V px `sh, rξs, ξq ´Bi V px 1 `sh, rξ 1 s, ξ 1 q ˘, χ E) ds, that }h ´1rpV px `he i , rξs, ξq ´V px, rξs, ξqq ´pV px 1 `he i , rξ 1 s, ξ 1 q ´V px 1 , rξ 1 s, ξ 1 qqs} 2 ď Cp|x ´x1 | `Φα pξ, ξ 1 qq, from which we get that, for any x P R d , any h " 0 and any µ P P 2 pR d q, there exists a version of the mapping R d Q v Þ Ñ h ´1rV px `he i , µ, vq ´V px, µ, vqs that is continuous on compact subsets of R d , uniformly in x P R d and in h " 0. As above, we deduce that the family pR d Q v Þ Ñ h ´1rV px `he i , µ, vq ´V px, µ, vqsq xPR d ,h "0 is relatively compact, for the topology of uniform convergence on compact subsets. Once again, following the same argument as above, this says that, for any x P R d , the functions pSupppµq Q v Þ Ñ h ´1rV px `he i , µ, vq ´V px, µ, vqs P R d q h "0 converge uniformly on compact subsets as h tends to 0 to some derivative function, which identies with Supppµq Q v Þ Ñ B i V px, µ, vq P R d . l

(3. 29 )From ( 3 .

 293 21), (3.23), (3.25), (3.27) and (3.28), we obtain, for any T ą 0, sup 0ďtďT

[ 5 ,

 5 Lemma 3.3] (see the adaptation of the proof in Subsection 6.1 in Appendix): Proposition 3.8. Consider a collection pV pµq : R d

4. 1 .

 1 Stability estimate for McKean-Vlasov linear FBSDEs. The strategy for investigating the derivatives of the solutions to (2.3) and (2.4) is standard. We identify the derivatives with the solutions of linearized systems, obtained by formal dierentiation of the coecients. For that reason, the analysis of the dierentiability relies on some preliminary stability estimates for linear FBSDEs. Unfortunately, because of the McKean-Vlasov structure of the coecients, we cannot borrow any estimate from the literature. We thus have to use a tailor-made version, which is the precise purpose of this subsection. 4.1.1. General set-up. Generally speaking, we are dealing with a linear FBSDE of the form

(

(4. 17 )

 17 From Cauchy-Schwarz' inequality and (ii) in Denition 4.5, we get that: Lemma 4.6. For any pair pX, χq and any p ě 1, N p,C M pX, χq ď K p }χ} p 2 .

( 4 .

 4 40)Recalling that Λ is a random variable Λ : Ω ˆΩ Q pω, ωq Þ Ñ Λpω, ωq on the product space pΩ ˆΩ, A b Â, P b Pq such that, for almost every ω P Ω, Λpω, ¨q P L 2 p Ω, Â, P; R `q with ÊrΛ 2 pω, ¨qs ď K 2 , we can bound the above right-hand side byÊ"`Λ ^Ψpς, xξyq ^xW y ˘xWy ı ď sup ! Ê" `xΛ 0 y ^Ψpς, xξyq ^xW y ˘xWy ı ; xΛ 0 y P L 2 p Ω, Â, P; R `q : Ê" xΛ 0 y 2 ‰ 1{2 ď K

r

  is continuous and Φ α is continuous at any point of the diagonal, we can nd a constant δ ą 0 such that sup pr,sqPrt,T s 2 :|s´r|ďδ Φ α `θ t,ξ,p0q r , θt,ξ,p0q s ˘ď ε.

  pr, sq P rt, T s 2 , Cauchy Schwarz' inequality yields ˇˇE " | Ĥ pw, θt,ξ,p0q s

4. 2 . 1 .

 21 First-order derivatives of the McKean-Vlasov system. As we already explained in Examples 4.2 and 4.3, the shape of the system (4.3) has been specically designed in order to investigate the derivative of the system of the original FBSDE in the direction of the measure. Thus, we shall make use of the results from Subsection 4.1, the constant

Σ

  p1q pr, θ n,λ,p0q r , xθ n,λ,p0q r yq `ϑn,λ,p0q r , xϑ n,λ,p0q r y ˘dW r

21 .

 21 For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p , such that, for T ď c p and with γ " c p in (4.[START_REF] Gangbo | Existence of a solution to an equation arising from the theory of Mean Field Games[END_REF])," M 2pE `Bχ θ t,x,rξs ˘‰1{2p ď C p }χ} 2 , (4.54)

  with the additional notation ζ :" d dλ |λ"0 ξ λ .

( 4 .

 4 73)Plugging (4.72) and (4.73) into (4.71), we easily deduce that the mapping (4.26) is dierentiable. Continuity of the partial derivatives is proved in the same way.

l 4 . 3 . 3 .

 433 Partial smoothness of B µ U . Strategy. Generally speaking, the strategy is the same

  r P rt, T s.

2 and H ww1 2 the related terms in ( 4 .

 24 74), we compute:

p2q a and G p2q a

 p2q , processes are estimated with S instead of H norms. Obviously, this does not aect (4.82) since Σ p2q a and G p2q a only involve the two rst coordinates of θ, ϑ 1 and ϑ 2 . The second main dierence comes from A 1 . Since neither σ nor g depend on the component Z, we can replace |θ s ´θ1 s | by |θ p0q s ´θp0q1 s | in the analysis of the term corresponding to A 1 . Choosing ε " sup sPrt,T s |θ p0q s ´θp0q1 s |, we get rid of the remaining term containing the indicator function of the event t|θ p0q s ´θp0q1

1 0

 1 By a standard Girsanov argument, the above decomposition of f R says that the FB-SDE(5.24) may be written, under a new probability, as a new FBSDE system with f pX t,x,pµuq uPrt,T s s , 0, µ r q as driver in the backward component and with b R `Xt,x,pµuq uPrt,T s r , Z t,x,pµuq uPrt,T s r ż B z f `Xt,x,pµuq uPrt,T s r , ϕ R pλZ t,x,pµuq uPrt,T s r q, µq∇ϕ R pλZ t,x,pµuq uPrt,T s r qdλ ˙:

  2) is obtained by means of the induction assumption (iii). Uniqueness follows from Theorem 2.8, observing that the quadratic term in the equation may be truncated (as any solution in the class Ť βě0 D β has a bounded gradient). l 5.3. Control of McKean-Vlasov equations. 5.3.1. General set-up. Another example taken from large population stochastic control

  Choosing Y of the form εϕpXq and Z of the form εψpXq, with Ppε " 1q " Ppε " ´1q " 1{2 and ε independent of X, and considering two bounded Borel measurable functions ϕ and ψ : R d Ñ R d , we deduce thatE " Tr `Bv B µ U `rXs ˘pXqϕpXq b ψpXq ˘‰ " E " Tr `Bv B µ U `rXs ˘pXqψpXq b ϕpXq ˘‰, (3.4)from which we deduce that B v B µ U prXsqpXq takes values in the set of symmetric matrices of size d. By continuity, it means that B v B µ U pµqpvq is a symmetric matrix for any v P R d when µ has the entire R d as support. By continuity in µ, we deduce that B v B µ U pµqpvq is a symmetric matrix for any v P R d and any µ P P 2 pR d q. Now, choosing Y and Z of the form ϕpXq and ψpXq respectively and plugging(3.4)

	the triplet p X, Ŷ , Ẑq denoting a copy of pX, Y, Zq on p Ω, Â, Pq and the tensorial product
	operating on R d .
		By Schwarz' Theorem, the roles of Z and Y can be exchanged, which means (choosing
	h " k " 0) that
		E " Tr `Bv B µ U `rXs ˘pXqZ b Y ˘‰ `EÊ " " E " Tr `Bv B µ U `rXs ˘pXqY b Z ˘‰ `EÊ Tr `B2 µ U `rXs ˘pX, XqZ b " Tr `B2 µ U `rXs ˘pX, XqY b Ŷ ˘‰	Ẑ˘‰	.	(3.3)
	into (3.3), we deduce that
	E	Ê" Tr `B2 µ U `rXs ˘pX, XqϕpXq b ψp Xq ˘‰ " E	Ê"	Tr `B2 µ U `rXs ˘pX, XqψpXq b ϕp Xq	˘‰
		" E	Ê"	Tr `B2 µ U `rXs ˘p X, Xqψp Xq b ϕpXq	˘‰
		" E	Ê"	Tr `"B 2 µ U `rXs ˘p X, Xq ‰ : ϕpXq b ψp Xq ˘‰,

  ϕ7µ denotes the image of µ by ϕ. The lifted version of U ‹ ϕ is nothing but U ˝ϕ, where (with an abuse of notation) ϕ is canonically lifted as

	Proposition 3.4. Assume that the chain rule (3.7) holds for any function U that is fully C 2 with rst and second order derivatives that are bounded and uniformly continuous (with
	respect to both the space and measure variables). Then Theorem 3.3 holds, in other words,
	the chain rule (3.7) is valid for any fully C 2 function U satisfying (3.6).
	Proof. [Proof of Proposition 3.4.] Let U be a fully C 2 function that satises (3.6). We
	`mollify' U in such a way that its mollication is bounded with bounded rst and second
	order derivatives. Let ϕ : R d Ñ R d be a smooth function with compact support and, for
	arbitrary µ P P 2 pR d q dene
	@µ P P 2 pR d q, `U ‹ ϕ ˘pµq :" U `ϕ7µ	˘,

where

  Without any loss of generality, we can assume that there exists a constant C such that |ϕ n pvq| ď C|v|, |B x ϕ n pvq| ď C and |B 2 xx rϕ n pvqs k | ď C , 1 ď k ď d , and v P R d and that ϕ n pvq " v for any n ě 1 and any v with |v| ď n. Then, for any µ P P 2 pR d q and any random variable X with µ as distribution, it holds

							(3.9)
	for any n ě 1 W 2 2 `ϕn 7µ, µ ˘ď E "	|ϕ n pXq ´X| 2 1 t|X|ěnu	‰	ď CE "	|X| 2 1 t|X|ěnu	‰	,
	which tends to 0 as n Ñ 8. By continuity of U and its partial derivatives and by (3.8),
	it is easy to deduce that, a.s.,

  any t ě 0 and any s P r0, ts such that Er|b s | 2 `|σ s | 4 s ă 8, lim

										.11)
	Indeed, by (3.8) and (3.9), it is enough to check that
	sup ně1 "ż R d	ˇˇB µ U `ϕn 7µ ˘pvq ˇˇ2d `ϕn 7µ ˘pvq	`żR d	ˇˇB v	"	 B µ U `ϕn 7µ ˘‰pvq ˇˇ2d `ϕn 7µ ˘pvq	ă 8,
	which follows directly from (3.6), noticing that the sequence pϕ n 7µq ně1 lives in a compact
	subset of P 2 pR d q as it is convergent.
	By (3.10) and (3.11) and by a standard uniform integrability argument, we deduce
	that, for nÑ`8	E "	B µ pU ‹ ϕ n qprXsqpXqb s	‰	" E "	B µ U prXsqpXqb s	‰	,
	lim nÑ`8	E Tr	"	B v `Bµ pU ‹ ϕ n qprXsq ˘pXqa s	‰(	" E Tr	" B v `Bµ U prXsq ˘pXqa s	‰(	.

  In particular, choosing A " t|X pµrq rPrt,T s s | ą R 4 u for some s P rt, T s, applying Markov inequality and using the fact that (3.45) is also satised by Er|X

					pµrq rPrt,T s s	| 2 s, we get that
	sup sPrt,Ss	E "	1 t|X s pµr q rPrt,T s	|ąR 4 u	|X s pµrq rPrt,T s

  rθ sq rPrt,T s P H 2 prt, T s; Rq is dierentiable and that the derivative reads as follows H p1q pr, θ r , xθ p0q r yqpϑ r , xϑ p0q r yq :" B w hpθ r , rθ p0q r sqϑ r `Ê In Example 4.2, the coecients B, Σ, F and G are obtained by replacing h by b, σ, f and g and by computing B p1q , Σ p1q , F p1q and G p1q accordingly. Leaving Example 4.2 and

	λ,p0q
	r
			"	B µ hpθ r , rθ p0q r sqpxθ p0q r yqxϑ p0q r y ‰	. (4.4)
	Of course, if h only acts on ppθ	p0q r , rθ	p0q r sqq rPrt,T s instead of ppθ r , rθ	p0q r sqq rPrt,T s , then dier-
	entiability holds in S 2 prt, T s; Rq.

  ) playing the role of L in pH1q. It is worth mentioning that the constant K has a major role in the sequel as it dictates the size of the time interval on which all the

	4.11) Conditions (4.7), (4.9), (4.10) and (4.11) must be compared with pH1q, the constant K Assumptions (4.7), (4.8), (4.9) and (4.11) read in the following way when, in the de-composition (4.4), h pV r , x V p0q r yq " B w hpθ r , rθ p0q r sq and Ĥ pV r , x V p0q r yq " B µ hpθ r , rθ p0q r sq: in (4.7Example 4.3. (Continuing Example 4.2) (1) Equation (4.7) expresses the fact that h is Lipschitz continuous with respect to

estimates derived in this section hold true.

The comparison between (4.7)(4.8)(4.9)(4.10)(4.11) and pH1q may be made more explicit within the framework of Example 4.2:

  Lipschitz continuity of h follows from(4.13)). This might seem rather weak and it might be tempting to expect an L 8 bound instead of a linear growth bound. As shown by the example in Remark 2.5, there is no way of guaranteeing This important feature explains why the space of boundary conditions we consider in the paper is not limited to functions with derivatives that are globally Lipschitz with respect to the measure argument. Because of the gap in the growth of the derivatives, we would fail to prove that the derivatives of the solution of the master equation (or equivalently of the decoupling eld of the FBSDEs (2.3)) are also globally Lipschitz with respect to the measure argument. Due to this lack of stability, we would not be able to extend the results from short to long time horizons. 4.1.2. Estimate of the solution. Part of our analysis relies on stability estimates for sys-

	tems of a more general form than (4.3), namely
	X s " η	`ż s	B `r, θr , x θp0q r y ˘`θ r , x θp0q r y ˘dr	`ż s	Σ `r, θp0q r , x θp0q r y ˘`θ p0q r , x θp0q r y ˘dW r ,
		t			t
	Y s " G `XT , x XT y ˘`X T , x XT y ˘`ż T	F `r, θr , x θr y ˘`θ r , x θp0q r y ˘dr	´ż T	Z r dW r , (4.15)
			s			s

that the derivative B µ h of the Lipschitz-continuous function h is bounded in L 8 , even when α " 0 (which is the strongest case) . Boundedness of the derivative only holds in L 2 , as is written in

(4.13)

.

the dierence between (4.15) and (4.3) being that the coecients (except the terminal boundary condition) may depend on other triplets θ, θ, θ and θ. We shall make use of the following denition, directly inspired from (4.7):

  Regarding the structure of the coecients, B, Σ, F and G, we also let

	R p a :" E t	"	γ 1{2 ˇˇG a pT q	ˇˇp `ˆż T	ˇˇpB a , F a qpsq ˇˇds ˙p	`ˆż T	ˇˇΣ a psq	ˇˇ2 ds	˙p{2 	.
				t		t

  Observe that, in(4.22), we used the supremum to get T p , which we bounded by γ 1{2 times γ p{2 . Making use of (4.7), we easily handle the term (4.21). In(4.18), it gives the contribution of the form γ 1{2 M 2p Et p θq, the γ 1{2 in front of M Et and the γ 1{2 in the denition of M 2pEt p θq arising as follows. When handling pb , f q, we can let a power 2 enter inside the time integral. This introduces the H 2 -norm of Z times an additional T less than γ, which can be split into γ 1{2 and γ 1{2 .Next we discuss the second term in(4.22). For this we use (4.7) and (4.8). With the shortened notation H " pB, F, Σq, we can indeed either say that Ĥ p θs , x θp0q s yq is bounded in L 2 or use the polynomial growth assumption. We get Λ s is a shortened notation for | Ĥ p θs , x θp0q s yq|. Now, using the conditional Cauchy-Schwarz inequality and the obvious bound E t rS 1 ^S2 s ď E t rS 1 s ^Et rS 2 s for two nonnegative random variables S 1 and S 2 , we obtain:Taking the power 2p and the conditional expectation E t , we get a term which is less than N 2p,C Multiplying by γ p{2 (see the prefactor in (4.22)), we get that it is less than N 2p,C Et p θp0q s , r| Xs | 2 `γ1{2 | Ys | 2 s 1{2 q which is less than N 2p,C

	ˇˇÊ "	Ĥ p θs , x θp0q s yqx θp0q s y ıˇˇˇď	Ê"! Λ s ^´C `C|x θp0q s y| α`1 `C} θp0q s } α`1 2	¯)ˇˇx θp0q s y	ˇˇı ,
	ˇˇÊ "	Ĥ p θs , x θp0q s yqx θp0q s y ıˇˇď
			Ê"! Êt	"	Λ 2 s	‰ 1{2 ^´C `C Êt	" |x θp0q s y| 2α`2 ‰ 1{2 `C} θp0q s } α`1 2	¯)Ê t	" ˇˇx θp0q s y	ˇˇ2	ı 1{2 ı .
		Et p	θp0q s ,	θp0q s q. Et p	θp0q
										˙p
										s	ˇˇ2 ds	(4.21)
						"
	`Γp γ 1{2	E t	" ˇˇÊ " Ĝ pX T , x XT yqx XT y ‰ˇˇ2 p	ı
			`γp{2 ess sup	E t	" ˇˇÊ "	p B , F , Σ qp θs , x θp0q s yqx θp0q s y	‰ˇˇ2 p	ı 	(4.22)
										sPrt,T s
	`Γp E t	"	γ 1{2 ˇˇG a pT q ˇˇ2p `ˆż T	ˇˇpB a , F a qpsq ˇˇds ˙2p	`ˆż T	ˇˇΣ a psq ˇˇ2 ds	˙p	.
										t	t

Γp E t "ˆż T t ˇˇpb , f qp θs , x θp0q s yq θs ˇˇds ˙2p `ˆż T t ˇˇσ p θp0q s , x θp0q s yq θ0 where s , pM 2 Et p θp0q qq 1{2 q.

  θ1 ˘(1{2 . step. Upper bound for the terms involving B , F , Σ or Ĝ . We can make use of the Lipschitz property (4.9) or of the L 2 bound (4.7). For a generic function Ĥ , which may be B , F or Σ , we get ^ş hdν that holds for a general measure ν with mass less than 1 and a general measurable nonnegative function h, we get

	Second ˇˇÊ "`Ĥ	p θs , x θp0q
	E t	"ˆż T	ˇˇÊ "`Ĥ	p θs , x θp0q s yq ´Ĥ p θ1 s , x θp0q1 s yq ˘x θp0q1 s y ‰ˇˇd s	˙2p 
			t
	ď E t	"ˆż T	ˇˇÊ "`Ĥ	p θs , x θp0q s yq ´Ĥ p θ1 s , x θp0q1 s yq ˘x θp0q1 s y	‰ˇˇ2	ds	˙p
				t
	ď C 1 sup sPrt,T s	} θp0q1 s } 2p 2	" 1 ^ˆE t	0 "ˆż T	| θs ´θ 1 s | 2 ds	˙p	`Φ2p

s yq ´Ĥ p θ1 s , x θp0q1 s qyq ˘x θp0q1 s y ‰ˇˇ2 ď C " 1 ^`| θs ´θ 1 s | 2 `Φ2 α p θp0q , θp0q1 ˘‰} θp0q1 s } 2 2 . (4.32)

Therefore, recalling the bound ş p1 ^hqdν ď 1 α `θ p0q , θp0q1 ˘˙* ,

(4.33)

  .[START_REF] Villani | Optimal Transport. Old and New[END_REF] Conclusion. In order to complete the proof of the rst part, notice that the terms labelled by a directly give the remainder ∆R 2p a in (4.27). The second part of the statement easily follows from Lemma 4.6.Remark 4.11. As the reader may guess, terms of the form M4p pϑ, θq and M4p p θ, θq in

(4.27)

will be handled by means of Corollary 4.8. However, we note that, in comparison with M4p , the `conditional' norm M 4p that is used in Corollary 4.8 incorporates an additional pre-factor γ 1{2 , see

(4.16)

. Roughly speaking, M4p pϑ, θq and M 4p

Et pϑq`pM 2 E p θp0q q 2p

  variable Λ 0 in the supremum being in L 2 pΩ, A, P; R `q and the function Φ α diering from the original one in (4.9) and (4.10) but satisfying the same properties on rL 2 pΩ, F t , P; R d qs 2 instead of rL 2 pΩ, A, P; R d qs 2 . Remark 4.13. Before we proceed with the proof of Corollary 4.12, we discuss what the assumptions we made on the structure of J ξ permit to say on the term N 2p,C

			Et pX, χq in
	(4.16). Recall indeed that
	N p,C Et pX, χq " sup
	2 a	˘1{2	ı * ,

the

  1 ,p0q } S p ,t `}θ ξ ´θξ 1 } H p ,t

			(4.43)
	ď C p	"	|ξ ´ξ1 | `W2 `rξs, rξ 1 s ˘‰,
	then, we can nd constants C 1 p such that, for all ξ and ξ 1 (with the notation (4.25)),

  Φ α`1 pt, ¨q : rL 2 pΩ, F t , P; R d qs 2 Ñ R `is continuous at any point of the diagonal, does not depend on p and satises (4.10) with α replaced by α `1. The restriction of Φ α`1 pt, ¨, ¨q to rL 2 pΩ, F 0 , P; R d qs 2 may be assumed to be independent of t P r0, T s.

	1 ˘ı1{2p	ď C p ´1 ^ |ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 q ( ¯`|χ| `}χ} 2	˘,
	Proof. The proof is a consequence of Corollary 4.12, with R 2p a " ∆R 2p a " 0. Exam-

where ple 4.15 (see in particular (4.46)) guarantees that the conditions of Corollary 4.12 are satised. We then deduce that (4.38) holds true. Existence of a function Φ α`1 pt, ¨, ¨q satisfying the prescription described in the statement then follows from Example 4.15. By Remark 4.16, we can assume that the restriction to rL 2 pΩ, F 0 , P; R d qs 2 is independent of t. l Remark 4.20. It is easy to derive from Lemma 4.19 that

  By Lemma 4.19, we can indeed bound the last term in (4.37) by By Remark 4.16, we can assume that the restriction of Φ α`1 pt, ¨, ¨q to rL 2 pΩ, F 0 , P; R d qs 2 is independent of t. l

			"
	sup	sup	E " `Λ0 ^Ψpς, ξq ˘´1 ^ |ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 q	( ¯`|χ| `}χ} 2	¯ı* .
	ςPO	}Λ 0 } 2 ďK
	Following Remark 4.20 to pass from Gâteaux to Fréchet, we deduce:
	Lemma 4.23.

with Ψ dened in (4.36) and Ψ given in this denition by

(4.46)

. Following Example 4.15 (see in particular (4.45)), we deduce that (4.55) holds true (modifying Φ α`1 pt, ¨, ¨q if necessary). For T ď c with c :" cpLq ą 0, t P r0, T s and x P R d , the function

  .56) where Φ α`1 : rL 2 pΩ, A, P; R d qs 2 Ñ R `is continuous at any point of the diagonal and satises (4.10), with α replaced by α `1.

Proof. Fréchet dierentiability is a consequence of the continuity Lemma 4.21 that permits to pass from Gâteaux to Fréchet on the model of Remark 4.20.

We then have B χ Y t,x,rξs t " ErB µ U pt, x, rξsqpξqχs. Combined with Lemma 4.21, this gives (4.56),

  By the smoothness property of B µ U ps, ¨q, the rst term in the right-hand side is bounded by CpEr|X t,x,rξs s ´x| 2 s 1{2 `Φα`1 pX t,ξ s , ξqq}χ} 2 , the constant C being allowed to increase from line to line. The coecients of (2.3) and (2.4) being at most of linear growth, we deduce from (4.1) that Er|X t,ξ s ´ξ| 2 s 1{2 and Er|X t,x,rξs s ´x| 2 s 1{2 are less than Cp1 `}ξ} 2 qps ´tq 1{2 and Cp1 `|x| `}ξ} 2 qps ´tq 1{2 respectively. Therefore, the rst term in the last line of (4.61) is bounded by

		(4.61)
	ξ s y ˘˘xχy	‰ .

  By pH1q and Lemmas 4.18 and 4.21 (with p " 1), it is bounded by Cps ´tq 1{2 }χ} 2 . Since B x U is bounded, the second term is less than CEr|B χ X By pH1q and Lemmas 4.18 and 4.21 again, it is less than Cps ´tq 1{2 }χ} 2 . For the third term, we rst apply Cauchy-Schwarz inequality to get that it is less than CEr|B χ X t,ξ s ´χ| 2 s 1{2 , recall (4.56). Then, by pH1q and Lemma 4.18, it is bounded by Cps ´tq 1{2 }χ} 2 . Continuity of

	ErY	t,x,rξs s	s, we obtain				t,x,rξs t	and ErU pt, X s t,x,rξs	, rX t,ξ s sqs "
			E	Ê"`B					s	,	"	X t,ξ s	‰˘`x X t,ξ s y ˘˘xχy ‰
			" E " B χ Y t t,x,rξs	´Bχ Y t,x,rξs s	‰	`E" B x U `s, X t,x,rξs s	, rX t,ξ s s ˘Bχ X t,x,rξs s	‰
			`E" E p	" B µ U `s, X t,x,rξs s	,	" X t,ξ s	‰˘`x X t,ξ s y ˘xB χ X t,ξ s ´χy ıı .
	The rst term is equal to E	ş s t F p1q pr, θ r t,x,rξs	, xθ	t,ξ,p0q r	yqpB χ θ	t,x,rξs r	, xB χ θ	t,ξ,p0q r	yqdr (with the
	notations of (4.4)). t,x,rξs s	|s "
	CEr|B χ X s t,x,rξs	´Bχ X t t,x,rξs	|s.

µ U pt, x, rξsq `xξy ˘´B µ U `s, X t,x,rξs

  .77) 4.3.4. Preliminary estimates. We start with the following bound of the remainder term Lemma 4.27. Given generic processes θ, θ, ϑ 1 , ϑ 2 , θ1 and θ2 , denote by H p2q For any p ě 1, we can nd a constant C p (independent of the processes) such that (using the notation M from (4.25))

	H a in (4.75): p2q
									a prq the
	term H a pθ r , x p2q	θp0q r y, ϑ 1 r , ϑ 2 r , x	θ1,p0q r	y, x	θ2,p0q
	E t	"	|G p2q a pT q| 2p `ˆż T	`|B p2q a psq| `|F p2q a psq| ˘ds	˙2p	`ˆż T	˙p 1{2p a psq| 2 ds |Σ p2q	(4.78)
				t					t

r yq in (4.75), H matching B, Σ, F or G.

  Lemma 4.28. Given processes θ, θ 1 , θ, θ1 , ϑ 1 , ϑ 11 , ϑ 2 , ϑ 21 , θ1 , θ11 , θ2 and θ21 , denote For any p ě 1, we can nd a constant C p (independent of the processes) such that, for any random variable ε with values in R `(with the notation M

	the terms H a pθ r , x p2q	θp0q r y, ϑ 1 r , ϑ 2 r , x	θ1,p0q r	y, x	θ2,p0q r	yq in (4.75) by H a prq and use a similar p2q
	denition for H a prq. from (4.25)), p2q1
	E t	"	|G p2q a pT q ´Gp2q1 a pT q| 2p `ˆż T	`|B p2q a psq ´Bp2q1 a psq| `|F p2q a psq ´F p2q1 a psq| ˘ds ˙2p
							t
			`ˆż T	|Σ p2q a psq ´Σp2q1 a psq| 2 ds
			t
								.80)
	We now handle H vµ a . By conditional Hölder inequality, we observe that, under condition (4.76), |H vµ a psq| ď CEr} θ1,p0q } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t s 1{2 , from which we get
			E t	"ˆż T t	|H vµ a psq|ds	˙2p 	ď C p E " } θ1,p0q } 2 S 4 ,t } θ2,p0q } 2 S 4 ,t	ı p	.

(4.81) By (4.79), (4.80) and (4.81) and with the notation (4.25), we get (4.78). The cases when H " Σ or G may be handled in the same way. l

  and 4.19, all the terms involving an M may be bounded in the same way. By (4.44) in Lemma 4.14, the same is true for the term involving Φ α . By Cauchy-Schwarz inequality and once again by Lemmas 4.18 and 4.19, the same bound holds for the terms integrated under E. In the end, the whole right-hand side in (4.82) may be bounded by C p p1 ^t|ξ ´ξ1 | `Φα`1 pt, ξ, ξ 1 quqp|χ| `}χ} 2 q (without the t when ξ, ξ 1 P L 2 pΩ, F 0 , P; R d q). In (4.38), this brings us to the case when the remainders are zero, but Φ α is replaced by Φ α`1 . Applying(4.45) in Example 4.15, we complete the proof of (4.92). The last part of the statement (choice of a version of Φ α`1 which is independent of t) follows from Remark 4.16. l 4.3.7. Study of the Non McKean-Vlasov system. We now repeat the same analysis but

  2 ζ,χ θ t,x,rξs and θ " B 2 ζ,χ θ t,ξ and with generic coecients H given by (compare if needed with (4.53)):h pV, x V p0q yq " B x hpV, r V p0q sq, Ĥ pV, x V p0q yq " B µ hpV, r V p0q sqpx V p0q yq, H a " Hp2q a , For any p ě 1, there exist two constants c p :" c p pLq ą 0 and C p such that, for T ď c p and with γ " c p in (4.16),

	where	Hp2q a prq is a variant of H a in (4.74) and reads: p2q
	Hp2q a prq :" H p2q a	`θt,x,rξs r	, xθ t,ξ,p0q r	y, B χ θ t,x,rξs r	, B ζ θ t,x,rξs r	, xB χ θ t,ξ,p0q r	y, xB ζ θ t,ξ,p0q r	y ˘.	(4.93)
	On the model of Lemmas 4.30 and 4.31, we claim (compare with Lemma 4.21):
	Lemma 4.32. " M 2p E	`B2 ζ,χ θ t,x,rξs ˘‰1{2p ď C p }χ} 2 ,

and

  rξsq is continuously dierentiable in the sense of Fréchet. Recalldeduce from Lemmas 4.18 and 4.19 that the gradient DUpt, x, ¨q satises (i) and (ii) in the statement of Theorem 3.6. Now, using the same sequence pξ λ q λPR as in 4.3.2, we λ playing the role of X λ in the statement of Theorem 3.6). We deduce that, for any pt, xq P r0, T sˆR d , the map P 2 pR d q Q µ Þ Ñ U pt, x, µq is partially C 2 . In particular, for any pt, x, µq P r0, T s ˆRd ˆP2 pR d q, we can nd a version of R d Q v Þ Ñ B µ U pt, x, µqpvq holds true for any e P R d and any ξ, χ P L 2 pΩ, F t , P; R d q, with ξ " µ, and for a prescribed random variable Z 1 independent of pξ, χq. From Lemma 4.32, we deduce thatEr|B v rB µ U pt, x, rξsqspξq| 2 s 1{2 ď C, Er|B v rB µ U pt, x, rξsqspξq ´Bv rB µ U pt, x 1 , rξ 1 sqspξ 1 q| 2 s 1{2 ď Cr|x ´x1 | `Φα`1 pξ, ξ 1 qs,the extension of Φ α`1 to the whole rL 2 pΩ, A, P; R d qs 2 being achieved as in the proof of Lemma 4.23.By means of Proposition 3.8, we deduce that, for given t P r0, T s and µ P P 2 pR d q, we can choose, for any

	ing the identity
	B χ Y	t,x,rξs t	" E " DUpt, x, rξsqpξqχ	‰ ,
	notice that
	d dλ |λ"0	E " DUpt, x, ξ λ qχ	‰	" B 2 ζ,χ Y	t,x,rξs t	,
	which satises (i) and (ii) in the statement of Theorem 3.6 thanks to Lemmas 4.30 and
	4.31 (with ξ that is continuously dierentiable, such a version being uniquely dened on the support
	of µ. Moreover, by (3.42), we have the relationship
	B 2 signpZ 1 qe,signpZ 1 qχ Y	t,x,rξs t	" E "

we Tr `Bv rB µ U pt, x, µqspξq ˘`χ b e ˘(‰ , which

  2 xx U and B x rB µ U s. l We now turn to Proof. [Proof of Theorem 2.7] We rst prove that U is a classical solution of the PDE (2.12). The main argument follows from (2.10), the idea being to apply the chain rule to pH0q and Theorems 4.25 and 4.33 provide estimates on the smoothness of b, σσ : , B µ U and B v rB µ U s. We deduce that we can nd a non-negative functional Φ on rL 2 pΩ, A, P; R d ˆRm ˆRmˆd qs 2 , continuous at any point of the diagonal, matching 0 on

	U pt `h, x, ¨q, which is licit thanks to Theorem 4.33. Following (2.9), we get
	U `t `h, x, rX t,ξ t`h s ˘´U `t `h, x, rξs "
	ż t`h	p E	" B µ U `t `h, x, rX t,ξ r s ˘`xX t,ξ r y	˘b`x θ t,ξ r y, rθ t,ξ,p0q r	s ˘ı dr
	t
	`1 2	t ż t`h	p E	" Trace	"	B v	"	B µ U	‰`t	`h, x, rX t,ξ r s ˘`xX t,ξ r y ˘`σσ : ˘`xθ t,ξ,p0q r	y, rθ t,ξ,p0q r	s ˘‰ı	dr.
	the diagonal, such that
	ˇˇˇU `t `h, x, rX t,ξ t`h s ˘´U `t `h, x, rξs hp
				E	" B µ U `t `h, x, rξs ˘`xξy ˘b`x θ t,ξ t y, rθ	t,ξ,p0q t	s ˘ı
														ˇˇď
			´h 2	p E	" Trace	"	B v	" B µ U	‰`t	`h, x, rξs ˘`xξy ˘`σσ : ˘`xθ	t,ξ,p0q t	y, rθ	t,ξ,p0q t	s ˘‰ı
			h sup	Φ `θt,ξ r , θ t,ξ t	˘.
			rPrt,t`hs

Assumption

  The remainder term ErR 2 a s is thus equal to γ 1{2 Er|B χ Y t,ξ S | 2 s, which is less than Cγ 1{2 Er|B χ X t,ξ S | 2 s. Therefore, choosing Γ 1 Cγ 1{2 " 1{2, we have, for S ´t ď c :" cpLq, G a pSq " ÊrB µ U pS, X We can apply once again Corollary 4.8, with p " 1, ϑ " B χ θ t,x,rξs , θ " B χ θ t,ξ and B, Σ and F given by (4.53). Recalling that B x U is bounded by Λ, we get for S ´t ď c :" cpΛ _ Lq,

				E "	sup	`|B χ X t,ξ s | 2 `|B χ Y t,ξ s | 2 ˘`ż S	|B χ Z t,ξ s | 2 ds	 1{2	ď C}χ} 2 .	(5.6)
					sPrt,Ss			t
	Now, consider the derivative process of the non McKean-Vlasov system (2.4). It
	satises a forward-backward system of the type (4.3). The boundary condition in the
	backward component may be expressed as
	B χ Y	t,x,rξs S	" B x U `S, X S t,x,rξs	, rX t,ξ S s ˘Bχ X t,x,ξ S	`Ê "	B µ U `S, X S t,x,rξs	, rX t,ξ S s ˘pxX t,ξ S yqxB χ X t,ξ S y	‰ .
	Under the notations (4.5) and (4.6), the above writing reads as the decomposition of
	the terminal condition in the form g pX S t,x,rξs t,x,rξs S , rX t,ξ S sqxB χ X t,ξ S ys. |B χ Y , rX t,ξ S sq " B x U pS, X S t,x,rξs t,x,rξs t | ď C ´E"ˇˇÊ rB µ U pS, X t,x,rξs S , rX t,ξ S sqpxX t,ξ S yqxB χ X t,ξ S ys , rX t,ξ S sq, Ĝ " 0 ˇˇ2 ‰ 1{2
						`E" sup	`|B χ X t,ξ s | 2 `|B χ Y t,ξ s | 2 ˘‰1{2	¯,
							sPrt,Ss
	the second part coming from the remainder term H a in (4.53) when H " B, Σ, F .
	Therefore, from the relationship B χ Y t t,x,rξs	" ÊrB µ U pt, x, rξsqpxξyqχs and from (5.6),
	we get
		Ê"	|B t,x,rξs S	, rX t,ξ S s ˘pxX t,ξ S yq| 2 ‰ 1{2	¯.

and µ U pt, x, rξsqpxξyq| 2 ‰ 1{2 ď C ´1 `EÊ " |B µ U `S, X

We deduce sup xPR d ,ξPL 2 pΩ,Ft,P

  T " B x GpX T , rX T sq, where H denotes the so-called extended Hamiltonian of the system:

	(5.9)
	with the boundary condition Y

dX t " b `Xt , rX t s, αpX t , rX t s, Y t q ˘dt `σpX t , rX t sqdW t dY t " ´Bx H `Xt , rX t s, Y t , αpX t , rX t s, Y t q ˘dt `Zt dW t ,

  For all x P R d , xx, B x F 0 p0, δ x qy ě ´cp1 `|x|q, xx, B x Gp0, δ x qy ě ´cp1 `|x|q, where δ x is the Dirac mass at point x. Moreover, the following Lasry-Lions monotonicity condition is in force:

  5.2.3. Direct approach. Theorem 5.3 is specically designed to handle the case when the coecients may be quadratic in x, provided that the extended Hamiltonian has a convex structure in px, αq. When the coecients are bounded in x and µ and σ is nondegenerate, we can give a direct proof of the solvability of the master equation(5.22) under the weaker assumption that the extended Hamiltonian is convex in α. The key point is to represent directly the value function V in(5.19) by means of a McKean-Vlasov FBSDE, and thus to avoid any further reference to the stochastic Pontryagin principle.

  The coecient σ has the form σ : R d Q x Þ Ñ σpxq P R dˆd , is bounded, twice dierentiable, with bounded and Lipschitz-continuous derivatives of order 1 and 2, and, for any x P R d , the matrix σpxq is invertible with sup xPR d |σ ´1pxq| ă 8.

The parameter k is equal to d and b may decomposed as bpx, αq " b 0 pxq `α, x P R d , α P R d , the function b 0 being bounded and twice continuously dierentiable with bounded and Lipschitz-continuous derivatives of order 1 and 2.

  ξ s , rX t,ξ s s ˘‰ " ´f `Xt,ξ s , rX t,ξ s s, α`X t,ξ s , B x V ps, X t,ξ s , rX t,ξ s sq ˘˘ds `Bx V ps, X t,ξ s , rX t,ξ s sqσpX t,ξ s qdW s , X t,ξ 1 s , rX t,ξ s s ˘‰ " " ´f `Xt,ξ 1 s , rX t,ξ s s, α`X t,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ s sq˘Bx V ps, X t,ξ 1 s , rX t,ξ s sq ´α `Xt,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq

	(5.25)
	and
	d " V `s, α`X
	t,ξ 1

  1 and the linear structure of the drift in α in(5.8)) thatE " V pt, ξ, rξsq ´V pt, ξ 1 , rξsq ‰By exchanging the roles of ξ and ξ 1 and then by summing up, we deduce thatE " V pt, ξ, rξsq ´V pt, ξ 1 , rξsq ´`V pt, ξ, rξ 1 sq ´V pt, ξ 1 , rξ 1 sq ˘‰ , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘´α `Xt,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ s sq ˘|2 ds , B x V ps, X t,ξ s , rX t,ξ s sq ˘´α `Xt,ξ s , B x V ps, X t,ξ s , rX t,ξ 1 s sq ˘|2 ds.Finally, rearranging the terms, we deduce from the Lasry-Lions condition thatE " V pt, ξ, rξsq ´V pt, ξ 1 , rξsq ´`V pt, ξ, rξ 1 sq ´V pt, ξ 1 , rξ 1 sq ˘‰ , B x V ps, X t,ξ 1 s , rX t,ξ 1s sq ˘´α `Xt,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ s sq ˘|2 ds

	ě E "	GpX t,ξ T , rX t,ξ T sq ´GpX t,ξ 1 T , rX t,ξ T sq ´`GpX t,ξ T , rX t,ξ 1 T sq ´GpX t,ξ 1 T , rX t,ξ 1 T sq ˘‰
		`E ż T	"`F 0 pX t,ξ s , rX t,ξ s sq ´F0 pX t,ξ 1 s , rX t,ξ s sq F0
					t
					pX t,ξ s , rX t,ξ 1 s sq ´F0 pX t,ξ 1 s , rX t,ξ 1 s sq ˘‰ds
		`λE	ż T	ˇˇα `Xt,ξ 1
					t
					ż T
		`λE	ˇˇα `Xt,ξ
					t
	ě λE	ż T	ˇˇα `Xt,ξ 1
			t
					ż T
		`λE	ˇˇα `Xt,ξ
					t
	´E ż T	" F 1	´Xt,ξ s ,	α`X t,ξ s , B x V ps, X t,ξ s , rX t,ξ s sq ˘F
		t
					1	´Xt,ξ 1 s ,	α`X t,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘¯ı ds
	ě E "	GpX t,ξ T , rX t,ξ T sq ´GpX t,ξ 1 T , rX t,ξ T sq ‰	t `E ż T	`F0 pX t,ξ s , rX t,ξ s sq ´F0 pX t,ξ 1 s , rX t,ξ s sq ˘ds
	`λE	ż T	ˇˇα `Xt,ξ 1
			t

s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘´α `Xt,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ s sq ˘|2 ds. s s s s , B x V ps, X t,ξ s , rX t,ξ s sq ˘´α `Xt,ξ s , B x V ps, X t,ξ s , rX t,ξ 1 s sq ˘|2 ds.

  1 P L 2 pΩ, F t , P; R d q, , B x V ps, X t,ξ s , rX t,ξ s sq ˘´α `Xt,ξ s , B x V ps, X t,ξ s , rX t,ξ 1 s sq ˘|2 ds ¯,(5.29) for a constant C that is independent of t, ξ and ξ 1 and the value of which is allowed to increase from line to line. In particular, using the Lipschitz property of α and once again the bound for B 2 xx V , we deduce that , B x V ps, X t,ξ s , rX t,ξ s sq ˘´α `Xt,ξ 1 s , B x V ps, X t,ξ 1 s , rX t,ξ 1 s sq ˘ˇ2 ds

	E "	sup	|X t,ξ s	´Xt,ξ 1 s | 2 ‰	ď C	´E" |ξ ´ξ1 | 2 ‰
	sPrt,T s
			`E ż T	ˇˇα `Xt,ξ
					t
		ż T
			ˇˇα `Xt,ξ
	E
		t
	ď C	´E"	|ξ ´ξ1 | 2 ‰
		`E ż T	ˇˇα `Xt,ξ
					t

s s s , B x V ps, X t,ξ s , rX t,ξ s sq ˘´α `Xt,ξ s , B x V ps, X t,ξ s , rX t,ξ 1 s sq ˘|2 ds ď C ´E" |ξ ´ξ1 | 2 ‰

`E" V pt, ξ, rξsq ´V pt, ξ 1 , rξsq ´`V pt, ξ, rξ 1 sq ´V pt, ξ 1 , rξ 1 sq ˘‰¯,

  , rX t,ξ s s, B x V ps, X t,ξ s , rX t,ξ s sq, αpX t,ξ s , B x V ps, X t,ξ s , rX t,ξ s sqq ˘ds|F t ı . , thanks to (5.29) and (5.30), and by the Lipschitz property of B x F and B x G in the variables x, µ and α, we get that, for any ξ,ξ 1 P L 2 pΩ, F t , P; R d q, V is smooth in x and B x V is Γ 1 -Lipschitz in x, we can write E " |B x V pt, ξ, rξsq ´Bx V pt, ξ, rξ 1 sq| 2 ‰ V `t, λξ `p1 ´λqξ 1 , rξs ˘´B x V `t, λξ `p1 ´λqξ 1 , rξ 1 s V `t, ξ, rξs ˘´B x V `t, ξ, rξ 1 s |B x V pt, ξ, rξsq ´Bx V pt, ξ, rξ 1 sq| 2 ‰ ď C}ξ 1 ´ξ} 2 2 ,the constant C being independent of t, ξ and ξ 1 . Plugging into (3.44), we can deduce

	T , rX t,ξ s s, B x V ps, X t,x,µ , rX t,ξ T s ż s , rX t,ξ s sq, αpX t,x,µ s , B x V ps, X t,x,µ s B x G `Xt,ξ B x H `Xt,x,µ s B x V pt, ξ, rξsq " E T t and thus " T , rX t,ξ T s ż T t B x H `Xt,ξ ´E" |ξ ´ξ1 | 2 ‰ `ż 1 0 E "`B ˘˘`ξ , rX t,ξ s sqq ˘ds ı , ´ξ1 ˘‰dλ ď C ´E" |ξ ´ξ1 | 2 ‰ `E"`B ˘˘`ξ ´ξ1 ˘‰¯. We nally get that E " that sPrt,T s Thereforeď C sup E " |X t,ξ s ´Xt,ξ 1 s | 2 ‰ ď C}ξ 1 ´ξ} 2 2 .

x, µq " E " B x G `Xt,x,µ s E " |B x V pt, ξ, rξsq ´Bx V pt, ξ 1 , rξ 1 sq| 2 ‰ ď C ´E" |ξ ´ξ1 | 2 ‰ `E" V pt, ξ, rξsq ´V pt, ξ 1 , rξsq ´`V pt, ξ, rξ 1 sq ´V pt, ξ 1 , rξ 1 sq ˘‰¯. Since x x

  pX t , rX t s, α t qdt  , over controlled McKean-Vlasov diusion processes of the form dX t " bpX t , rX t s, α t qdt `σdW t , t P r0, T s,

  to (69) therein, we see from the assumption imposed on V that the bound ˆˇU `z0 `r ´2r d e d ˘´pU pz 0 `rq ˇˇ2 dr, where V n is a mollication of V that satises (3.32) with respect to a constant C n that converges to C as n tends to the innity. Dividing by h d and following the lines of the original argument, we get, for a given z 0 P R d , Dividing by |rBU d {Bz d spU pz 0 qq| and letting n tend to the innity, we complete the proof.6.2. Dierentiability lemma. Lemma 6.1. Consider a function V : R d ˆP2 pR d q ˆRd Ñ R d such that, for any ξ, χ P L 2 pΩ, A, P; R d q, the mapping R d Q x Þ Ñ ErxV px, rξs, ξq, χys is dierentiable (where x¨, ¨y denotes the inner product in R d ). Assume moreover that there exist a constant C ě 0 and a function Φ α as in pH1q such that, for all x, x 1 P R d and ξ, ξ 1 , χ P L 2 pΩ, A, P; R d q: , rξ 1 s, ξ 1 q, χy ‰ˇˇď C `|x ´x1 | `Φα pξ, ξ 1 q ˘}χ} 2 .

										˙2α 
										|x| 2 dµpxq
										R d
	ˇˇB Bx d V n	`U pz 0 q	˘BU d Bz d	`z0 ˘ˇˇ2 ď C 2 n	"	1 `2|U `z0 ˘|2α `ˆż R d	|x| 2 dµpxq ˙2α 	ˇˇB Bz d U d	`U pz 0 q ˘ˇˇ2 .
	ˇˇd dx	E "	xV px, rξs, ξq, χy ‰ˇˇď	C}χ} 2 ,
	ˇˇd dx	E "	xV px, rξs, ξq, χy ‰	´d dx	E " xV px 1

becomes ż |r|ăh ˇˇV n " U `z0 `r ´2r d e d ˘‰ ´Vn `U pz 0 `rq ˘ˇ2 dr ď C 2 n ż |r|ăh " 1 `|U `z0 `r ´2r d e d ˘|2α `|U pz 0 `rq| 2α `ˆż

  Now, for h " 0, denoting by e i the ith vector of the canonical basis, px `he i , rξs, ξq ´V px, rξs, ξq, χ ˘´V i x,ξ , χ

								d q such that
			d dx i	E "	xV px, rξs, ξq, χy ‰	" E "	xV i x,ξ , χy ‰	.
	"A						Eı
	E	h ´1`V
	"	ż 1 0 ´d dx i	E "	xV px `she i , rξs, ξq, χy ‰	´d dx i	E "	xV px, rξs, ξq, χy ‰ ¯ds.

As far as we understand the sketch of the proof in[START_REF] Lions | Cours au collège de France[END_REF], the underlying arguments are reminiscent of the way in which we use convexity in the rst class of examples.

For any matrix a we will denote its transpose by a : and its trace by Trpaq.

For notational convenience, the lifting procedure is done onto the same probability space that carries the driving Brownian motion W . Alternatively, one can use an arbitrary rich enough atomless probability space, see[4] and Section 3 for details.

This means that for two families pX λ q λPR and pX λ,1 q λPR with X 0 " X 0,1 and rd{dλs |λ"0 X λ " rd{dλs |λ"0 X λ,1 , the derivatives rd{dλs |λ"0 rDUpX λ q ¨χs and rd{dλs |λ"0 rDUpX λ,1 q ¨χs are the same (the variable χ being given).

From[START_REF] Villani | Optimal Transport. Old and New[END_REF] Theorem 6.9], µn converges weakly to µ. Using the Skorokhod representation theorem, we can nd a sequence pξnq converging almost surely to ξ. The convergence holds also in L 2 since this sequence is uniformly square integrable, recall[START_REF] Villani | Optimal Transport. Old and New[END_REF] Denition 6.8(iii)].

Acknowledgement. The authors would like to thank Pierre Cardaliaguet for fruitful discussions. The nancial support of a CNRS-Royal Society International exchange grant is also acknowledged.

And, the Lasry-Lions monotonicity condition in the last line of pH4q(iii) holds true.

We here prove that Theorem 5.5. For a given T ą 0 and under pH5q, the master PDE (5.22) has a unique classical solution in the space

Proof. In comparison with the proof of Theorem 5.3, the diculty here is that we do not have an a priori existence and uniqueness result for the McKean-Vlasov FBSDE system representing the master PDE (5.22). In order to proceed, we thus revisit the proof of Proposition 5.2 and show, by induction, that there exist an integer N ě 1 and a constant Λ ě 0 such that, with δ " T {N , the following holds true for any n P t1, . . . , N u:

pI n q : There exists a mapping V : rT ´nδ, T sˆR d ˆP2 pR d q Q pt, x, µq Þ Ñ V pt, x, µq P R that belongs to Ť βě0 D β prT ´nδ, T sq such that (i) for any t P rT ´nδ, T s, the function V pt, ¨, ¨q satises the same assumption as g in pH0q(i), pH1q pH2q, but with the constant L replaced by Λ and L and α being replaced by some Ln and αn ;

(ii) V satises the master PDE (5.22) on rT ´nδ, T s ˆRd ˆP2 pR d q.

First step. We start with the following observation. Equation (5.22) is of the type (2.12), with m " 1, bpx, y, z, νq " b 0 pxq `αpx, zσ ´1pxqq 12 , σpx, νq " σpxq, f px, y, z, νq " F 0 px, µq `F1 px, αpx, zσ ´1pxqqq and gpx, µq " Gpx, µq (recall that the product zσ ´1pxq makes sense since z reads as an element of R 1ˆm , that is a row vector). Since b does not rely on y and ν, we shall write bpx, zq for bpx, y, z, νq. Similarly, since f is independent of the variable y and depends on the variable ν P P 2 pR d ˆRq through its rst marginal µ P P 2 pR d q only (recall that, formally, ν is understood as the joint marginal law of the process pX, Y q), we shall write f px, z, µq for f px, y, z, νq. Now, recalling (5.15), we know that px, zq Þ Ñ αpx, zq is twice dierentiable with respect to px, zq with bounded and Lipschitz-continuous derivatives of order 1 and 2. In particular, we can nd a constant C such that, for all x, z P R d and µ P P 2 pR d q, |B z f px, z, µq| ď Cp1 `|z|q, (5.23) which plays an important role below.

Of course, the assumption pH0q(i) is not satised because of the quadratic growth of f in the variable z. In order to apply Theorem 2.7, we shall make use of a truncation argument. Considering a smooth function ϕ R : R d Ñ R d that matches the identity on the ball of center 0 and of radius R, that is zero outside the ball of center 0 and of radius 2R and that satises |∇ϕ

In particular, for any pt, xq P r0, T s ˆRd and any ow of probability measures pµ u q uPrt,T s with values in P 2 pR d q, we know from [START_REF] Delarue | On the existence and uniqueness of solutions to FBSDEs in a non-degenerate case[END_REF]

x,pµuq uPrt,T s r dW r , s P rt, T s,

(5.24) 12 Pay attention that the letter b is used both to denote the rst-order coecient in (2.12) and the drift in (5.8). We feel that the reader can easily make the distinction between the two of them.

Below, we make use of the stochastic Pontryagin principle in order to characterize the optimal paths. Although the form of the Pontryagin principle is dierent from what it is in mean-eld games, it imposes, in a rather similar way, restrictive conditions on the structure of the SDE (5.32), among which the fact that σ has to be constant. The Hamiltonian is dened in the same way as before, see (5.10), but the FBSDE derived from the stochastic Pontryagin principle has a more complicated form (see [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF]):

B µ Gpv, µqpxqdµpvq, with x P R d , ν P P 2 pR d ˆRd q, µ P P 2 pR d q and π 1 :

Existence and uniqueness of a solution to (5.33) have been established under the following assumption (see [START_REF] Carmona | Forward-Backward Stochastic Dierential Equations and Controlled McKean Vlasov Dynamics[END_REF]):

Assumption(pH6q(i)). The drift b is of the linear form bpx, µ, αq " b 0 x`b 1 ş R d vdµpvqb 2 α. The cost functions F and G are locally Lipschitz continuous in px, µ, αq, the local Lipschitz constant being at most of linear growth in |x|, p ş R d |v| 2 dµpvqq 1{2 and |α|. Moreover, F and G are also C 1 in px, µ, αq, the derivative in px, αq being Lipschitz continuous in px, µ, αq and the functions B µ F and B µ G satisfying (with h " F and w " px, αq or h " g and w " x)

Finally, there exists λ ą 0 such that F px 1 , µ 1 , α 1 q ´F px, µ, αq ´xx 1 ´x, B x F px, µ, αqy

(5.34)

for any pair pξ, ξ 1 q with µ and µ 1 as marginal distributions, where x, x 1 P R d , µ, µ 1 P P 2 pR d q and α, α 1 P R k . In a similar way, the function px, µq Þ Ñ Gpx, µq is convex in the joint variable px, µq.

Of course, the Hamiltonian is convex in α under pH6q(i) so that the minimizer (5.11) is well-dened. By (5.15) and by a suitable version of the implicit function theorem, the function α inherits the smoothness of B α H. For instance, assume that Under pH6q(i-ii-iii), by Lemma 5.6, the function R d ˆP2 pR d q ˆRd Q px, µ, yq Þ Ñ B x Hpx, µ, αpx, µ, yqq satises pH0q(i), pH1q and pH2q. We now discuss h. By linearity of b, we rst observe that (recalling that B µ r

The smoothness of the rst term is easily handled, the smoothness of the second one in

x as well. The diculty is to dierentiate the second one with respect to ν. We get

where the `0' indicates that the derivative in the direction w is zero. We let the reader check the required conditions for the derivative in the direction ν in pH1q and pH2q are indeed satised. Derivatives in the direction x are easily handled.

We deduce that Proposition 5.2 applies. As for mean-eld games, the master PDE satised by U is not the `natural' equation associated with the optimization problem.

Following the previous subsection, we thus dene

where ξ " µ, pX t,ξ s q sPrt,T s denotes the forward component in (5.33), under the initial condition X t,ξ t " ξ, and pX t,x,µ s q sPrt,T s denotes the corresponding solution of (2.4).

As in the proof of Theorem 5.3, we are willing to apply the results from Section 4 in order to investigate the smoothness of V . Again, this requires some precaution as the coecients may be of quadratic growth in the space variable and in the measure argument. Proceeding as in the proof of Theorem 5.3, we have 13 Theorem 5.7. Under pH6q(iiii), the function V satises the statement of Theorem 5.3, with the same master equation expect that U inside is the decoupling eld of (5.33).

On the model of Remark 5.4, the identication of U pt, x, µq in terms of V pt, x, µq now reads U pt, x, µq " B x V pt, x, µq `żR d B µ V pt, x 1 , µqpxqdµpx 1 q, (5.35) which can be proved by dierentiating the map L 2 pΩ, F t , P; R d q Q ξ Þ Ñ ErV pt, ξ, rξsqs P R in the direction χ P L 2 pΩ, F t , P; R d q. By the same kind of expansion as in Remark 5.4, we get

‰ " E " U pt, ξ, rξsqχ ‰ , 13 Pay attention that there is no need for an analogue of pH4q(iv), since pH4q(iv) is necessarily true under pH6q(iiii), with F0px, µq in (5.21) replaced by F0px, µ, αpx, µ, U pt, x, µqqq, the constants appearing in pH0q(i)pH1qpH2q being uniform in t P r0, T s.