Schauder Estimates for a Class of Potential Mean Field Games of Controls

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Schauder Estimates for a Class of Potential Mean Field Games of Controls

Joseph Frédéric Bonnans, Saeed Hadikhanloo, Laurent Pfeiffer

To cite this version:

Joseph Frédéric Bonnans, Saeed Hadikhanloo, Laurent Pfeiffer. Schauder Estimates for a Class of

Potential Mean Field Games of Controls. Applied Mathematics and Optimization, Springer Verlag

(Germany), 2019, pp.34. �hal-02048437�

Schauder Estimates for a Class of Potential Mean Field Games of Controls ^∗

J. Fr´ ed´ eric Bonnans ^† Saeed Hadikhanloo ^‡ Laurent Pfeiffer ^§ February 25, 2019

Abstract

An existence result for a class of mean field games of controls is provided. In the considered model, the cost functional to be minimized by each agent involves a price depending at a given time on the controls of all agents and a congestion term. The existence of a classical solution is demonstrated with the Leray-Schauder theorem; the proof relies in particular on a priori bounds for the solution, which are obtained with the help of a potential formulation of the problem.

Keywords: Mean field games of controls, extended mean field games, strongly coupled mean field games, potential formulation, H¨ older estimates.

AMS Classification: 91A13, 49N70.

1 Introduction

The goal of this work is to prove the existence and uniqueness of a classical solution to the following system of partial differential equations:



 

 

 

 



 

 

 

 



(i) −∂ _t u − σ∆u + H(x, t, ∇u(x, t) + φ(x, t) ^| P (t)) = f (x, t, m(t)) (x, t) ∈ Q,

(ii) ∂ t m − σ∆m + div(vm) = 0 (x, t) ∈ Q,

(iii) P (t) = Ψ t, R

T

φ(x, t)v(x, t)m(x, t) dx

t ∈ [0, T ], (iv) v(x, t) = −H _p (x, t, ∇u(x, t) + φ(x, t) ^| P (t)) (x, t) ∈ Q, (v) m(x, 0) = m 0 (x), u(x, T ) = g(x) x ∈ T ^d ,

(MFGC) where

u = u(x, t) ∈ R , m = m(x, t) ∈ R , v = v(x, t) ∈ R ^d , P = P (t) ∈ R ^k ,

∗

The first author acknowledges support from the FiME Lab (Institut Europlace de Finance). The two first authors acknowledges support from the PGMO project ‘Optimal control of conservation equations’, itself supported by iCODE (IDEX Paris-Saclay) and the Hadamard Mathematics LabEx.

†

Inria and CMAP, Ecole Polytechnique, France. E-mail: frederic.bonnans@inria.fr

‡

Inria and CMAP, Ecole Polytechnique, France. E-mail: saeed.hadikhanloo@inria.fr

§

Institute of Mathematics, University of Graz, Austria. E-mail: laurent.pfeiffer@uni-graz.at

with (x, t) ∈ Q := T ^d × [0, T ]. The parameters T > 0, σ > 0 are given and

H : (x, t, p) ∈ Q × R ^d → R, Ψ : (t, z) ∈ [0, T ] × R ^k → R ^k , φ : (x, t) ∈ Q → R ^k×d , f : (x, t, m) ∈ Q × D ₁ (T ^d ) → R, m 0 : x ∈ T ^d → R + , g : x ∈ T ^d → R

are given data. The set D ₁ ( T ^d ) is defined as D ₁ ( T ^d ) =

n

m ∈ L ^∞ ( T ^d ) | m ≥ 0, Z

T

m(x) dx = 1 o

. (1)

Here we work with Z ^d -periodic data and we set the state set as the d-dimensional torus T ^d , that is a quotient set R ^d /Z ^d . The Hamiltonian H is assumed to be such that H(x, t, p) = L ^∗ (x, t, −p), for some mapping L, where L ^∗ (x, t, −p) denotes the Fenchel transform with respect to p. The mapping L is assumed to be convex in its third variable.

The function u, as a solution to the Hamilton-Jacobi-Bellman (HJB) in equation (i)(MFGC) is the value function corresponding to the stochastic optimal control problem:

inf α E h Z _T

0

L(X _t , t, α _t ) + hφ(X _t , t) ^| P (t), α _t i + f(X _t , t, m(t)) dt + g(X _T ) i

, (2)

subject to the stochastic dynamics dX _t = α _t dt+ √

2σ dB _t , X ₀ = x ∈ T ^d . The feedback law v given by (iv)(MFGC) is then optimal for this stochastic optimal control problem. Equation (ii)(MFGC) is the Fokker-Planck equation which describes the evolution of m t = L(X _t ) when the optimal feedback law is employed. At last, (iii)(MFGC) makes the quantity P (t) endogenous.

An interpretation of the system (MFGC) is as follows. Consider a stock trading market. A typical trader, with an initial level of stock X 0 = x, controls its level of stock (X t ) t∈[0,T ] through the purchasing rate α _t with stochastic dynamic dX _t = α _t dt + √

2σdB _t . The agent aims at minimizing the expected cost (2) where P (t) is the price of the stock at time t. The agent is considered to be infinitesimal and has no impact on P (t), so it assumes the price as given in its optimization problem. On the other hand, in the equilibrium configuration, the price P (t) (t ∈ [0, T ]) becomes endogenous and indeed, is a function of the optimal behaviour of the whole population of agents as formulated in (iii)(MFGC). The expression D(t) := R

T

φ(x, t)v(x, t)m(x, t) dx can be considered as a weigted net demand formulation and the relation P = Ψ(D) is the result of supply-demand relation which determines the price of the good at the market. Thus, this system captures an equi- librium configuration. Similar models have been proposed in the electrical engineering literature, see for example [2, 10, 11] and the references therein.

In most mean field game models, the individual players interact through their position only,

that is, via the variable m. The problem that we consider belongs to the more general class

of problems, called extended mean field games, for which the players interact through the joint

probability distribution µ of states and controls. Several existence results have been obtained for

such models: in [13] for stationary mean field games, in [15] for deterministic mean field games. In

[6, Section 5], a class of problems where µ enters in the drift and the integral cost of the agents

is considered. We adopt the terminology mean field games of controls employed by the authors of

the latter reference. Let us mention that our existence proof is different from the one of [6], which

includes control bounds. In [3, Section 1], a model where the drift of the players depends on µ is

analyzed. In [14], a mean field game model is considered where at all time t, the average control

(with respect to all players) is prescribed. We finally mention that extended mean field games

have been studied with a probabilistic approach in [1, 8] and in [7, Section 4.6], and that a class of linear-quadratic extended mean field games has been analyzed in [19].

A difficulty in the study of mean field games of controls is the fact that the control variable, at a given time t, cannot be expressed in an explicit fashion as a function of m(·, t) and u(·, t).

Instead, one has to analyze the well-posedness and the stability of a fixed point equation (see for example [6, Lemma 5.2]). In our model, if we combine (iii) and (iv)(MFGC), we obtain the fixed point equation

v = −H _p (∇u + Ψ(∫ φvm)) (3)

for the control variable v. A central idea of the present article is the following: equation (3) is equivalent to the optimality conditions of a convex optimization problem, when L is convex and Ψ is the gradient of a convex function Φ. This observation allows to show the existence and uniqueness of a solution v (to equation (3)) and to investigate its dependence with respect to ∇u and m in a natural way. More precisely, we prove that this dependence is locally H¨ older continuous.

The existence of a classical solution of (MFGC) is established with the Leray-Schauder theorem and classical estimates for parabolic equations. A similar approach has been employed in [16] and [17] for the analysis of a mean field game problem proposed by Chan and Sircar in [9]. In this model, each agent exploits an exhaustible resource and fixes its price. The evolution of the capacity of a given producer depends on the price set by the producer, but also on the average price (with respect to all producers).

The application of the Leray-Schauder theorem relies on a priori bounds for fixed points. These bounds are obtained in particular with a potential formulation of the mean field game problem:

we prove that all solutions to (MFGC) are also solutions to an optimal control problem of the Fokker-Planck equation. We are not aware of any other publication making use of such a potential formulation for a mean field game of controls, with the exception of [17] for the Chan and Sircar model. Let us mention that besides the derivation of a priori bounds, the potential formulation of the problem can be very helpful for the numerical resolution of the problem and the analysis of learning procedures (which are out of the scope of the present work).

The article is structured as follows. We list in Section 2 the assumptions employed all along.

The main result (Theorem 3.1) is stated in Section 3. We provide in Section 4 a first incomplete potential formulation of the problem, incomplete in so far as the term f (m) is not integrated. We also introduce some auxiliary mappings, which allow to express P and v as functions of m and u.

We give some regularity properties for these mappings in Section 5. In Section 6 we establish some a priori bounds for solutions to the coupled system. We finally prove our main result in Section 7.

In the last section, we give a full potential formulation of the problem, prove the uniqueness of the solution to (MFGC) and prove that (u, P, f(m)) is the solution to an optimal control problem of the HJB equation, under an additional monotonicity condition on f .

2 Assumptions on data and parabolic estimates

2.1 Notation and assumptions

Let us introduce the main notation used in the article. Recall that D ₁ (T ^d ) was defined in (1). For

all m ∈ D ₁ ( T ^d ), for all measurable functions v : T ^d → R ^d such that |v(·)| ² m(·) is integrable, the

following inequality holds true, Z

T

v(x)m(x) dx

2

≤ Z

T

|v(x)| ² m(x) dx. (4) by the Cauchy-Schwarz inequality.

The gradient of the data functions with respect to some variable is denoted with an index, for example, H _p denotes the gradient of H with respect to p. The same notation is used for the Hessian matrix. The gradient of u with respect to x is denoted by ∇u. Let us mention that very often, the variables x and t are omitted, to alleviate the calculations. We also denote by R

φvm the integral R

T

φvm dx when used as a second argument of Ψ. For a given normed space X, the ball of center 0 and radius R is denoted B(X, R).

Along the article, we use the following H¨ older spaces: C ^α (Q), C ^2+α ( T ^d ), and C ^2+α,1+α/2 (Q), defined as usual with α ∈ (0, 1). Sobolev spaces are denoted by W ^k,p , the order of derivation k being possibly non-integral. We fix now a real number p such that

p > d + 2.

We will also make use of the following Banach space:

W ^2,1,p (Q) = L ^p (0, T ; W ^2,p ( T ^d )) ∩ W ^1,p (Q).

Convexity assumptions We collect below the required assumptions on the data. As announced in the introduction, H is related to the convex conjugate of a mapping L : Q × R ^d → R as follows:

H(x, t, p) = L ^∗ (x, t, −p) = sup

v∈ R

−hp, vi − L(x, t, v). (5) The mapping L is assumed to be strongly convex in its third variable, uniformly in x and t, that is, we assume that L is differentiable with respect to v and that there exists C > 0 such that

hL _v (x, t, v ₂ ) − L _v (x, t, v ₁ ), v ₂ − v ₁ i ≥ 1

C |v ₂ − v ₁ | ² , (A1) for all (x, t) ∈ Q and for all v ₁ and v ₂ ∈ R ^d . This ensures that H takes finite values and that H is continuously differentiable with respect to p, as can be easily checked. Moreover, the supremum in (5) is reached for a unique v, which is then given by v = −H _p (x, t, p), i.e.

H(x, t, p) + L(x, t, v) + hp, vi = 0 ⇐⇒ v = −H _p (x, t, p), ∀(x, t, p, v) ∈ Q × R ^d × R ^d . (6) We also assume that Ψ has a potential, that is, there exists a mapping Φ : [0, T ] × R ^k → R, differentiable in its second argument, such that

Ψ(t, z) = Φ z (t, z), ∀(t, z) ∈ [0, T ] × R ^k . (7)

Regularity assumptions We assume that L v is differentiable with respect to x and v and that φ

is differentiable with respect to x. All along the article, we make use of the following assumptions.

Growth assumptions: there exists C > 0 such that for all (x, t) ∈ Q, y ∈ T ^d , v ∈ R ^d , z ∈ R ^k , and m ∈ D ₁ ( T ^d ),

• L(x, t, v) ≤ C|v| ² + C (A2)

• |L(y, t, v) − L(x, t, v)| ≤ C|y − x|(1 + |v| ² ) (A3)

• |Ψ(t, z)| ≤ C|z| + C (A4)

• |f (x, t, m)| ≤ C. (A5)

H¨ older continuity assumptions:

• For all R > 0, there exists α ∈ (0, 1) such that



 

 



 

 



L ∈ C ^α (B R ), L _v ∈ C ^α (B _R , R ^d ), L vx ∈ C ^α (B R , R ^d×d ), L vv ∈ C ^α (B _R , R ^d×d ),



 

 

Ψ ∈ C ^α (B _R ⁰ , R ^d ), φ ∈ C ^α (Q, R ^k×d ), D x φ ∈ C ^α (Q, R ^k×d×d ),

(A6)

where B R = Q × B ( R ^d , R) and B _R ⁰ = [0, T ] × B ( R ^k , R).

• There exists α ∈ (0, 1) and C > 0 such that

|f(x ₂ , t ₂ , m ₂ ) − f(x ₁ , t ₁ , m ₁ )| ≤ C |x ₂ − x ₁ | + |t ₂ − t ₁ | ^α + km ₂ − m ₁ k ^α _L

( T

)

, (A7) for all (x 1 , t 1 ) and (x 2 , t 2 ) ∈ Q and for all m 1 and m 2 ∈ D ₁ ( T ^d ).

• There exists α ∈ (0, 1) such that m ₀ ∈ C ^2+α ( T ^d ) and g ∈ C ^2+α ( T ^d ). (A8) We finally assume that

m ₀ (x) ≥ 0, ∀x ∈ T ^d and Z

T

m ₀ (x) dx = 1. (A9)

Let us mention here that the variables C > 0 and α ∈ (0, 1) used all along the article are generic constants. The value of C may increase from an inequality to the next one and the value of the exponent α may decrease.

Some lower bounds for L and for Φ can be easily deduced from the convexity assumptions. By Assumption (A6), L(x, t, 0) and L _v (x, t, 0) are bounded. It follows then from the strong convexity assumption (A1) and (A2) that there exists a constant C > 0 such that

1

C |v| ² − C ≤ L(x, t, v), for all (x, t, v) ∈ Q × R ^d . (8) Without loss of generality, we can assume that Φ(t, 0) = 0, for all t ∈ [0, T ]. Since Φ is convex, we have that Φ(t, z) ≥ hΨ(t, 0), zi, for all z ∈ R ^k . We deduce then from Assumption (A4) that

Φ(t, z) ≥ −C|z|, for all z ∈ R ^k , (9)

where C is independent of t and z.

Some regularity properties for the Hamiltonian can be deduced from the convexity assumption

(A1) and the H¨ older continuity of L and its derivatives (Assumption (A6)). They are collected in

the following lemma.

Lemma 2.1. The Hamiltonian H is differentiable with respect to p and H _p is differentiable with respect to x and p. Moreover, for all R > 0, there exists α ∈ (0, 1) such that

H ∈ C ^α (B _R ), H _p ∈ C ^α (B _R , R ^d ), H px ∈ C ^α (B R , R ^d×d ), H pp ∈ C ^α (B R , R ^d×d ).

Proof. For a given (x, t, p) ∈ Q × R ^d , there exists a unique v := v(x, t, p) maximizing the function v ∈ R ^d 7→ −hp, vi − L(x, t, p), which is strongly concave by (A1). It is then easy to deduce from (8) and the boundedness of L(x, t, 0) that there exists a constant C, independent of (x, t, p), such that

|v(x, t, p)| ≤ C(|p| + 1). (10) For all (x, t, p) ∈ Q × R ^d ,

p + L v (x, t, v(x, t, p)) = 0. (11)

Since L _v is continuously differentiable with respect to x and v, we obtain with the inverse mapping theorem that v(x, t, p) is continuously differentiable with respect to x and p. Let R > 0 and let (x 1 , t 1 , p 1 ) and (x 2 , t 2 , p 2 ) ∈ Q × B R . Let v i = v(x i , t i , p i ) for i = 1, 2. By (10), we have |v _i | ≤ C, where C does not depend on x _i , t _i , and p _i (but depends on R). Moreover, we have

hp ₂ − p 1 , v 2 − v 1 i + hL _v (x 2 , t 2 , v 2 ) − L v (x 1 , t 1 , v 2 ), v 2 − v 1 i

+ hL _v (x ₁ , t ₁ , v ₂ ) − L _v (x ₁ , t ₁ , v ₁ ), v ₂ − v ₁ i = 0.

We deduce from (A1), Young’s inequality, and (A6) that there exists C > 0 and α ∈ (0, 1), both independent of x _i , t _i , and p _i (but dependent of R) such that

1

C |v ₂ − v 1 | ² ≤ |hp ₂ − p 1 , v 2 − v 1 i| + |hL _v (x 2 , t 2 , v 2 ) − L v (x 1 , t 1 , v 2 ), v 2 − v 1 i|

≤ 1

2ε |p ₂ − p 1 | ² + ε|v ₂ − v 1 | ² + C

ε |x ₂ − x 1 | ^α + |t ₂ − t 1 | ^α ,

for all ε > 0. Taking ε = _2C ¹ , we deduce that the mapping (x, t, p) ∈ B _R 7→ v(x, t, p) is H¨ older continuous. Since L is H¨ older continuous on bounded sets, we obtain that the Hamiltonian H(x, t, p) = −hp, v(x, t, p)i − L(x, t, v(x, t, p)) is H¨ older continuous on B R .

One can easily check that H p (x, t, p) = −v(x, t, p), which proves that H p is H¨ older continuous on B _R . Finally, differentiating relation (11) with respect to x and p, we obtain that

D x v(x, t, p) = − L vv (x, t, v(x, t, p)) ⁻¹ L vx (x, t, v(x, t, p)) D _p v(x, t, p) = − L _vv (x, t, v(x, t, p)) ⁻¹ .

We deduce then with Assumption (A6) that D x v(x, t, p) and D p v(x, t, p) (and thus H px and H pp ) are H¨ older continuous on B _R , as was to be proved.

An example of coupling term We finish this subsection with an example of a mapping f satisfying the regularity assumptions (A5) and (A7). Let ϕ ∈ L ^∞ ( R ^d ) be a given Lipschitz con- tinuous mapping, with modulus C 1 . Let us set C 2 = kϕk _L

( R

) . Let K : Q × [−C ₂ , C 2 ] → R be a measurable mapping satisfying the following assumptions:

1. The mapping x ∈ T ^d 7→ K(x, 0, 0) lies in L ¹ ( T ^d ).

2. There exist a mapping C ₃ ∈ L ¹ ( T ^d ) and α ∈ (0, 1) such that for a.e. x ∈ T ^d , for all t ₁ and t ₂ ∈ [0, T ] and for all w ₁ and w ₂ ∈ [−C ₂ , C ₂ ],

|K(x, t ₂ , w 2 ) − K(x, t 1 , w 1 )| ≤ C 3 (x) |t ₂ − t 1 | ^α + |w ₂ − w 1 | ^α .

Let us set ˜ ϕ(x) := ϕ(−x). We identify m ∈ L ^∞ ( T ^d ), with its extension by 0 over R ^d so that the convolution product below is well-defined:

m ∗ ϕ(x) :=

Z

R

m(x − y)ϕ(y)dy, x ∈ T ^d . (12)

We keep in mind that m ∗ ϕ is a function over T ^d . Then

km ∗ ϕk ∞ ≤ kϕk ∞ , for all m ∈ D ₁ (T ^d ). (13) In a similar way we can define

f K (x, t, m) = (K(·, t, m ∗ ϕ(·)) ∗ ϕ)(x), ˜ (14) and we have that

kf _K (x, t, m)k _∞ ≤ kK(·, t, m ∗ ϕ)k ₁ k ϕk ˜ _∞ ≤ k|K(·, t, m ∗ ϕ)k _∞ kϕk _∞ (15) The following lemma is inspired from [4, Example 1.1].

Lemma 2.2. The above mapping f K satisfies Assumptions (A5) and (A7).

Proof. Assumption (A5) follows from (15). We next prove (A7). Let (x ₁ , t ₁ ) and (x ₂ , t ₂ ) ∈ Q, let m ₁ and m ₂ ∈ D ₁ ( T ^d ). Then

|f _K (x ₂ , t ₂ , m ₂ ) − f _K (x ₁ , t ₂ , m ₂ )| ≤ kK(y, t ₂ , m ₂ ∗ ϕ(y))k ₁ kϕ(x ₂ − ·) − ϕ(x ₁ − ·)k ∞

≤ C ₁ C|x ₂ − x ₁ |.

Also,

|f _K (x ₁ , t ₂ , m ₂ ) − f _K (x ₁ , t ₁ , m ₁ )| = kK(y, t ₂ , m ₂ ∗ ϕ(y)) − K(y, t ₁ , m ₁ ∗ ϕ(y))k ₁ kϕk ∞

≤ C ₂ kC ₃ k _L

( T

) |t ₂ − t ₁ | ^α + k(m ₂ − m ₁ ) ∗ ϕk ^α _L

( T

)

. Finally, we have k(m ₂ − m ₁ ) ∗ ϕk _L

( T

) ≤ km ₂ − m ₁ k _L

( T

) kϕk _L

( T

) and thus, assumption (A7) follows.

3 Main result and general approach

Theorem 3.1. There exists α ∈ (0, 1) such that (MFGC) has a classical solution (m, u, P, v ), with



 

 



 

 



m ∈ C ^2+α,1+α/2 (Q), u ∈ C ^2+α,1+α/2 (Q), P ∈ C ^α (0, T ; R ^k ),

v ∈ C ^α (Q, R ^d ), D _x v ∈ C ^α (Q, R ^d×d ).

(16)

The result is obtained with the Leray-Schauder theorem, recalled below.

Theorem 3.2 (Leray-Schauder, [12], Theorem 11.6). Let X be a Banach space and let T : X × [0, 1] → X be a continuous and compact mapping. Assume that T (x, 0) = 0 for all x ∈ X and assume there exists C > 0 such that kxk _X < C for all (x, τ ) ∈ X × [0, 1] such that T (x, τ ) = x.

Then, there exists x ∈ X such that T (x, 1) = x.

The application of the Leray-Schauder theorem and the construction of T will be detailed in Section 7. Let us mention that the set of fixed points of T (·, τ ), for τ ∈ [0, 1], will coincide with the set of solutions of the following parametrization of (MFGC):



 

 

 

 



 

 

 

 



(i) −∂ _t u − σ∆u + τ H(x, t, ∇u(x, t) + φ(x, t) ^| P (t)) = τ f (x, t, m(t)) (x, t) ∈ Q,

(ii) ∂ _t m − σ∆m + τ div(mv) = 0 (x, t) ∈ Q,

(iii) P (t) = Ψ t, R

T

φ(x, t)v(x, t)m(x, t) dx

t ∈ [0, T ], (iv) v(x, t) = −H _p (x, t, ∇u(x, t) + φ(x, t) ^| P (t)) (x, t) ∈ Q, (v) m(x, 0) = m 0 (x), u(x, T ) = τ g(x) x ∈ T ^d ,

(MFGC τ ) Of course, (MFGC _τ ) corresponds to (MFGC) for τ = 1. Let us introduce the space X and X ⁰ , used for the formulation of the fixed-point equation:

X := W ^2,1,p (Q) 2

, X ⁰ := X × L ^∞ (0, T ; R ^k ) × L ^∞ (Q, R ^d ).

The HJB equation (i) and the Fokker-Planck equation (ii) are classically understood in the viscosity and weak sense, respectively. However, due to the choice of the solution spaces, we may interpret these equations as equalities in L ^p (Q). A first and important step of our analysis is the construction of auxiliary mappings allowing to express v and P as functions of m and u. These mappings cannot be obtained in a straightforward way, since in (iii), P depends on v and in (iv), v depends on P.

Lemma 3.1. Let τ ∈ [0, 1], let (m, v) ∈ W ^2,1,p (Q) × L ^∞ (Q, R ^d ) be a weak solution to the Fokker- Planck equation ∂ t m − σ∆m + τ div(vm) = 0, m(·, 0) = m 0 (·). Then m ≥ 0 and for all t ∈ [0, T ], R

T

m(x, t) dx = 1.

Proof. Multiply (MFGC τ )(ii) by µ(x, t) := min(0, m(x, t)). Use ∇µ(x, t) = 1 {m(x,t)<0} ∇m(x, t), so that integrating (by parts) over Q t := T ^d × (0, t), since v is essentially bounded, we get that

1 2

Z

T

µ(x, t) ² dx + σ Z Z

Q

|∇µ(x, s)| ² dx ds = τ Z Z

Q

hv, ∇µim dx ds

= τ Z Z

Q

hv, ∇µiµ dx ds ≤ C Z

Q

|µ| ² dx ds + σ Z Z

Q

|∇µ| ² dx ds,

so that after cancellation of the contribution of ∇µ, we obtain, applying Gronwall’s lemma to a(t) := R

T

µ(x, t) ² , that a(t) = 0 for all t which means that m is non-negative. Moreover, for all t ∈ [0, T ],

Z

T

m(x, t) dx = Z

T

m(x, 0) dx + Z Z

Q

σ∇m − τ div(vm) dx ds.

Integrating by parts the double integral we see that it is equal to 0, and we conclude by noting that R

T

m(x, 0) dx = R

T

m 0 (x) dx = 1.

4 Potential formulation

In this section, we first establish a potential formulation of the mean field game problem (MFGC τ ), that is to say, we prove that for (u τ , m τ , v τ , P τ ) ∈ X ⁰ satisfying (MFGC τ ), (m τ , v τ ) is a solution to an optimal control problem. We prove then that for all t, v _τ (·, t) is the unique solution of some optimization problem, which will enable us to construct the announced auxiliary mappings.

Let us introduce the cost functional B : W ^2,1,p (Q) × L ^∞ (Q, R ^d ) × L ^∞ (Q) → R , defined by B(m, v; ˜ f ) =

Z Z

Q

L(x, t, v(x, t)) + ˜ f (x, t)

m(x, t) dx dt + Z

T

g(x)m(x, T ) dx +

Z _T

0

Φ t,

Z

T

φ(x, t)v(x, t)m(x, t) dx

dt. (17)

We have the following result.

Lemma 4.1. For all τ ∈ [0, 1], for all (u τ , m τ , v τ , P τ ) ∈ X ⁰ satisfying (MFGC τ ), the pair (m τ , v τ ) is the solution to the following optimization problem:

min

m∈W

(Q) v∈L

(Q, R

)

B(m, v; ˜ f _τ ), subject to:

( ∂ t m − σ∆m + τ div(vm) = 0,

m(x, 0) = m ₀ (x), (18) where f ˜ τ (x, t) = f (x, t, m τ (t)).

Remark 4.1. Let us emphasize that the above optimal control problem is only an incomplete potential formulation, since the term ˜ f τ still depends on m τ .

Proof of Lemma 4.1. Let us consider the case where τ ∈ (0, 1]. Let (m, v) ∈ W ^2,1,p (Q)×L ^∞ (Q, R ^d ) be a feasible pair. For all (x, t) ∈ Q, v τ = −H _p (∇u _τ + φ ^| P τ ). Therefore, by (5) and (6), we have that

L(v) ≥ − H(∇u _τ + φ ^| P τ ) − h∇u _τ + φ ^| P τ , vi , L(v τ ) = − H(∇u _τ + φ ^| P τ ) − h∇u _τ + φ ^| P τ , v τ i , for all (x, t) ∈ Q. Moreover, by Lemma 3.1, m ≥ 0 and m _τ ≥ 0. Therefore,

L(v)m − L(v _τ )m _τ ≥ −H(∇u _τ + φ ^| P _τ )(m − m _τ ) − h∇u _τ + φ ^| P _τ , vm − v _τ m _τ i . (19) Using (i)(MFGC _τ ), we obtain

L(v)m − L(v τ )m τ ≥ 1

τ (−∂ _t u τ − σ∆u τ − τ f ˜ τ )(m − m τ ) − h∇u _τ + φ ^| P τ , vm − v τ m τ i . After integration with respect to x, we obtain that for all t,

Z

T

(L(v)m − L(v τ )m τ ) + ˜ f τ (m − m τ ) dx

≥ 1 τ

Z

T

(−∂ _t u τ − σ∆u τ )(m − m τ ) dx − Z

T

h∇u _τ , vm − v τ m τ i dx − hP _τ , R

φ(vm − v τ m τ )i.

We obtain with the convexity of Φ and (iii)(MFGC _τ ) that Φ( R

φmv) − Φ( R

φv _τ m _τ ) ≥ hΨ( R

φm _τ v _τ ), R

φ(vm − v _τ m _τ )i

= hP _τ , R

φ(mv − m τ v τ )i. (20)

Using the previous calculations to bound B(m, v; ˜ f _τ ) − B (m _τ , v _τ ; ˜ f _τ ) from below, we observe that the term hP _τ , R

φ(m − m τ v τ )i cancels out and obtain that B(m, v; ˜ f _τ ) − B(m _τ , v _τ ; ˜ f _τ ) ≥

Z Z

Q

1

τ (−∂ _t u _τ − σ∆u _τ ) − h∇u _τ , mv − m _τ v _τ i dx dt +

Z

T

g(x)(m(x, T ) − m τ (x, T )) dx.

Integrating by parts and using (ii)(MFGC τ ), we finally obtain that B(m, v; ˜ f _τ ) − B(m _τ , v _τ ; ˜ f _τ ) ≥ 1

τ Z

T

u _τ (0, x)(m(0, x) − m _τ (0, x)) dx = 0,

as was to be proved. We do not detail the proof for the case τ = 0, which is actually simpler.

Indeed, for τ = 0, the solution to the Fokker-Planck equation is independent of v and thus m = m _τ in the above calculations.

We have proved that the pair (m _τ , v _τ ) is the solution to an optimal control problem. Therefore, for all t, v τ (·, t) minimizes the Hamiltonian associated with problem (18). Let us introduce some notation, in order to exploit this property. For m ∈ D ₁ ( T ^d ), we denote by L ² _m ( T ^d , R ^d ) the Hilbert space of measurable mappings v : T ^d → R ^d such that R

T

|v| ² m < ∞, equipped with the scalar product R

T

hv ₁ , v ₂ im. An element of L ² _m ( T ^d ) is an equivalent class of functions equal m almost everywhere. Note that L ^∞ ( T ^d ) ⊂ L ² _m ( T ^d ).

For t ∈ [0, T ], for m ∈ D ₁ ( T ^d ), and for w ∈ L ² _m ( T ^d ), we consider the mapping v ∈ L ² _m (T ^d , R ^d ) 7→ J(v; t, m, w) := Φ t, R

φvm +

Z

T

L(v) + hw, vi m dx.

Combining inequalities (19) and (20) (with m = m τ ), we directly obtain that for all t ∈ [0, T ], for all v ∈ L ² _m ( T ^d , R ^d ) with m = m _τ (·, t),

J v; t, m _τ (t), ∇u _τ (t)

≥ J v _τ (t); t, m _τ (t), ∇u _τ (t) .

The following lemma will enable us to express P _τ (t) and v _τ (·, t) as functions of m _τ (·, t) and u τ (·, t). The key idea is, roughly speaking, to prove the existence and uniqueness of a minimizer to J (·; t, m, w).

Lemma 4.2. For all t ∈ [0, T ], for all m ∈ D ₁ ( T ^d ), for all R > 0, and for all w ∈ L ^∞ ( T ^d , R ^d ) such that kwk _L

( T

, R

) ≤ R, there exists a unique pair (v, P ) ∈ L ^∞ ( T ^d , R ^d ) × R ^k , such that

( v(x) = −H _p (x, t, w(x) + φ(x, t) ^| P ), for a.e. x ∈ T ^d , P = Ψ(t, R

φvm). (21)

The pair (v, P ) is then denoted (v(t, m, w), P(t, m, w)). Moreover, we have

kv(t, m, w)k _L

( T

, R

) ≤ C and |P(t, m, w)| ≤ C, (22)

where the constant C is independent of t, m, and w (but depends on R).

Proof. If the pair (v, P ) satifies (21), then

v = −H _p (w + φ ^| Ψ( R

φvm)) a.e. on T ^d . (23)

One can easily check that for proving the existence and uniqueness of a pair (v, P ) satisfying (21), it is sufficient to prove the existence and uniqueness of v ∈ L ^∞ ( T ^d , R ^d ) satisfying (23). For future reference, let us mention that by (6), relation (23) is equivalent to

φ ^| (x, t)Ψ(t, R

φvm) + L _v (x, t, v(x)) + w(x) = 0, for a.e. x ∈ T ^d . (24) Step 1 : existence and uniqueness of a minimizer of J (·; t, m, w) in L ² _m ( T ^d , R ^d ).

In view of (A1), v 7→ R

T

L(v)m dx is strongly convex over L ² _m ( T ^d , R ^d ). Since the sum of a l.s.c. convex function and of a l.s.c. strongly convex function is l.s.c. and strongly convex, so is J (·; t, m, w). Since it is continuous, it possesses a unique minimizer ¯ v in L ² _m ( T ^d , R ^d ). We obtain

Ck¯ vk ² _L

m

( T

, R

) − C ≤ J (¯ v; t, m, w)) ≤ J(0; t, m, w)) = C, (25) so that k¯ vk ² _L

m

( T

, R

) ≤ C, with C independent of t, m, and w, but depending on R, as all constants C used in the proof.

Step 2: existence of v(t, m, w) and a priori bound.

One can check that J (·; t, m, w) is differentiable in L ^∞ ( T ^d ). Since ¯ v is optimal, DJ(¯ v; t, m, w) = 0 and thus

φ ^| (x, t)Ψ(t, R

φ(x ⁰ )¯ v(x ⁰ )m(x ⁰ ) dx ⁰ ) + L _v (x, t, ¯ v(x)) + w(x)

m(x) = 0, for a.e. x ∈ T ^d . Using then the equivalence of (23) and (24), we obtain that

m(x) > 0 = ⇒ v(x) = ¯ −H _p x, t, w(x) + φ ^| (x, t)Ψ(t, R

φ(x ⁰ , t)¯ v(x ⁰ )m(x ⁰ ) dx ⁰ )

, for a.e. x ∈ T ^d . Consider now the measurable function v defined by

v(x) = −H _p x, t, w(x) + φ ^| (x, t)Ψ(t, R

φ(x ⁰ , t)¯ v(x ⁰ )m(x ⁰ ) dx ⁰

, for a.e. x ∈ T ^d .

The two functions v and ¯ v may not be equal for a.e. x if m(x) = 0 on a subset of T ^d of non-zero mea- sure. Still they are equal in L ² _m (T ^d , R ^d ), which ensures in particular that R

φ(x ⁰ , t)¯ v(x ⁰ )m(x ⁰ ) dx ⁰ = R φ(x ⁰ , t)v(x ⁰ )m(x ⁰ ) dx ⁰ and finally that v satisfies (23) and lies in L ^∞ ( T ^d , R ^d ), as a consequence of the continuity of H _p (proved in Lemma 2.1). We also have that k¯ vk _L

m

( T

, R

) = kvk _L

m

( T

, R

) ≤ C, by (25). Using the Cauchy-Schwarz inequality and Assumption (A6), we obtain that | R

φvm| ≤ C.

We obtain then with Assumption (A4) that for P = Ψ( R

φvm), we have |P| ≤ C. Using Assump- tion (A6) and the continuity of H _p , we finally obtain that kvk _L

( T

, R

) ≤ C. Thus the bound (22) is satisfied.

Step 3: uniqueness of v(t, m, w).

Let v ₁ and v ₂ ∈ L ^∞ ( T ^d , R ^d ) satisfy (23). Then DJ(v ₁ ; t, m, w) = DJ (v ₂ ; t, m, w) = 0, proving that v 1 and v 2 are minimizers of J (·; t, m, w) and thus are equal in L ² _m (T ^d , R ^d ). It follows that R φ(x ⁰ , t)v 1 (x ⁰ )m(x ⁰ ) dx ⁰ = R

φ(x ⁰ , t)v 2 (x ⁰ )m(x ⁰ ) dx ⁰ and finally that v 1 = v 2 , by (23).

5 Regularity results for the auxiliary mappings

We provide in this section some regularity results for the mappings v and P. We begin by proving that P(·, ·, ·) is locally H¨ older continuous. For this purpose, we perform a stability analysis of the optimality condition (24).

Lemma 5.1. Let t ₁ and t ₂ ∈ [0, T ], let w ₁ and w ₂ ∈ L ^∞ ( T ^d , R ^d ), and let m ₁ and m ₂ ∈ D ₁ ( T ^d ).

Let R > 0 be such that kw ₁ k _L

( T

, R

) ≤ R and kw ₂ k _L

( T

, R

) ≤ R. Then, there exist a constant C > 0 and an exponent α ∈ (0, 1), both independent of t ₁ , t ₂ , w ₁ , w ₂ , m ₁ , and m ₂ but depending on R, such that

|P(t ₂ , m ₂ , w ₂ ) − P(t ₁ , m ₁ , w ₁ )| ≤ C |t ₂ − t ₁ | ^α + kw ₂ − w ₁ k ^α _L

( T

, R

) + km ₂ − m ₁ k ^α _L

₍

T

)

. (26) Proof. Note that all constants C > 0 and all exponents α ∈ (0, 1) involved below are independent of t 1 , t 2 , w 1 , w 2 , m 1 , and m 2 . They are also independent of x ∈ T ^d and ε > 0. For i = 1, 2, we set v i = v(t i , m i , w i ) and φ i = φ(·, t _i ) ∈ L ^∞ ( T ^d ). By (22), we have

kv _i k _L

( T

, R

) ≤ C. (27) By the optimality condition (24), we have that

φ ^| _i Ψ t i , R φ i v i m i

+ L v (t i , v i ) + w i = 0, for a.e. x ∈ T ^d . (28) Consider the difference of (28) for i = 2 with (28) for i = 1. Integrating with respect to x the scalar product of the obtained difference with v ₂ m ₂ − v ₁ m ₁ , we obtain that

(a ₁ ) + (a ₂ ) + (a ₃ ) = 0, where

(a ₁ ) = Z

T

φ ^| ₂ Ψ(t ₂ , R

φ ₂ v ₂ m ₂ ) − φ ^| ₁ Ψ(t ₁ , R

φ ₁ v ₁ m ₁ ), v ₂ m ₂ − v ₁ m ₁ dx, (a 2 ) =

Z

T

hL _v (t 2 , v 2 ) − L v (t 1 , v 1 ), v 2 m 2 − v 1 m 1 i dx, (a 3 ) =

Z

T

hw ₂ − w 1 , v 2 m 2 − v 1 m 1 i dx.

We look for a lower estimate of these three terms. Let us mention that the term v ₂ m ₂ − v ₁ m ₁ , appearing in the three terms, will be estimated only at the end.

Estimation from below of (a 1 ). We have (a 1 ) = (a 11 ) + (a 12 ), where (a ₁₁ ) =

Z

T

φ ^| ₂ Ψ(t ₂ , R

φ ₂ v ₂ m ₂ ) − φ ^| ₁ Ψ(t ₁ , R

φ ₁ v ₂ m ₂ ), v ₂ m ₂ − v ₁ m ₁ i dx (a ₁₂ ) =

Z

T

φ ^| ₁ Ψ(t ₁ , R

φ ₁ v ₂ m ₂ ) − φ ^| ₁ Ψ(t ₁ , R

φ ₁ v ₁ m ₁ ), v ₂ m ₂ − v ₁ m ₁ dx.

By monotonicity of Ψ, we have that (a ₁₂ ) = D

Ψ(t ₁ , R

φ ₁ v ₂ m ₂ ) − Ψ(t ₁ , R

φ ₁ v ₁ m ₁ ), Z

T

φ ₁ v ₂ m ₂ − φ ₁ v ₁ m ₁ dx E

≥ 0.

Let us consider (a ₁₁ ). We set

( Ψ i = Ψ(t i , R

φ i v 2 m 2 ), for i = 1, 2, ξ(x) = φ 2 (x) ^| Ψ 2 − φ 1 (x) ^| Ψ 1 ,

so that (a 11 ) = R

T

hξ, v ₂ m 2 − v 1 m 1 i dx. Using Assumption (A6), one can check that |Ψ _i | ≤ C and that

Ψ 2 − Ψ 1

≤ C|t ₂ − t 1 | ^α . Since ξ = (φ 2 − φ 1 ) ^| Ψ 2 + φ ^| ₁ (Ψ 2 − Ψ 1 ), we obtain with Assumption (A6) again that

kξk _L

( T

, R

) ≤ C |Ψ ₂ − Ψ ₁ | + kφ ₂ − φ ₁ k _L

( T

, R

)

≤ C|t ₂ − t ₁ | ^α and further with Young’s inequality that

|(a ₁₁ )| ≤ C

ε |t ₂ − t 1 | ^α + ε

2 kv ₂ m 2 − v 1 m 1 k ² _L

( T

, R

) . Estimation from below of (a ₂ ). We have (a ₂ ) = (a ₂₁ ) + (a ₂₂ ) + (a ₂₃ ), where

(a 21 ) = Z

T

hL _v (t 2 , v 2 ) − L v (t 1 , v 2 ), v 2 m 2 − v 1 m 1 )i dx (a 22 ) =

Z

T

hL _v (t 1 , v 2 ) − L v (t 1 , v 1 ), v 2 (m 2 − m 1 )i dx (a 23 ) =

Z

T

hL _v (t 1 , v 2 ) − L v (t 1 , v 1 ), (v 2 − v 1 )m 1 i dx.

As a consequence of (27), assumption (A6), and Young’s inequality, we have that

|(a ₂₁ )| ≤ 1

2ε kL _v (t 2 , v 2 (·) − L v (t 1 , v 2 (·))k ² _L

(T

,R

) + ε

2 kv ₂ m 2 − v 1 m 1 k ² _L

(T

,R

)

≤ C

ε |t ₂ − t ₁ | ^α + ε

2 kv ₂ m ₂ − v ₁ m ₁ k ² _L

₍

T

, R

) . By (27) and assumption (A6), |L _v (t ₁ , x, v _i (x))| ≤ C, therefore

|(a ₂₂ )| ≤ Ckm ₂ − m 1 k _L

( T

, R

) . Finally, we have

(a 23 ) ≥ 1 C

Z

T

|v ₂ − v 1 | ² m 1 dx, by assumption (A1) and since m ₁ ≥ 0.

Estimation from below of (a 3 ). Using (30) and Young’s inequality, we obtain that

|(a ₃ )| ≤ 1

2ε kw ₂ − w ₁ k ² _L

( T

, R

) + ε

2 kv ₂ m ₂ − v ₁ m ₁ k ² _L

₍

T

, R

) . Conclusion. We have proved that

1 C

Z

T

|v ₂ − v 1 | ² m 1 dx ≤ (a 23 ) = (a 2 ) − (a 21 ) − (a 22 )

= − (a 1 ) − (a 21 ) − (a 22 ) − (a 3 )

≤ − (a ₁₁ ) − (a ₂₁ ) − (a ₂₂ ) − (a ₃ )

≤ C

ε |t ₂ − t 1 | ^α + 1

2ε kw ₂ − w 1 k ² _L

( T

, R

) + Ckm ₂ − m 1 k _L

( T

, R

)

+ 3

2 εkv ₂ m 2 − v 1 m 1 k ² _L

( T

; R

) . (29)

Let us estimate kv ₂ m ₂ − v ₁ m ₁ k _L

( T

; R

) . Using the Cauchy-Schwarz inequality, we obtain that kv ₂ m ₂ − v ₁ m ₁ k _L

( T

, R

) ≤ kv ₂ (m ₂ − m ₁ )k _L

( T

, R

) + k(v ₂ − v ₁ )m ₁ k _L

( T

, R

)

≤ Ckm ₂ − m ₁ k _L

( T

, R

) + Z

T

|v ₂ − v ₁ | ² m ₁ dx 1/2

. (30)

Injecting this inequality in (29) and taking ε = _3C ¹ , we obtain that Z

T

|v ₂ − v 1 | ² m 1 ≤ C

|t ₂ − t 1 | ^α + km ₂ − m 1 k _L

( T

) + kw ₂ − w 1 k ² _L

( T

, R

)

. (31)

Let us prove (26). We have Z

T

φ 2 v 2 m 2 dx − Z

T

φ 1 v 1 m 1 dx = Z

T

(φ 2 − φ 1 )v 2 m 2 dx +

Z

T

φ 1 v 2 (m 2 − m 1 ) dx + Z

T

φ 1 (v 2 − v 1 )m 1 dx.

Therefore, using assumption (A6) and (31), we obtain that

Z

T

φ ₂ v ₂ m ₂ dx − Z

T

φ ₁ v ₁ m ₁ dx

≤ C

kφ ₂ − φ 1 k _L

( T

, R

) + km ₂ − m 1 k _L

( T

) + Z

T

|v ₂ − v 1 | ² m 1 dx 1/2

≤ C

|t ₂ − t 1 | ^α + km ₂ − m 1 k ^1/2 _L

₍

T

) + kw ₂ − w 1 k _L

(T

,R

)

. Inequality (26) follows, using assumption (A6).

Given m ∈ L ^∞ (0, T ; D ₁ ( T ^d )) and w ∈ L ^∞ (Q), we consider the Nemytskii operators associated with v and P, that we still denote by v and P without risk of confusion:

v(m, w) ∈ L ^∞ (Q, R ^d ), v(m, w)(x, t) = v(t, m(·, t), w(·, t))(x), P(m, w) ∈ L ^∞ (0, T ; R ^k ), P(m, w)(t) = P(t, m(·, t), w(·, t)),

for all (x, t) ∈ Q. We use now Lemma 5.1 to prove regularity properties of the Nemytskii operators v and P. We recall that X = W ^2,1,p (Q) 2

. Lemma 5.2. For all R > 0, the mapping

(m, w) ∈ L ^∞ (0, T ; D ₁ (T ^d )) × B L ^∞ (Q, R ^d ), R) 7→ P(m, w) ∈ L ^∞ (0, T ; R ^k ) (32) and the mapping

(u, m) ∈ B(W ^2,1,p (Q), R) × L ^∞ (0, T ; D ₁ ( T ^d )) 7→ v(m, ∇u) ∈ L ^∞ (Q, R ^d ) ∩ L ^p (0, T ; W ^1,p ( T ^d )) (33) are both H¨ older continuous, that is, there exist α ∈ (0, 1) and C > 0 such that

kP(m ₂ , w ₂ ) − P(m ₁ , w ₁ )k _L

(0,T ; R

) ≤ C km ₂ − m ₁ k ^α _L

(Q) + kw ₂ − w ₁ k ^α _L

(Q)

kv(m ₂ , ∇u ₂ ) − v(m ₁ , ∇u ₁ )k _L

(Q, R

)∩L

(0,T ;W

( T

)) ≤ C ku ₂ − u ₁ k ^α _W

(Q) + km ₂ − m ₁ k ^α _L

(Q)

,

for all m ₁ and m ₂ ∈ L ^∞ (0, T ; D ₁ ( T ^d )), for all w ₁ and w ₂ ∈ B(L ^∞ (Q, R ^d ), R), and for all u ₁ and

u 2 in B (W ^2,1,p (Q), R).

Proof. The H¨ older continuity of the first mapping is a direct consequence of Lemma 5.1. As a consequence, the mapping

(u, m) ∈ B (W ^2,1,p (Q), R) × L ^∞ (0, T ; D ₁ ( T ^d )) 7→ ∇u + φ ^| P(m, ∇u) ∈ L ^∞ (Q, R ^d ) is H¨ older continuous. Using then the relations



 

 

v(m, ∇u) = −H _p (∇u + φ ^| P(m, ∇u)), D x v(m, ∇u) = −H _px (∇u + φ ^| P(m, ∇u))

−H _pp (∇u + φ ^| P(m, ∇u))(∇ ² u + Dφ ^| P(m, ∇u)),

(34)

and the H¨ older continuity of H _p , H _px , and H _pp on bounded sets (Lemma 2.1), we obtain that the second mapping is H¨ older continuous.

Remark 5.1. As a consequence of Lemma 5.2, the images of the mappings given by (32) and (33) are bounded. This fact will be used in the steps 3 and 5 of the proof of Proposition 6.1.

Lemma 5.3. Let R > 0 and β ∈ (0, 1). Then, there exists α ∈ (0, 1) and C > 0 such that for all u ∈ B (W ^2,1,p (Q), R) and for all m ∈ B(C ^β (Q), R) ∩ L ^∞ (0, T ; D ₁ ( T ^d )),

kP(m, ∇u)k _C

(0,T;R

) ≤ C.

Proof. We recall that by Lemma A.2, k∇uk _C

(Q,R

) ≤ Ckuk _W

(Q) . We obtain then the bound on kP(m, ∇u)k _C

(0,T ; R

) with Lemma 5.1.

Lemma 5.4. Let R > 0 and β ∈ (0, 1). There exist α ∈ (0, 1) and C > 0 such that for all u ∈ B (C ^2+β,1+β/2 (Q), R) and for all m ∈ B(C ^β (Q), R) ∩ L ^∞ (0, T ; D ₁ ( T ^d )),

( kv(m, ∇u)k _C

(Q, R

) ≤ C kD _x v(m, ∇u)k _C

(Q,R

) ≤ C.

Proof. The result follows from relations (34), Lemma 5.3, and from the H¨ older continuity of H _p , H px , and H pp on bounded sets.

6 A priori estimates for fixed points

Proposition 6.1. There exist a constant C > 0 and an exponent α ∈ (0, 1) such that for all τ ∈ [0, 1], for all (u τ , m τ , v τ , P τ ) ∈ X ⁰ satisfying (MFGC τ ),

m τ ∈ C ^2+α,1+α/2 (Q), km _τ k _C

(Q) ≤ C, u _τ ∈ C ^2+α,1+α/2 (Q), ku _τ k _C

(Q) ≤ C, P τ ∈ C ^α (0, T ; R ^k ), kP _τ k _C

(0,T;R

) ≤ C

v τ ∈ C ^α (Q, R ^d ), kv _τ k _C

(Q,R

) ≤ C

D x v τ ∈ C ^α (Q, R ^d×d ), kD _x v τ k _C

(Q,R

) ≤ C.

Proof. The proof is decomposed in 10 steps. Let us fix τ ∈ [0, 1] and (u _τ , m _τ , v _τ , P _τ ) ∈ X ⁰ sat- isfying (MFGC _τ ). All constants C and all exponents α ∈ (0, 1) involved below are independent of (u τ , m τ , v τ , P τ ) and τ . Let us recall that ˜ f τ ∈ L ^∞ (Q) has been defined in Lemma 4.1 by f ˜ _τ (x, t) = f (x, t, m _τ (t)).

Step 1: kP _τ k _L

(0,T ; R

) ≤ C.

Let us take v ⁰ = 0 and let m ⁰ be the solution to the heat equation ∂ _t m ⁰ − σ∆m ⁰ = 0, m ⁰ (x, 0) = m 0 (x). By Lemma 4.1, B(m τ , v τ ; ˜ f τ ) ≤ B(m ⁰ , v ⁰ ; ˜ f τ ). Since kφk _L

(Q, R

) ≤ C, we have for all ε > 0 and for all t ∈ [0, T ] that

Z

T

φv _τ m _τ dx ≤ C

Z

T

|v _τ |m _τ dx ≤ C Z

T

|v _τ | ² m _τ dx 1/2

≤ C ε + Cε

Z

T

|v _τ | ² m _τ dx, by the Cauchy-Schwarz inequality and Young’s inequality. The constant C is also independent of ε. Using then the lower bounds (8) and (9) and assumptions (A5) and (A8), we obtain that

C ≥ B(m ⁰ , v ⁰ ; ˜ f _τ ) ≥ B (m _τ , v _τ ; ˜ f _τ )

≥ Z Z

Q

1

C |v _τ | ² m τ dx dt − C Z

Q

φv τ m τ dx dt − C

≥ 1

C − Cε Z Z

Q

|v _τ | ² m _τ dx dt − C 1 + 1

ε

. Taking ε = 1/(2C ² ), we deduce that RR

Q |v _τ | ² m τ dx dt ≤ C. Using then assumption (A4), the boundedness of φ, the Cauchy-Schwarz inequality and the estimate obtained previously, we deduce that

kP _τ k _L

(0,T ; R

) = Z T

0

|Ψ(t, R

φv _τ m _τ )| ² dt ≤ C Z T

0

Z

T

φv _τ m _τ dx

2

dt

≤ C Z Z

Q

|v _τ | ² m _τ dx dt ≤ C, (35) as was to be proved.

Step 2: ku _τ k _L

(Q) ≤ C, k∇u _τ k _L

(Q, R

) ≤ C.

The argument is classical. We have that u _τ is the unique solution to the HJB equation (i)(MFGC _τ ).

It is therefore the value function associated with the following stochastic optimal control problem:

u τ (x, t) = τ

inf

α∈L

F

(t,T; R

) J τ (x, t, α)

, (36)

where

J _τ (x, t, α) := E h Z T

t

L(X _s , s, α _s ) + hφ(X _s , s) ^| P _τ (s), α _s i + ˜ f _τ (X _s , s)

ds + g(X _T ) i

, (37) and (X s ) s∈[t,T] is the solution to the stochastic dynamic dX s = τ α s ds + √

2σdB s , X t = x. Here, L ² _F (t, T ; R ^d ) denotes the set of stochastic processes on (t, T ), with values in R ^d , adapted to the filtration F generated by the Brownian motion (B s ) _s∈[0,T] , and such that E R T

t |α(s)| ² ds

< ∞.

Then, the boundedness of u _τ from above can be immediately obtained by choosing α = 0 in (36)

and using the boundedness of g. We can as well bound u _τ from below since for all (x, s) ∈ Q and for all α ∈ R ^d , we have

L(x, s, α) + hφ(x, s) ^| P _τ (s), αi ≥ 1

C |α| ² − kφk _L

(Q, R

) |P _τ (s)||α| − C

≥ 1

C |α| ² − C|P _τ (s)| ² − C,

for some constant C independent of (x, s), α, and P τ (s). We already know from the previous step that kP _τ k _L

(0,T; R

) ≤ C. So we can conclude that u τ is also bounded from below, and thus ku _τ k _L

_(Q) ≤ C. We also deduce from the above inequality that for all α ∈ L ² _F (t, T ; R ^d ),

E Z T

t

|α _s | ² ds ≤ C J τ (x, t, α) + 1

. (38)

Let us bound ∇u _τ . Choose ε ∈ (0, 1). For arbitrary (x, t), take an ε-optimal stochastic optimal control ˜ α for (36). We can deduce from the boundedness of the map u _τ and inequality (38) that

E Z T

t

|˜ α s | ² ds ≤ C J τ (x, t, α) + 1

≤ C(u τ (x, t) + ε + 1) ≤ C, (39) where C is independent of (τ, x, t) and ε. Let y ∈ T ^d . Set

dX s = τ α ˜ s ds + √

2σdB s , X t = x, and Y s = X s − x + y, (40) then obviously dY s = ˜ α s ds + √

2σdB s , Y t = y. We have u τ (x, t) + ε ≥ τ E

h Z T t

L(X s , s, α ˜ s ) + hP _τ (s), φ(X s , s) ^| α ˜ s i + ˜ f τ (X s , s)

ds + g(X T ) i

, u _τ (y, t) ≤ τ E

h Z _T

t

L(Y _s , s, α ˜ _s ) + hφ(Y _s , s) ^| P _τ (s), α ˜ _s i + ˜ f _τ (Y _s , s)

ds + g(Y _T ) i . Therefore,

u τ (y, t) − u τ (x, t) ≤ ε + |(a)| + |(b)| + |(c)| + |(d)|, where (a), (b), (c), (d) are given by

(a) = τ E Z _T

t

L(Y s , s, α ˜ s ) − L(X s , s, α ˜ s ) ds, (b) = τ E

Z T t

φ(Y s , s) − φ(X s , s) |

P τ (s), α ˜ s i ds, (c) = τ E

g(Y T ) − g(X T ) , (d) = τ E

Z T t

f ˜ _τ (Y _s , s) − f(X ˜ _s , s) ds.

First, we have

|(a)| ≤ τ E Z T

t

L(Y _s , s, α ˜ _s ) − L(X _s , s, α ˜ _s ) ds

≤ C|y − x|

1 + E h Z T

t

|˜ α s | ² ds i

≤ C|y − x|,

as a consequence of assumption (A3) and (39). Then, using assumption (A6), (35), and (39), we obtain

|(b)| ≤ τ E Z _T

t

|φ(Y _s , s) − φ(X _s , s)||P _τ (s)|| α(s)| ˜ ds

≤ C|y − x| kP _τ k _L

(0,T;R

) E Z T

t

|˜ α(s)| ² ds ≤ C|y − x|.

By assumption (A8), |(c)| ≤ E

|g(Y _T ) − g(X T )|

≤ C|y − x|. Finally, since ˜ f τ is a Lipschitz function (by assumption (A7)),

|(d)| ≤ τ E Z T

t

f ˜ τ (Y s , s) − f ˜ τ (X s , s)

ds ≤ C|y − x|. (41) Letting ε → 0, we obtain that u τ (y, t) − u τ (x, t) ≤ C|y − x|. Exchanging x and y, we obtain that u _τ is Lipschitz continuous with modulus C and finally that k∇u _τ k _L

(Q, R

) ≤ C.

Step 3: kP _τ k _L

(0,T; R

) ≤ C.

By Lemma 3.1, m τ ∈ L ^∞ (0, T ; D ₁ ( T ^d )). We have that k∇u _τ k _L

(Q, R

) ≤ C and P τ = P(m τ , ∇u _τ ).

The bound on kP _τ k _L

(0,T ; R

) follows then from Lemma 5.2 and Remark 5.1.

Step 4: ku _τ k _W

(Q) ≤ C.

By assumption (A6), φ is bounded. We have proved that kP _τ k _L

(0,T ;R

) ≤ C and by Lemma 2.1, the Hamiltonian H is continuous. Therefore, kH(∇u _τ + φ ^| P τ )k _L

_(Q) ≤ C. By assumption (A5), kτ f ˜ τ k _L

_(Q) ≤ C. It follows that u τ , as the solution to the HJB equation (i)(MFGC τ ), is the solution to a parabolic equation with bounded coefficients. Thus, by Theorem A.4, ku _τ k _W

(Q) ≤ C. We also obtain with Lemma A.2 that ku _τ k _C

(Q) ≤ C and k∇u _τ k _C

(Q,R

) ≤ C.

Step 5: kv _τ k _L

(Q,R

) ≤ C, kD _x v τ k _L

(Q,R

) ≤ C.

We have proved that v τ = v(m τ , ∇u _τ ) and ku _τ k _W

(Q) ≤ C. The estimate follows directly with Lemma 5.2 and Remark 5.1.

Step 6: km _τ k _C

(Q) ≤ C.

The Fokker-Planck equation can be written in the form of a parabolic equation with coefficients in L ^p :

∂ _t m _τ − σ∆m _τ + τ hv _τ , ∇m _τ i + τ m _τ div(v _τ ) = 0,

since kD _x v _τ k _L

(Q, R

) ≤ C. Combining Theorem A.2 and lemma A.2, we get that km _τ k _C

(Q) ≤ C.

Step 7: kP _τ k _C

(0,T;R

) ≤ C.

We already know that ku _τ k _W

(Q) ≤ C, that km _τ k _C

(Q) ≤ C, and that m _τ ∈ L ^∞ (0, T ; D ₁ ( T ^d )).

Thus Lemma 5.3 applies and we obtain that kP _τ k _C

(0,T ;R

) ≤ C.

Step 8: ku _τ k _C

(Q) ≤ C.

We have proved that k∇u _τ k _C

(Q, R

) ≤ C and kP _τ k _C

(0,T ; R

) ≤ C. Moreover, we have assumed that φ is H¨ older continuous and know that H is H¨ older continuous on bounded sets. It follows that

kH(∇u _τ + φ ^| P _τ )k _C

(Q) ≤ C.

It follows from assumption (A7) that τ f ˜ _τ is H¨ older continuous. Since g ∈ C ^2+α ( T ^d ), we finally obtain that ku _τ k _C

(Q) ≤ C, by Theorem A.5.

Step 9: kv _τ k _C

(0,T ;R

) ≤ C and kD _x v _τ k _C

(0,T ;R

) ≤ C.

We have ku _τ k _C

(Q) ≤ C and km _τ k _C

(Q) ≤ C. Thus Lemma 5.4 applies and the announced estimates hold true.

Step 10: km _τ k _C

(Q) ≤ C.

A direct consequence of Step 9 is that m τ is the solution to a parabolic equation with H¨ older continuous coefficients. Therefore, by Theorem A.5, km _τ k _C

(Q) ≤ C, which concludes the proof of the proposition.

7 Application of the Leray-Schauder theorem

Proof of Theorem 3.1. Step 1: construction of T .

Let us define the mapping T : X × [0, 1] → X which is used for the application of the Leray- Schauder theorem. A difficulty is that the auxiliary mappings P and v are only defined for m ∈ L ^∞ (0, T ; D ₁ ( T ^d )). Therefore we need a kind of projection operator on this set. Note that R

T

1 dx = 1. We consider the mapping

ρ : m ∈ L ^∞ (Q) 7→ ρ(m) ∈ L ^∞ (0, T ; D ₁ ( T ^d )), defined by

ρ(m) = m ₊ (x, t) max(1, R

m + (y, t) dy) + 1 −

R m + (y, t) dy max(1, R

m + (y, t) dy) ,

where m ₊ (x, t) = max(0, m(x, t)). The well-posedness of the mapping can be easily checked, as well as the following properties:

• For all m ∈ L ^∞ (0, T ; D ₁ ( T ^d )), ρ(m) = m.

• The mapping ρ is Lipschitz continuous, from L ^∞ (Q) to L ^∞ (Q).

• For all α ∈ (0, 1), there exists a constant C > 0 such that if m ∈ C ^α (Q), then ρ(m) ∈ C ^α (Q) and kρ(m)k _C

(Q) ≤ Ckmk _C

(Q) .

For proving the first property, we suggest to consider separately the two cases: R

m + (y, t) dy < 1 and R

m ₊ (y, t) dy ≥ 1.

For a given (u, m, τ ) ∈ X × [0, 1], the pair (˜ u, m) = ˜ T (u, m, τ ) is defined as follows: ˜ u is the solution to

( −∂ _t u ˜ − σ∆˜ u + τ H(∇u + φ ^| P(ρ(m), ∇u)) = τ f (x, t, ρ(m(t))) (x, t) ∈ Q,

˜

u(x, T ) = τ g(x) x ∈ T ^d , and ˜ m is the solution to

( ∂ _t m ˜ − σ∆ ˜ m + τ div(v(ρ(m), ∇u)m) = 0 (x, t) ∈ Q,

˜

m(x, 0) = m 0 (x) x ∈ T ^d .