Second order mean field games with degenerate diffusion and local coupling

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1 Nonlinear Differ. Equ. Appl. 22 (2015), c 2015 Springer Basel /15/ publishe online April 26, 2015 DOI /s Nonlinear Differential Equations an Applications NoDEA Secon orer mean fiel games with egenerate iffusion an local coupling Pierre Caraliaguet, P. Jameson Graber, Alessio Porretta an Daniela Tonon Abstract. We analyze a (possibly egenerate) secon orer mean fiel games system of partial ifferential equations. The istinguishing features of the moel consiere are (1) that it is not uniformly parabolic, incluing the first orer case as a possibility, an (2) the coupling is a local operator on the ensity. As a result we look for weak, not smooth, solutions. Our main result is the existence an uniqueness of suitably efine weak solutions, which are characterize as minimizers of two optimal control problems. We also show that such solutions are stable with respect to the ata, so that in particular the egenerate case can be approximate by a uniformly parabolic (viscous) perturbation. Mathematics Subject Classification. 35K55, 49N Introuction This paper is evote to the analysis of secon orer mean fiel games systems with a local coupling. The general form of these systems is: (i) t φ A ij ij φ + H(x, Dφ) =f(x, m(t, x)) (ii) t m ij (A ij m) iv(md p H(x, Dφ)) = 0 (1) (iii) m(0) = m 0,φ(x, T )=φ T (x) where A : R R is symmetric an nonnegative, the Hamiltonian H : R R R is convex in the secon variable, the coupling f : R [0, + ) [0, + ) is increasing with respect to the secon variable, m 0 is a probability ensity an φ T : R R is a given function. The functions H an f, an the matrix A, coul as well epen on time, but since this oes not give any aitional ifficulty, we will avoi it just to simplify notations. Mean fiel game systems (MFG systems) have been introuce simultaneously by Lasry an Lions [17 19,21] an Huang et al. [15] to escribe Nash equilibria in ifferential games with infinitely many players. The first unknown

2 1288 P. Caraliaguet et al. NoDEA φ = φ(t, x) is the value function of an optimal control problem of a typical small player. In this control problem, the ynamics is given by the controlle stochastic ifferential equation X s = v s s +Σ(X s )B s, where (v s ) is the control, (B s ) is a Brownian motion an ΣΣ T = A. The cost is given by [ ] T E H (X s, v s )+f(x s,m(s, X s )) s + φ T (X T ) 0 where H is the Fenchel conjugate of H with respect to the secon variable. For each time t [0,T] the quantity m(t, x) enotes the ensity of population of small players at position x. In the control problem the term involving f formalizes the fact that the cost of the player epens on this ensity m.as φ is the value function of this control problem, the optimal control of a typical small player is formally given by the feeback (t, x) D p H(x, Dφ(t, x)). Hence the secon equation (1)-(ii) is the Kolmogorov equation of the process (X s ) when the small player plays in an optimal way. By the mean fiel approach, this equation also escribes the evolution of the whole population ensity as all players play in an optimal way. MFG systems with uniformly parabolic iffusions typically A ij ij φ = Δφ have been the object of several contributions, either by PDE methos (see, e.g., [7,12,13,17 19,21,23]) or by stochastic techniques (see, e.g., [2,15]): in this setting one often expects the solutions to be smooth, at least if the coupling is nonlocal an regularizing or if it has a small growth. This is still the case for a logarithmic coupling: f(m) =ln(m), as recently prove in [11]. The case of local couplings with an arbitrary growth has been iscusse in [7] for purely quaratic hamiltonians (i.e. H = Dφ 2 ), in which case solutions are prove to be smooth, an in [23] for general hamiltonians, by proving existence an uniqueness of weak solutions. Here we concentrate on egenerate parabolic equations. In this case the usual fixe point techniques use to prove the existence of solutions in the uniformly parabolic setting break own by lack of regularity. One then has to rely on convex optimization methos: this iea, which goes back to the analysis of some optimal transport problems (see [1,5]), has alreay been use to stuy first orer MFG systems (i.e., A 0): see [3,4,14]. However it was not clear in these papers wether the weak solution was stable with respect to viscous approximation, i.e., if we coul obtain weak solutions of the first orer MFG systems by passing to the limit in uniformly parabolic ones. This issue has partially motivate our stuy. In this paper we show the existence an uniqueness of a weak solution for the egenerate mean fiel game system (1) as well as the stability of solutions with respect to perturbation of the ata: this inclues of course stability by viscous approximation. Concerning existence an uniqueness of solutions, the paper improves the existing results in two irections. First we consier non uniformly parabolic

3 Vol. 22 (2015) Secon orer mean fiel games secon orer MFG systems, which have never been consiere before. The introuction of secon orer erivatives inuces several issues: in particular, in contrast with the first orer equations, we o not expect the function φ to be BV (as in [4,14]), which obliges us to be very careful about trace properties. Seconly an this is new even for first orer MFG systems we rop a restriction between the growth conition of H an the growth conition of f, restriction which was manatory in the previous papers: see [3,4]. To overcome the ifficulty, we provie new integral estimates for subsolutions of Hamilton Jacobi equations with unboune right-han sie (Theorems 3.1 an 3.3). We think that these results are of inepenent interest. With these estimates in han, the structure of proof for the existence an uniqueness follows roughly the lines alreay evelope in [3 5, 14]: basically it amounts to show that the MFG system can be viewe as an optimality conition for two convex problems, the first one being an optimal control of Hamilton Jacobi equation, the secon one an optimal control problem for the Fokker Planck equation (see Sect. 4 for etails). A byprouct of this approach is the stability of weak solutions with respect to the ata (Theorem 6.5), which can be obtaine by Γ convergence techniques. The paper is organize as follows. First we introuce the notation an assumptions neee throughout the paper (Sect. 2). Then (Sect. 3) we give our new estimates for subsolutions of Hamilton Jacobi equations with a superlinear growth in the graient variable an an unboune right-han sie. In Sect. 4, we introuce the two optimal control problems an show that they are in uality while in Sect. 5 we show that the optimal control problem for the Hamilton Jacobi equation has a relaxe solution. Section 6 is evote to the analysis of the MFG system (existence, uniqueness an characterization). In the last section we iscuss the stability of solutions. 2. Notations an assumptions Notations We enote by x, y the Eucliean scalar prouct of two vectors x, y R an by x the Eucliean norm of x. We use conventions on repeate inices: for instance, if a, b R, we often write a i b i for the scalar prouct a, b. More generally, if A an B are two square symmetric matrices of size, we write A ij B ij for Tr(AB). To avoi further ifficulties arising from bounary issues, we work in the flat imensional torus T = R \Z. We enote by P (T )thesetofborel probability measures over T. It is enowe with the Kantorovich-Rubinstein istance (which metricizes the weak-* convergence): 1 (m, m ):=sup φ(m m ) φ T where the supremum is taken over the set of Lipschitz continuous maps φ : T R which are Lipschitz continuous of constant 1. For k, n N an T > 0, we enote by C k ([0,T] T, R n ) the space of maps φ = φ(t, x) of class C k in time an space with values in R n.forp [1, ]

4 1290 P. Caraliaguet et al. NoDEA an T>0, we enote by L p (T )anl p ((0,T) T )thesetofp integrable maps over T an [0,T] T respectively. We often abbreviate L p (T )an L p ((0,T) T )intol p. We enote by f p the L p norm of a map f L p. Assumptions We now collect the assumptions on the coupling f, the Hamiltonian H an the initial an terminal conitions m 0 an φ T. These conitions are suppose to hol throughout the paper. (H1) (Conitions on the coupling) the coupling f : T [0, + ) R is continuous in both variables, increasing with respect to the secon variable m, an there exist q>1anc 1 such that 1 m q 1 C 1 f(x, m) C 1 m q 1 + C 1 m 0. (2) C 1 Moreover we ask the following normalization conition to hol: f(x, 0) = 0 x T. (3) We enote by p the conjugate of q: 1/p +1/q =1. (H2) (Conitions on the Hamiltonian) The Hamiltonian H : T R R is continuous in both variables, convex an ifferentiable in the secon variable, with D p H continuous in both variables, an has a superlinear growth in the graient variable: there exist r>1anc 2 > 0 such that 1 ξ r C 2 H(x, ξ) C 2 rc 2 r ξ r + C 2 (x, ξ) T R. (4) We note for later use that the Fenchel conjugate H of H with respect to the secon variable, i.e., H (x, ξ) :=sup ζ R ξ,ζ H(x, ζ), is continuous an satisfies similar inequalities 1 r ξ r C 2 H (x, ξ) C 2 C 2 r ξ r + C 2 (x, ξ) T R, (5) where r is the conjugate of r: 1 r + 1 r =1. (H3) (Conitions on A) there exists a Lipschitz continuous map Σ : T R D such that ΣΣ T = A :letc 3 be a constant such that Σ(x) Σ(y) C 3 x y x, y T. (6) Moreover we suppose that either r p or A is constant. (7) We recall that p is the conjugate of q. (H4) (Conitions on the initial an terminal ata) φ T : T R is of class C 2, while m 0 : T R is a C 1 positive ensity (namely m 0 > 0an m T 0 x = 1). Conition (3) is just a normalization conition, which we may assume without loss of generality. Inee, if all the conitions (H1)...(H4) but (3) hol, then one just nees to replace f(x, m) byf(x, m) f(x, 0) an H(x, p) by H(x, p) f(x, 0): the new H an f still satisfy the above conitions (H1)...(H4) with (3). Note that assumption (H1) oes not allow couplings which are not boune below, in contrast with [4,11].

5 Vol. 22 (2015) Secon orer mean fiel games Let us set m f(x, τ)τ if m 0 F (x, m) = 0 + otherwise Then F is continuous on T (0, + ), ifferentiable an strictly convex in m an satisfies 1 m q C 1 F (x, m) C 1 qc 1 q m q + C 1 m 0 (8) (changing the constant C 1 if necessary). Let F be the Fenchel conjugate of F with respect to the secon variable. Note that F (x, a) =0fora 0 because F (x, m) is nonnegative an equal to + for m<0. Moreover, 1 a p C 1 F (x, a) C 1 pc 1 p a p + C 1 a 0. (9) Some comments on the growth conition on the ata are now in orer. If A is constant (as, for instance, in the first orer case), we o not require any relation between the growth of H an the growth of F : both maps just nee to be superlinear. This is an extension compare to [3,4]. The price to pay is that we can no longer guarantee the bouneness of solutions. When the iffusion epens on the space variable, we nee the relation r p, where p is the growth of F : this is require by a regularization proceure in Lemma 5.3. Note that our growth conitions prevent couplings f with a slow growth; this is ue to the possibly egenerate character of the iffusion, such restrictions are not neee in the uniformly parabolic case (see [11 13, 23]). 3. Basic estimates on solutions of Hamilton Jacobi equations In this section we prove estimates in Lebesgue spaces for subsolutions of Hamilton Jacobi equations of the form { (i) t φ A ij (x) ij φ + H(x, Dφ) α(t, x) (10) (ii) φ(x, T ) φ T (x) in terms of Lebesgue norms of α an φ T. We assume that (4) an (6) hol, an (10) is unerstoo in the sense of istributions. This means that Dφ L r an, for any nonnegative test function ζ Cc ((0,T] T ), T ζ(t )φ T + φ t ζ + Dζ, ADφ + ζ( i A ij j φ + H(x, Dφ)) T 0 T T αζ. 0 T The estimates will be a consequence of the ivergence structure of secon orer terms.

6 1292 P. Caraliaguet et al. NoDEA Theorem 3.1. Assume that φ L r ((0,T); W 1,r (T )) is a nonnegative function satisfying, in istributional sense, { (i) t φ i (A ij (x) j φ)+c 0 Dφ r α(t, x) (11) (ii) φ(x, T ) φ T (x) for some nonnegative, boune Lipschitz matrix A ij,ansomer>1, c 0 > 0, α L p ((0,T) T ) an φ T L (T ). Then, there exists a constant C = C(p,, r, c 0,T, α Lp ((0,T ) T ), φ T Lη (T )) such that φ L ((0,T ),L η (T )) + φ L γ ((0,T ) T ) C where η = (r(p 1)+1) r(p 1) an γ = rp(1+) r(p 1) if p<1+ r p>1+ r. We note for later use that γ>r. an η = γ =+ if Proof. Up to a rescaling, we may assume that c 0 = 1. We first claim that, for any real function g W 1, (R) which is nonecreasing, an nonnegative in R +,wehave T G(φ(τ)) x + Dφ r g(φ) xt T τ T T αg(φ) xt + G(φ T ) x (12) τ T T for a.e. τ (0,T), where G(r) = r g(s) s. 0 There are several possible ways to justify (12), one is to use regularization. We first exten φ to (0,T +1] T by efining φ = φ T on [T,T + 1]. Then it still hols in the sense of istributions ) t φ i (Ãij (t, x) j φ + Dφ r χ (0,T ) α(t, x) (13) where Ãij(t, x) =A ij (x)χ (0,T ) (t) an α(t, x) =α(t, x)χ (0,T ) (t). Let ξ be a stanar convolution kernel in (t, x) efine on R +1, with support in the unit ball of R +1 an ξ ɛ (t, x) =ξ((t, x)/ɛ)/(ɛ) +1, ξ ɛ 0, R +1 ξ ɛ (t, x)tx =1, for all ɛ>0. Let φ ɛ = ξ ɛ φ an α ɛ = ξ ɛ α. Then φ ɛ,α ɛ are C an converge to φ, α in their respective Lebesgue spaces. Convolving ξ ɛ with (13) we obtain on (ɛ, T +1) T : t φ ɛ i (Ãij (t, x) j φ ɛ ) + ξ ɛ (D φ) r α ɛ (t, x)+r ɛ where D φ = Dφχ (0,T ) an we ) use the fact that Dφ ) Dφ r is convex, an where R ɛ = i (Ãij (t, x) j φ ɛ + ξ ɛ i (Ãij (t, x) j φ. Using the notation (cf. [9]) [ξ ɛ,c](f) :=ξ ɛ (cf) c(ξ ɛ f) we can rewrite R ɛ as R ɛ = [ξ ɛ, i à ij ]( j φ)+[ξ ɛ, Ãij i ]( j φ). Invoking [9, Lemma II.1], we have that R ɛ 0inL r, since Dφ L r an A ij is Lipschitz.

7 Vol. 22 (2015) Secon orer mean fiel games Multiplying by g(φ ɛ ) an integrating over [τ,t + ɛ] T,forτ (ɛ, T ), it follows T G(φ ɛ (τ))x G(φ ɛ (T + ɛ))x + A ij (x) j φ ɛ g (φ ɛ ) i φ ɛ xt T T τ T T +ɛ T +ɛ + ξ ɛ (D φ) r g(φ ɛ )xt g(φ ɛ )α ɛ (t, x)xt τ T τ T T +ɛ + g(φ ɛ )R ɛ xt. τ T Since T A τ T ij (x) j φ ɛ g (φ ɛ ) i φ ɛ xt 0, an since φ ɛ (T + ɛ) =ξ ɛ φ T,we obtain T +ɛ G(φ ɛ (τ))x G(ξ ɛ φ T )x + ξ ɛ (D φ) r g(φ ɛ )xt T T τ T T +ɛ T +ɛ g(φ ɛ )α ɛ (t, x)xt + g(φ ɛ )R ɛ xt. τ T τ T Since g is boune, while R ɛ an α ɛ converge in L r ((0,T) T ) an in L p ((0,T) T ) respectively, we can pass to the limit as ɛ goes to zero an for almost every τ we get (12). Now we procee with the esire estimate. First we observe that, up to replacing g(r) with g(r k), we can assume that φ is boune an that g may be any C 1 function. In particular, we take g(φ) =φ (σ 1)r for σ>1, obtaining 1 φ(τ) (σ 1)r+1 x + 1 T (σ 1)r +1 T σ r Dφ σ r xt τ T T αφ (σ 1)r 1 xt + φ (σ 1)r+1 T x. τ T (σ 1)r +1 T Let us enote henceforth by c possibly ifferent constants only epening on r, σ, an T. By arbitrariness of τ, the previous inequality implies φ σ (σ 1)r+1 σ c L ((0,T );L (σ 1)r+1 σ (T )) T 0 + Dφ σ r L r ((0,T ) T ) αφ (σ 1)r xt + c φ (σ 1)r+1 T x. T T On the other han, by interpolation we have (see e.g. [8, Proposition 3.1, Chapter 1]) v q L q ((0,T ) T ) c v ηr L ((0,T );L η (T )) Dv r L r ((0,T ) T ) where q = r +η (14) for any v L r ((0,T); W 1,r (T )) such that T v(t) x = 0 a.e. in (0,T). So we euce that

8 1294 P. Caraliaguet et al. NoDEA { T φ σ q L q ((0,T ) T ) c αφ (σ 1)r xt + 0 T T ( ) q + c φ(t) σ x t 0 T for η = (σ 1)r+1 σ an q = r η+. We choose σ such that σq =(σ 1)rp an therefore, by Höler inequality, we conclue φ σ q L q ((0,T ) T ) c r α 1+ L p ((0,T ) T ) φσ ( + c φ (σ 1)r+1 T x T q p (1+ r ) L q ((0,T ) T ) ) 1+ r φ (σ 1)r+1 T T 0 x } 1+ r T ( ) q + c φ(t) σ x t. T Since φ(t) x is estimate in terms of α T L 1 ((0,T ) T ) an φ T L 1 (T ),the last term can be absorbe into the left-han sie up to a constant C = C( α L1 ((0,T ) T ), φ T L1 (T )). Moreover, since p<1+ r,wehave q p (1+ r ) < q. Hence we en up with an estimate φ σ q L q ((0,T ) T ) C( α L p ((0,T ) T ), φ T L (σ 1)r+1 (T )). Computing the value of σ in terms of r an p we get qσ = rp(1 + ) r(p 1) an (σ 1)r +1= (r(p 1) + 1) r(p 1) so the first part of the Theorem is prove. Finally, we prove the L estimate by using a strategy which goes back to [24]. To this purpose, we replace φ with φ k an use (12) with g(s) =(s + ) r ; for any k φ T L (T ) we obtain T T [(φ k) + (τ)] σ+1 x+ D(φ k) σ + r xt α (φ k) σ + xt T τ T τ T with σ = r. Using as before the embeing (14) weget { T (φ k) σ + q L q ((0,T ) T ) c α (φ k) σ + xt 0 T T ( ) q + c (φ k) σ + x t 0 T { T c α (φ k) σ + xt 0 T T + c {x : φ(t) >k} q 1 where, using that σ = r,wehave q = r σ+1 σ 0 + = r } 1+ r } 1+ r T (φ k) σq + xt, ( +2 1 ). (15) r

9 Vol. 22 (2015) Secon orer mean fiel games Notice that {x : φ(t) > k} is uniformly small provie k is large, only epening on α L 1 an φ T L 1. Therefore, absorbing the last term in the left-han sie we euce { T } 1+ r (φ k) σ + q L q ((0,T ) T ) c α (φ k) σ + xt 0 T for some c = c( α L1 ((0,T ) T ), φ T L1 (T )). One can check that, since r>1, (15) implies q>1+ r an, in particular, 1 q + 1 p < 1. Thus, by Höler inequality we get (φ k) σ + q L q ((0,T ) T ) c (φ k) σ + 1+ r r L q ((0,T ) T ) α 1+ L p ((0,T ) T ) A k (1 1 q 1 p )(1+ r ), where A k := {(t, x) :φ(t, x) >k}. Since, for any h>kwe have T (φ k) σq + xt A h (h k) σq, 0 T we en up with the inequality A h 1 1 q (1+ r ) (h k) σq σ(1+ r ) (φ k) σ + q (1+ r ) L q ((0,T ) T ) which means that c α 1+ r L p ((0,T ) T ) A k (1 1 q 1 p )(1+ r ) A h C A k β (h k) δ h>k φ T L (T ) for some C = C( α Lp ((0,T ) T )), some δ>0an with β = (1 1 q 1 p )(1+ r ). 1 1 q (1+ r ) One can check that β>1since p>1+ r. Therefore, by a classical iteration lemma (see e.g. [24]), it follows that A k0 = 0 for some (explicit) k 0 > 0, which in particular implies the esire boun in terms of α Lp ((0,T ) T ) an φ T L (T ). Remark 3.2. The assumption that φ is nonnegative can be roppe an in this case the estimates are given on φ + ; inee, if φ satisfies (11), then φ + also oes. This can be seen in the previous proof by taking g = g(r + ), with g(0) = 0. Let us also stress that the Lipschitz continuity of the matrix A was only use to recover the estimate from the istributional formulation (namely, to be sure that φ is limit of solutions of smooth approximating problems). The constant C of the estimate, however, oes not epen on A in any way; in particular, the estimate will hol uniformly for any viscous approximation to possibly less regular matrices. As a corollary, we euce the following result for problem (10).

10 1296 P. Caraliaguet et al. NoDEA Theorem 3.3. Assume that (4) an (6) hol true an let φ satisfy (10) with α L p ((0,T) T ), φ T L (T ).Then,φ + satisfies the estimates of Theorem 3.1. In particular, if φ is boune below, we have φ L ((0,T ),L η (T )) + φ L γ ((0,T ) T ) C where η = (r(p 1)+1) r(p 1) an γ = rp(1+) r(p 1) if p<1+ r an η = γ =+ if p>1+ r, with a constant C epening on T,p,,r,C 2,C 3 [appearing in (4) an (6)] an on α L p ((0,T ) T ), φ T L η (T ) an φ L (T ). 4. Two optimization problems Mean fiel games systems with local coupling can be stuie as an optimality conition between two problems in uality. The first optimization problem is escribe as follows: let us enote by K 0 the set of maps φ C 2 ([0,T] T ) such that φ(t,x)=φ T (x) an efine, on K 0, the functional T A(φ) = 0 F (x, t φ(t, x) A ij ij φ + H(x, Dφ(t, x))) xt T φ(0,x)m 0 (x). T (16) Then the problem consists in optimizing inf A(φ). φ K 0 (17) For the secon optimization problem, let K 1 be the set of pairs (m, w) L 1 ((0,T) T ) L 1 ((0,T) T, R ) such that m(t, x) 0 a.e., with T m(t, x)x = 1 for a.e. t (0,T), an which satisfy in the sense of istributions the continuity equation t m ij (A ij (x)m)+iv(w) =0in(0,T) T, m(0) = m 0. (18) On the set K 1, let us efine the following functional T ( ) B(m, w) = m(t, x)h w(t, x) x, + F (x, m(t, x)) xt 0 T m(t, x) + φ T (x)m(t,x)x T where, for m(t, x) = 0, we impose that ( ) { m(t, x)h w(t, x) + if w(t, x) 0 x, = m(t, x) 0 if w(t, x) =0. Since H an F are boune from below an m 0 a.e., the first integral in B(m, w) is well efine in R {+ }. In orer to give a meaning to the last integral φ T T (x)m(t,x)x,we use the stanar fact that the measure m(t) is actually efine for any t (see Lemma 4.1 below). Inee, let us efine v(t, x) = w(t,x) m(t,x) if m(t, x) > 0an

11 Vol. 22 (2015) Secon orer mean fiel games v(t, x) = 0 otherwise. Thanks to the growth of H (see (5)), B(m, w) is infinite if m v r / L 1 (xt). Therefore, we can assume without loss of generality that m v r L 1 (xt), or, equivalently, that v L r (mxt). In this case equation (18) can be rewritten as a Kolmogorov equation t m ij (A ij (x)m) iv(mv) =0in(0,T) T, m(0) = m 0. (19) Lemma 4.1. There is a constant C, epening on w/m L r an on A, m such that 1 (m(t), m(t )) C t t r t, t [0, T]. For the sake of completeness, we give the proof here. Proof. We first exten the pair (m, w) to [ 1,T] T by efining m = m 0 on [ 1, 0] an w(s, x) =0for(s, x) ( 1, 0) T. Note that t m ij (Ãij(t, x)m)+iv(w) = 0 hols in the sense of istributions on ( 1,T) T, where Ãij(t, x) =A ij (x) ift (0,T)anÃij(t, x) = 0 otherwise. Let ξ be a stanar convolution kernel in (t, x), with a support containe in the unit ball of R +1 an ξ ɛ (t, x) =ξ((t, x)/ɛ)/(ɛ) +1, ξ ɛ 0, R +1 ξ ɛ (t, x)tx =1, for all ɛ>0. Let m ɛ = ξ ɛ m an w ɛ = ξ ɛ w. Then m ɛ,w ɛ are C an T m ɛ (t, x)x = 1 for all t ( 1+ɛ, T ɛ) anɛ>0 small enough. Convolving ξ ɛ with (18), we obtain t m ɛ ij (ξ ɛ (Ãij(t, x)m)) + iv(w ɛ )=0in( 1/2,T ɛ) T, with m ɛ ( 1/2,x)= R ξ ɛ (s, x y)m 0 (y)ys. T The equation can be rewritten as t m ɛ ij (Ãɛ ij(t, x)m ɛ )) iv(m ɛ v ɛ )=0 in( 1/2,T ɛ) T (20) where Ãɛ ij an v ɛ = wɛ m ɛ. Let us consier the following stochastic ifferential equations efine for all ɛ>0 { X ɛ t = v ɛ (t, Xt ɛ )t +Σ ɛ (Xt ɛ )Bt ɛ t [ 1/2,T ɛ], (21) ξɛ (Ãijm) = m ɛ X 1/2 ɛ = Zɛ 1/2 where Bt ɛ is a stanar -imensional Brownian motion over some probability space (Ω, A, P), Σ ɛ Σ T ɛ = Ãɛ, an the initial conition Z 1/2 ɛ L1 (T )is ranom, inepenent of (Bt ɛ ) an with law m ɛ ( 1/2, ). For all ɛ>0, the vector fiel v ɛ is continuous, uniformly Lipschitz continuous in space an boune. Therefore, there exists a unique solution to (21). Moreover, as a consequence of Ito s formula, we have that, if the ensity L(Z0)=ξ ɛ ɛ m 0, then m ɛ (t) =L(Xt ɛ ) solves (20) in the sense of istributions. Let 1 be the Kantorovich-Rubinstein istance on P (T ) an γ ɛ Π(m ɛ (t),m ɛ (s)) the law of the pair (Xt ɛ,xs) ɛ for 0 s < t T, where Π(m ɛ (t),m ɛ (s)) is the set of Borel probability measures μ on T T such

12 1298 P. Caraliaguet et al. NoDEA that μ(a T )=m ɛ (t, A) anμ(t A) =m ɛ (s, A) for any Borel set A T. We have 1 (m ɛ (t),m ɛ (s)) x y γ ɛ (x, y) =E[ Xt ɛ Xs ]. ɛ T T Moreover, t [ t ] E[ Xt ɛ Xs ] ɛ E[ v ɛ (τ,xτ ɛ ) τ]+e Σ ɛ (Xτ ɛ )B τ s s t ( [ t ]) 1/2 v ɛ (τ,x) m ɛ (τ,x)xτ + E Σ ɛ Σ ɛ (Xτ ɛ )τ s T s t v ɛ (τ,x) m ɛ (τ,x)xτ + A C t s 1 2. T s Recalling the efinition of v ɛ, we have that m ɛ v ɛ r = wɛ r m r 1 belongs to ɛ L 1 ([0,T] T ) for all ɛ>0. Inee, the function (m, w) w r m r 1 is convex an w r m r 1 belongs to L 1 ([0,T] T ). Thus T T ξ ɛ w r T ( ) 0 (ξ ɛ m) r 1 xτ ξ ɛ w r w r xτ 0 T m r 1 m r 1. 1 Therefore, using Höler inequality, ( t ) 1 1 (m ɛ (t),m ɛ (s)) v ɛ (τ,x) r r m ɛ (τ,x)xτ s T ( t ) 1 r m ɛ (τ,x)xτ + A C t s 1 2 s T 1 w r r t s 1 r + A t s 1 2. m r 1 1 Letting ɛ 0wehavem ɛ m in L 1 ([0,T] T ) an for a.e. τ [0,T], m ɛ (τ) m(τ) inl 1 (T ), moreover for a.e. 0 s<t T lim 1(m ɛ (t),m ɛ (s)) = 1 (m(t),m(s)). ɛ 0 Thus for a.e. 0 s<t T 1 (m(t),m(s)) C t s 1 r + A t s 1 2. The secon optimal control problem is the following: inf B(m, w). (22) (m,w) K 1

13 Vol. 22 (2015) Secon orer mean fiel games Lemma 4.2. We have inf A(φ) = min B(m, w). φ K 0 (m,w) K 1 Moreover, the minimum in the right-han sie is achieve by a unique pair (m, w) K 1 satisfying (m, w) L q ((0,T) T ) L r q r +q 1 ((0,T) T ). r Remark 4.3. Note that q r +q 1 > 1 because r > 1anq>1. Proof. The strategy of proof which is very close to the corresponing one in [3 5] consists in applying the Fenchel-Rockafellar uality theorem (cf. e.g., [10]). In orer to o so, it is better to reformulate the first optimization problem (17) in a more suitable form. Let E 0 = C 2 ([0,T] T )ane 1 = C 0 ([0,T] T, R) C 0 ([0,T] T, R ). We efine on E 0 the functional F(φ) = m 0 (x)φ(0,x)x + χ S (φ), T where χ S is the characteristic function of the set S = {φ E 0,φ(T, ) =φ T }, i.e., χ S (φ) =0ifφ S an + otherwise. For (a, b) E 1, we efine T G(a, b) = F (x, a(t, x)+h(x, b(t, x))) xt. 0 T The functional F is convex an lower semi-continuous on E 0 while G is convex an continuous on E 1.LetΛ:E 0 E 1 be the boune linear operator efine by Λ(φ) =( t φ + A ij ij φ, Dφ). We can observe that inf A(φ) = inf {F(φ)+G(Λ(φ))}. φ K 0 φ E 0 It is easy to verify that the qualification hypothesis, that ensures the stability of the above optimization problem, hols. Inee, there is a map φ such that F(φ) < + an such that G is continuous at Λ(φ): it is enough to take φ(t, x) =φ T (x). Therefore we can apply the Fenchel-Rockafellar uality theorem, which states that inf {F(φ)+G(Λ(φ))} = max { F (Λ (m, w)) G ( (m, w))} φ E 0 (m,w) E 1 where E 1 is the ual space of E 1, i.e., the set of vector value Raon measures (m, w) over[0,t] T with values in R R, E 0 is the ual space of E 0, Λ : E 1 E 0 is the ual operator of Λ an F an G are the convex conjugates of F an G respectively. By a irect computation we have F (Λ (m, w)) φ T (x)m(t,x) if t m ij (A ij m)+iv(w)=0,m(0)=m 0 = T + otherwise where the equation t m ij (A ij m)+iv(w) =0,m(0) = m 0 hols in the sense of istributions. Following [3], we have G (m, w) =+ if (m, w) / L 1 an, if (m, w) L 1,

14 1300 P. Caraliaguet et al. NoDEA where G (m, w) = T 0 T K (x, m(t, x),w(t, x))tx, F (x, m) mh (x, w m ) if m>0 K (x, m, w) = 0 if m =0,w =0 + otherwhise is the convex conjugate of Therefore K(x, a, b) =F (x, a + H(x, b)) (x, a, b) T R R. max { F (Λ (m, w)) G ( (m, w))} (m,w) E 1 { T = max F (x, m) mh (x, w 0 T m ) tx T φ T (x)m(t,x) x where the last maximum is taken over the L 1 maps (m, w) such that m 0 a.e. an t m ij (A ij m)+iv(w) =0,m(0) = m 0 hols in the sense of istributions. Since m T 0 = 1, it follows that m(t) = T 1 for any t [0,T]. Thus the pair (m, w) belongs to the set K 1 an the first part of the statement is prove. Take now an optimal (m, w) K 1 in the above system. Observe that ue to optimality we have w(t, x) = 0 for all (t, x) [0,T] T such that m(t, x) = 0. The growth conitions (4) an (8) imply T C F (x, m)+mh (x, w 0 T m ) tx + φ T (x)m(t,x) x T T ( 1 0 T C m q + m w ) r C(m +1) xt φ T. C m Therefore m L q.moreover,byhöler inequality, we also have T r q w r r q +q 1 = w r +q 1 0 T {m>0} ( r 1 ) q r m +q 1 w r r +q 1 q C m r 1 r q {m>0} so that w L r +q 1. Finally, a minimizer to (22) shoul be unique, because the set K 1 is convex an the maps F (x, ) anh (x, ) are strictly convex: thus m is unique an so is w m in {m >0}. Asw =0in{m =0}, uniqueness of w follows as well. }

15 Vol. 22 (2015) Secon orer mean fiel games Analysis of the optimal control of the HJ equation In general, we o not expect problem (17) to have a solution. In this section we exhibit a relaxation for (17) (Proposition 5.2) an show that this relaxe problem has at least one solution (Proposition 5.4) The relaxe problem Recall that the exponents η>1anγ>1 are efine in Theorem 3.3. Let K be the set of pairs (φ, α) L γ ((0,T) T ) L p ((0,T) T ) such that Dφ L r ((0,T) T ) an which satisfy in the sense of istributions t φ A ij (x) ij φ + H(x, Dφ) α, φ(t, ) φ T (23) (for the precise meaning of the inequality, see the beginning of Sect. 3). The following statement explains that φ has a trace in a weak sense. Lemma 5.1. Let (φ, α) K. Then, for any Lipschitz continuous map ζ : T R, the map t T ζ(x)φ(t, x)x has a BV representative on [0,T]. Moreover, if we enote by T ζ(x)φ(t +,x)x its right limit at t [0,T), then the map ζ T ζ(x)φ(t +,x)x is continuous in L η (T ). As a consequence, for any nonnegative C 1 map ϑ :[0,T] T R, one can write the integration by parts formula: for any 0 t 1 t 2 T, [ ] t2 t2 ϑφ + φ t ϑ + Dϑ, ADφ + ϑ( i A ij j φ + H(x, Dφ)) T t 1 t 1 T t2 αϑ. t 1 T Proof of Lemma 5.1. One easily checks that, for any Lipschitz continuous, nonnegative map ζ : T R, ζφ(t)+ Dζ, ADφ(t) + ζ( i A ij j φ + H(x, Dφ) α) 0, t T T hols in the sense of istributions. As the secon integral is in L 1 ((0,T)), the map t ζφ(t) is BV. If now ζ is Lipschitz continuous an changes sign, one T can write ζ = ζ + ζ an the map t ζφ(t) = ζ + φ(t) ζ φ(t) is T T T still BV. The continuity with respect to ζ comes from the L ((0,T),L η (T )) estimate on φ given in Theorem 3.3. We exten the functional A to K by setting T A(φ, α) = F (x, α(t, x)) xt φ(0,x)m 0 (x) x 0 T T The next proposition explains that the problem (φ, α) K. inf A(φ, α) (24) (φ,α) K

16 1302 P. Caraliaguet et al. NoDEA is the relaxe problem of (17). For this we first note that inf A(φ, α) = inf A(φ, α) (φ,α) K (φ,α) K, α 0 a.e. (25) because one can always replace α by α 0 since F (x, α) =0forα 0. Proposition 5.2. We have inf A(φ) = inf A(φ, α). φ K 0 (φ,α) K The proof requires the following inequality: Lemma 5.3. Let (φ, α) Kan (m, w) K 1. Assume that mh (, w/m) L 1 ((0,T) T ) an m L q ((0,T) T ).Then [ ] T T mφ + m (α + H (x, w ) T t t T m ) 0 (26) an [ ] t t mφ + m (α + H (x, w ) T 0 0 T m ) 0. Moreover, if equality hols in the inequality (26) for t =0, then w = md p H(x, Dφ) a.e. Proof. We first exten the pair (m, w) to[ 1,T +1] T by efining m = m 0 on [ 1, 0], m = m(t )on[t,t +1] an w(s, x) =0for(s, x) ( 1, 0) (T,T + 1) T. Note that t m ij (Ãij(t, x)m)+iv(w) =0on( 1,T+1) T, where à ij (t, x) =A ij (x) ift (0,T)anÃij(t, x) = 0 otherwise. Let ξ ɛ = ξ ɛ (t, x) be a smooth convolution kernel with support in B ɛ ; we smoothen the pair (m, w) in a stanar way into (m ɛ,w ɛ ). Then (m ɛ,w ɛ ) solves t m ɛ ij (Ãijm ɛ )+iv(w ɛ )= i R ɛ in ( 1/2,T +1/2) (27) in the sense of istributions, where R ɛ := [ξ ɛ, j à ij ](m)+[ξ ɛ, Ãij j ](m). (28) Here we use again the commutator notation (cf. [9]) [ξ ɛ,c](f) :=ξ ɛ (cf) c(ξ ɛ f). (29) Invoking [9, Lemma II.1], we have that R ɛ 0inL q, since m L q an à ij W 1,. Let us fix time t (0,T)atwhichφ(t + )=φ(t )=φ(t) inl γ (T) an m ɛ (t) converges to m(t). By the inequality satisfie by (φ, α), we have T t φ t m ɛ + i φ j (Ãijm ɛ )+m ɛ H(x, Dφ)+ T T m ɛ (t)φ(t) m ɛ (T )φ T T t αm ɛ. T By (27) wehave T T φ t m ɛ + i φ j (Ãijm ɛ )= T i φr ɛ + Dφ, w ɛ. T t t

17 Vol. 22 (2015) Secon orer mean fiel games On the other han, by convexity of H, T ( m ɛ H x, w ) ɛ t T m ɛ T t T w ɛ,dφ + m ɛ H(x, Dφ). (30) Collecting the above (in)equalities we obtain T ( m ɛ (t)φ(t) m ɛ (T )φ T + m ɛ (α + H x, w )) ɛ + j φr ɛ. T T t T m ɛ By assumption (7) which states that r p, an since Dφ L r,wehave j φr ɛ 0asɛ 0. Following the proof of Lemma 2.7 in [4] wehave T t T m ɛ H ( x, w ) T ( ɛ mh x, w ) m ɛ t T m as ɛ 0. The continuity of t m(t) inp (T ) given by Lemma 4.1 implies the convergence m ɛ (T )φ T m(t )φ T. T T Recalling that φ L γ ((0,T) T ), m L q ((0,T) T )anγ p, m ɛ φ strongly converges to mφ in L 1 ((0,T) T ) an, therefore, up to a subsequence, T m ɛ (t)φ(t) T m(t)φ(t) a.e. Hence m(t)φ(t) m(t )φ T + T T T t T m ( ( α + H x, w )). m We can argue similarly in the time interval [0,t] an obtain t ( ( m 0 φ(0) m(t)φ(t)+ m α + H x, w )). T T 0 T m Let us assume finally that the following equality hols: [ T mφ ] T 0 T ( ( + m α + H x, w )) 0 T m = 0. Then there is an equality in inequality (26) for almost all t.fixsuchat (0,T) an let { ( ( E σ (t) := (s, y),s [t, T ], m H y, w ) ) } + H(x, Dφ) w, Dφ + σ. m

18 1304 P. Caraliaguet et al. NoDEA If E σ (t) > 0, then for ɛ>0 small enough, the set E ɛ,σ (t):={(s, y), s [t, T ], m ɛ (H (y, w ɛ )+H(x, Dφ)) w ɛ,dφ +σ/2} m ɛ has a measure larger than E σ (t) /2. Coming back to inequality (30), we have T m ɛ H (x, w T ɛ ) w ɛ,dφ + m ɛ H(x, Dφ) E σ (t) σ/4 t T m ɛ t T Then inequality (26) becomes T ( ( m(t)φ(t) m(t )φ T + m α + H x, w )) E σ (t) σ/4, T T t T m which contraicts the fact that there is an equality in (26). So E σ (t) =0for any σ an for a.e. t, which shows that m(h (y, w m )+H(x, Dφ)) = w, Dφ a.e. Thus w = md p H(x, Dφ) hols a.e. in {m > 0} an, as w = 0 in {m =0}, a.e. in (0,T) T. Proof of Proposition 5.2. We follow the argument evelope by Graber in [14]. Inequality inf φ K0 A(φ) inf (φ,α) K A(φ, α) being obvious, let us check the reverse one. Let (φ, α) K. For any (m, w) K 1 with mh (, w m ) L1,we have, by Lemma 5.3, T A(φ, α) αm F (m) m 0 φ(0) 0 T T T mh (x, w 0 T m ) F (m) m(t )φ T = B(m, w) T Taking the sup with respect to (m, w) in the right-han sie we obtain thanks to Lemma 4.2: A(φ, α) inf B(m, w) = inf A(φ). (m,w) K 1 φ K Existence of a solution for the relaxe problem The next proposition explains the interest of consiering the relaxe problem (24) instea of the original one (17). Proposition 5.4. The relaxe problem (24) has at least one solution (φ, α) K which is boune below by a constant epening on φ T C 2,on A ij C 0 an on H(,Dφ T ). Proof. We start with the construction of a suitable minimizing sequence. Let ( φ n ) be a minimizing sequence for problem (17) an let us set α n (t, x) = max{0 ; t φn (t, x) A ij ij φn (t, x)+h(x, D φ n (t, x))}. (31) By Proposition 5.2 an the fact that F (x, α) =0ifα 0, the pair ( φ n,α n ) is also a minimizing sequence of (24). Let ψ be the unique viscosity solution to t ψ A ij (x) ij ψ + H(x, Dψ) =0, ψ(t, ) =φ T.

19 Vol. 22 (2015) Secon orer mean fiel games As φ T is C 2, ψ(t, x) φ T (x) C(T t), where the constant C epens on φ T C 2,on A ij C 0 an on H(,Dφ T ).Letφ n be the (continuous) viscosity solution to t φ n A ij (x) ij φ n + H(x, Dφ n )=α n, φ n (T, ) =φ T. (32) By comparison, φ n φ n ψ. AsH is convex, (32) hols in the sense of istributions (see [16]). Therefore the sequence (φ n,α n ) is still minimizing, with the following boun below for (φ n ): φ n (t, x) φ T (x) C(T t). (33) Step 1 We claim that (α n ) is boune in L p ((0,T) T ). For this, we integrate (32) against m 0 on (0,T) T φ n (0)m T i m 0 A ij j φ n +( j A ij )m 0 j φ n + m 0 H(x, Dφ n ) T y T T m 0 α n + φ T m 0. 0 T T As (1/C 0 ) m 0 C 0 for some C 0 > 0, Dm 0 < + an H is coercive, we get φ n (0)m T Dφ n r C 0 α n p + C. (34) T C 0 T On the other han, as (φ n ) is a minimizing sequence an F is coercive, 1 T C α n p p φ n (0)m 0 F (x, α n ) φ n (0)m 0 + C C. T 0 T T Aing the previous inequalities, we get 1 C α n p p + 1 T Dφ n r C 0 α n p + C, C 0 T so that (α n ) is boune in L p ((0,T) T ) while (Dφ n ) is boune in L r. Step 2 We show here that (φ n,α n ) has a limit. As (α n ) is boune in L p an (φ n ) is uniformly boune below thanks to (33), Theorem 3.3 implies that (φ n ) is boune in L γ. So we can assume with loss of generality that α n ᾱ in L p, φ n φ in L γ an Dφ n D φ in L r where, in view of the convexity of H, the pair ( φ, ᾱ) belongs to K. Step 3 We now prove that ( φ, ᾱ) is a minimizer. By weak lower semicontinuity arguments, we have T T lim inf F (x, α n ) F (x, ᾱ). n 0 T 0 T Let ζ n (t) = m T 0 φ n (t) an ζ(t) = m T 0 φ(t). Then (ζn ) converges weak* to ζ in L thanks to Theorem 3.3. As t ζ n(t)+ Dm 0,ADφ n (t) + m 0 ( i A ij j φ n + H(x, Dφ n ) α n ) 0, T

20 1306 P. Caraliaguet et al. NoDEA we also have by coercivity of H an thanks to the boun on (α n ): Letting n + : ζ n (0) Ct 1 p ζ n (t) t [0,T]. lim sup ζ n (0) Ct 1 p ζ(t) a.e. t [0,T], n so that lim sup n ζ n (0) m T 0 φ(0). Hence T T lim inf F (x, α n ) m 0 φ n (0) F (x, ᾱ) n 0 T T 0 T an ( φ, ᾱ) is a minimum. T m 0 φ(0) Remark 5.5. If r>2anp>1+/r, then by [6] the sequence (φ n ) built at the beginning of the proof is uniformly Höler continuous. Hence so is φ. 6. Existence an uniqueness of a solution for the MFG system In this section we show that the MFG system (1) has a unique weak solution an prove the stability of this solution with respect to the ata Definition of weak solutions The variational metho escribe above provies weak solutions for the MFG system. By a weak solution, we mean the following: Definition 6.1. We say that a pair (φ, m) L γ ((0,T) T ) L q ((0,T) T ) is a weak solution to (1) if (i) the following integrability conitions hol: Dφ L r,mh (,D p H(,Dφ)) L 1 an md p H(,Dφ)) L 1. (ii) Equation (1)-(i) hols in the following sense: inequality t φ i (A ij (x) j φ)+( i A ij ) j φ+h(x, Dφ) f(x, m) in (0,T) T, (35) with φ(t, ) φ T, hols in the sense of istributions, (iii) Equation (1)-(ii) hols: t m ij (A ij (x)m) iv(md p H(x, Dφ)))=0 in (0,T) T, m(0)=m 0 (36) in the sense of istributions, (iv) The following equality hols: T m(t, x)(f(x, m(t, x)) + H (x, D p H(x, Dφ)(t, x))) xt 0 T + m(t,x)φ T (x) m 0 (x)φ(0,x)x =0. (37) T Notice that last term in (37) is well efine ue to Lemma 5.1.

21 Vol. 22 (2015) Secon orer mean fiel games Our main result is the following existence an uniqueness theorem: Theorem 6.2. There exists a weak solution (φ, m) to the MFG system (1). Moreover this solution is unique in the following sense: if (φ, m) an (φ,m ) are two solutions, then m = m a.e. an φ = φ in {m >0}. Finally, there exists a solution which is boune below by a constant epening on φ T C 2,on A ij C 0 an on H(,Dφ T ). Remark 6.3. Uner the assumptions of Remark 5.5, i.e., if r>2anp> 1 + /r, theφ-component of the solution is locally Höler continuous Existence of a weak solution The first step towars the proof of Theorem 6.2 consists in showing a one-toone equivalence between solutions of the MFG system an the two optimization problems (22) an (24). Theorem 6.4. Let ( m, w) K 1 be a minimizer of (22) an ( φ, ᾱ) Kbe a minimizer of (24). Then( φ, m) is a weak solution of the mean fiel games system (1) an w = md p H(,D φ) while ᾱ = f(, m) a.e.. Conversely, any weak solution ( φ, m) of (1) is such that the pair ( m, md p H (,D φ)) is the minimizer of (22) while ( φ, f(, m)) is a minimizer of (24). Proof. Let ( m, w) K 1 be a minimizer of Problem (22) an ( φ, ᾱ) Kbe a minimizer of Problem (24). Due to Lemma 4.2 an Proposition 5.2, wehave T ( F (x, ᾱ)+f (x, m)+ mh x, w m ) xt 0 T + φ T m(t ) φ(0)m 0 x =0. T We show that ᾱ = f(x, m). Inee, by convexity of F, F (x, ᾱ(t, x)) + F (x, m(t, x)) ᾱ(t, x) m(t, x) 0, (38) hence T ( ᾱ(t, x) m(t, x)+ mh x, w m ) xt + φ T m(t ) φ(0)m 0 x 0. 0 T T Thanks to Lemma 5.3, the above inequality is in fact an equality, w = md p H(,D φ) a.e. an the equality hols almost everywhere in Eq. (38). Therefore, ᾱ(t, x) =f(x, m(t, x)) (39) almost everywhere an (37) hols: T ( f m + mh x, w m ) xt + φ T m(t ) φ(0)m 0 x =0. 0 T T In particular mh (,D p H(,D φ)) L 1. Moreover, since ( φ, ᾱ) Kan Eq. (39) hols, we have t φ Aij ij φ + H(x, D φ) f(x, m) in the sense of istributions an φ(t ) φ T. Furthermore, since ( φ, ᾱ) Kan w = md p H(,D φ), we have that md p H(,D φ) L 1 an (36) hols in the sense of istributions.

22 1308 P. Caraliaguet et al. NoDEA Therefore ( φ, m) is a solution in the sense of Definition 6.1. Suppose now that ( φ, m) is a weak solution of (1) as in Definition 6.1. Set w = md p H(,D φ) an ᾱ(t, x) =f(x, m(t, x)). By efinition of weak solution w, ᾱ L 1, m L q an φ L γ. Moreover, since m L q, the growth conition (8) implies that ᾱ L p. Therefore ( m, w) K 1 an ( φ, ᾱ) K. It remains to show that ( φ, ᾱ) minimizes A an ( m, w) minimizes B. Let ( φ, ᾱ ) K. By the convexity of F in the secon variable, we have T A( φ, ᾱ )= F (x, ᾱ (t, x))xt (0,x)m 0 (x)x 0 T T T 0 F (x, ᾱ(t, x)) + α F (x, ᾱ(t, x))(ᾱ (t, x) ᾱ(t, x))xt T φ (0,x)m 0 (x)x T T 0 F (x, ᾱ(t, x)) + m(t, x)(ᾱ (t, x) ᾱ(t, x))xt T φ (0,x)m 0 (x)x, T T A( φ, ᾱ)+ m(t, x)(ᾱ (t, x) ᾱ(t, x))xt 0 T + ( φ(0,x) φ (0,x))m 0 (x)x. T Due to Eq. (37) an Lemma 5.3 applie to ( φ, ᾱ )an( m, w) wehave T m(t, x)(ᾱ (t, x) ᾱ(t, x))xt + ( φ(0,x) φ (0,x))m 0 (x)x 0 T T T ( ) = m(t, x)ᾱ (t, x)+ m(t, x)h w(t, x) x, xt 0 T m(t, x) + φ T (x) m(t,x) φ (0,x)m 0 (x)x 0. T Hence, A( φ, ᾱ ) A( φ, ᾱ), an ( φ, ᾱ) is a minimizer of A. The argument for ( m, w) is similar. Let ( m, w ) minimize B. Then because F is convex in the secon variable, we have ( ) B( m, w )= φ T m (T )+ m H x, w T m + F (x, m ) ( ) φ T m (T )+ m H x, w T m +F (x, m)+f(x, m)( m m) ( ) = φ T m (T )+ m H x, w T m + F (x, m)+ᾱ( m m)

23 Vol. 22 (2015) Secon orer mean fiel games = B( m, w)+ φ T m (T ) m 0 φ(0) + T B( m, w). ( ) m H x, w m +ᾱ m Here we use Eq. (37) in the next to last line, an we applie Lemma 5.3 to ( φ, ᾱ) an( m, w ) in the last line. Therefore ( m, w) is a minimizer of B Uniqueness of the weak solution Proof of Theorem 6.2 (uniqueness part). Let ( φ, m) be a weak solution to (1). In view of Theorem 6.4, the pair ( m, md p H(,D φ)) is the minimizer of (22) while ( φ, f(, m)) is a solution of (24). In particular, m is unique because of the uniqueness of the solution of (22). Let now (φ 1, m) an (φ 2, m) be two weak solutions of (1), an set ᾱ = f(, m). Let φ = φ 1 φ 2. Assume for now that φ is a subsolution of (23) in the sense of istributions. Then ( φ, ᾱ) K, an so because φ(0)m0 φ 1 (0)m 0 we have that ( φ, ᾱ) is also a solution of (24). Inee, one euces from Lemma 5.3 that for a.e. t [0,T], ( φ, ᾱ) an (φ 1, ᾱ), are both minimizers of the problem T inf F (x, α) m(t)φ(t). (φ,α) K t T T In particular, m(t) φ(t) = m(t)φ T T 1 (t). As φ 1 φ, this implies that φ 1 = φ a.e. in { m >0}. The same argument, replacing φ 1 with φ 2, shows that φ 2 = φ a.e. in { m >0}, an uniqueness is prove. The main ifficulty is to show that φ = φ 1 φ 2 is inee a subsolution of (23) in the sense of istributions, i.e. T ζ(t )φ T + φ t ζ + Dζ, AD φ + ζ( i A ij j φ + H(x, D φ)) T 0 T T ᾱζ (40) 0 T for any nonnegative smooth map ζ with support in (0,T] T. Let ɛ>0. Introuce the following translation an extension of (φ k, ᾱ), k =1, 2: { φk (t +2ɛ, x) if t [ 2ɛ, T 2ɛ) φ k (t, x) = (41) φ T (x) if t [T 2ɛ, T +2ɛ] an { ᾱ(t +2ɛ, x) if t [ 2ɛ, T 2ɛ) α(t, x) = (42) λ if t [T 2ɛ, T +2ɛ] where λ = max x H(x, Dφ T (x)) + A ij (x) ij φ T (x). Then we have that t φk A ij ij φk + H(x, D φ k ) α (43) in the sense of istributions on ( 2ɛ, T +2ɛ) T.

24 1310 P. Caraliaguet et al. NoDEA For now we will fix a smooth vector fiel ψ on [0,T] T. Notice that t φk A ij ij φk + ψ D φ k α + H (x, ψ) (44) in the sense of istributions on ( 2ɛ, T +2ɛ) T. Let ξ 1 be a smooth convolution kernel in R +1 with support in the unit ball, with ξ 1 0an ξ 1 = 1. Then efine the stanar mollifier sequence ξ ɛ (t, x) =ɛ 1 ξ 1 ((t, x)/ɛ). Set φ ɛ k = ξɛ φ k an α ɛ = ξ ɛ α. By taking the convolution we have, in a pointwise sense, t φ ɛ k A ij ij φ ɛ k + ψ Dφ ɛ k α ɛ + ξ ɛ H (,ψ)+r k ɛ S k ɛ (45) on [0,T] T, where R k ɛ := [ξ ɛ,a ij j ]( i φk ), S k ɛ := [ξ ɛ,ψ](d φ k ). (46) Here we use the same commutator notation as in (29). Invoking [9, Lemma II.1], we have that Rɛ k an Sɛ k, k =1, 2 are smooth functions which converge to zero in L r, since A ij W 1, is given an ψ may also be chosen in W 1,. Define R ɛ := max{rɛ 1 Sɛ 1,Rɛ 2 Sɛ 2 }. This, too, converges to zero in L r. Moreover, for k =1, 2 t φ ɛ k A ij ij φ ɛ k + ψ Dφ ɛ k α ɛ + ξ ɛ H (,ψ)+r ɛ (47) hols in a pointwise sense, hence also in a viscosity sense. By stanar results, (47) hols also for φ ɛ := φ ɛ 1 φ ɛ 2 in a viscosity sense. The result of [16] implies that it also hols in the sense of istributions, that is, for any smooth map ζ with support in (0,T] T we have T ζ(t )φ ɛ (T )+ φ ɛ t ζ + Dζ, ADφ ɛ + ζ( i A ij j φ ɛ + Dφ ɛ ψ) T 0 T T ζ(α ɛ + ξ ɛ H (,ψ)+r ɛ ). (48) 0 T By construction, φ ɛ (T )=φ T for all ɛ>0. Observe that φ ɛ φ in L γ an Dφ ɛ D φ in L r, as these sequences are only slight aaptations of classical convolutions of φ an D φ. Finally, note that α ɛ ᾱ in L p, while ξ ɛ H (,ψ) H (,ψ) uniformly. Letting ɛ 0+, we are left with T ζ(t )φ T + φ t ζ + Dζ, AD φ + ζ( i A ij j φ + ψ D φ) T 0 T T ζ(ᾱ + H (,ψ)). (49) 0 T Now since ψ is an arbitrary smooth vector fiel, we may take a sequence that approximates p H(x, D φ) inl r. By the convexity of H(x, ) this yiels (40), as esire.

25 Vol. 22 (2015) Secon orer mean fiel games Stability We now consier the stability of solutions with respect to the ata: assume that (A n ), (H n ), (f n ), (m n 0 ) an (φ n T )convergetoa, H, f, m 0 an φ T respectively. We investigate the convergence of the corresponing solution (φ n,m n )tothe solution (φ, m). For this we assume that the A n, H n, f n, m n 0 an φ n T satisfy conitions (H1)...(H4) uniformly with respect to n. More precisely we suppose the following: (H1 ) The f n : T [0, + ) R are continuous in both variables, increasing with respect to the secon variable m, an there exist q>1anc 1 such that 1 m q 1 C 1 f n (x, m) C 1 m q 1 + C 1 m 0, n N. C 1 an f n (x, 0) = 0 x T, n N. Moreover we suppose that (f n ) converges locally uniformly to f which satisfies (H1). (H2 ) The Hamiltonians H n : T R R are continuous in both variables, convex an ifferentiable in the secon variable, with D p H n continuous in both variables, an have a superlinear growth in the graient variable: there exist r>1anc 2 > 0 such that 1 ξ r C 2 H n (x, ξ) C 2 rc 2 r ξ r + C 2 (x, ξ) T R, n N. We also suppose that the (H n ) converge locally uniformly to H which satisfies conition (H2). (H3 ) There exists a constant C 3 > 0 an continuous maps Σ n : T R D such that Σ n (Σ n ) T = A n an Σ n (x) Σ n (y) C 3 x y x, y T, n N with either r p or the A n are constant in space-time for all n N. We suppose that the (A n ) converge locally uniformly to A. (H4 ) The φ n T : T R converge to φ in C 2 (T ), while the m n 0 : T R converge to m 0 in C 1 an are uniformly boune below: there exists a constant C 4 > 0 such that m n 0 C 4 for all n. Note that the limit A, H, f, m 0 an φ T satisfies assumptions (H1)...(H4). Theorem 6.5. Let (φ n,m n ) be a weak solution of (1) associate with A n, H n, f n an with the initial an terminal conitions m n 0 an φ n T. Assume also that the sequence (φ n ) is uniformly boune below. Then (m n ) converges strongly to m in L q while φ n converges weakly an up to a subsequence to a map φ in L γ, where the pair ( φ, m) is a weak solution to (1). Note that the existence of a solution (φ n,m n ), such that φ n is boune by below, is ensure by Theorem 6.2.

26 1312 P. Caraliaguet et al. NoDEA The result is a simple consequence of Theorem 6.4 an of the Γ- convergence of the corresponing variational problems. Proof. Let us set w n = m n D p H n (,Dφ n )) an α n = f(,m n ). Let K n be the set of pairs (φ, α) L γ ((0,T) T ) L p ((0,T) T ) such that Dφ L r ((0,T) T ) an which satisfy in the sense of istributions t φ A n ij(x) ij φ + H n (x, Dφ) α, φ(t, ) φ n T. (50) We consier the functionals A n an B n efine on K n an K 1 respectively by T A n (φ, α) = (F n ) (x, α(t, x)) xt φ(0,x)m n 0 (x) x 0 T T an T ( ) B n (m, w) = m(t, x)(h n ) w(t, x) x, + F n (x, m(t, x)) xt 0 T m(t, x) + φ n T (x)m(t,x)x T where (H n ), F n an (F n ) are efine from H n an f n as usual. Accoring to the secon part of Theorem 6.4, the pair (m n,w n ) is a minimizer of problem (22) associate with B n, while the pair (φ n,α n ) is a minimizer of problem (24) associate with A n. We claim that lim sup inf n + K An inf A an lim sup inf B n inf B. (51) n K n + K 1 K 1 Let us explain the proof for A n, the proof for B n being similar. For this we consier the class K2 n of C 2 maps such that ψ(t, ) φ n T. Following Proposition 5.2, wehave T inf (F n ) (x, t ψ A n ij(x) ij ψ + H n (x, Dψ))xt ψ K2 n 0 T ψ(0)m n 0 x =inf T K An. n Let ψ C 2 with ψ(t, ) =φ T. As the map ψ n := ψ φ n T φ T belongs to K2 n,wehave T lim sup inf n + K An lim sup (F n ) (x, t ψ n A n ij ij ψ n n n + 0 T + H n (x, Dψ n ))xt ψ n (0)m n 0 x T T = lim sup (F n ) (x, t ψ A n ij ij ψ n + 0 T + H n (x, Dψ))xt ψ(0)m n 0 x + φ n T φ T T

27 Vol. 22 (2015) Secon orer mean fiel games T F (x, t ψ A ij ij ψ + H(x, Dψ))xt 0 T ψ(0)m 0 x. T This proves claim (51). Combining (51) with Lemma 4.2 an Proposition 5.2, we have lim sup inf n + K An inf A = inf B lim sup inf B n n K K 1 n + K 1 = lim inf( inf B n ) lim inf n + K 1 inf n + K An. n Since the left-han sie is not larger then the right-han sie, the above inequalities are in fact equalities, which shows that inf A = inf B = lim inf K K 1 n + K An = lim inf B n. (52) n n + K 1 We now show that the sequence (m n,w n ) converges to a minimizer of B. Using the estimates establishe for the proof of Proposition 5.2, we have m n L q + w n r q L r +q 1 + w n /m n L r C. (53) mn By Lemma 4.1, this implies that the maps t m n (t) are uniformly Höler continuous in P (T ). Hence there is a subsequence of (m n,w n ) (still enote in the same way) which converges weakly in L q L r q r +q 1 to some ( m, w) K1 an which is such that (m n (t)) converges to ( m(t)) in C 0 ([0,T],P(T )). Then, for any ξ L ((0,T) T, R )anσ L ((0,T) T )wehave T lim inf n + Bn (m n,w n ) lim inf ξ,w n H(x, ξ) n + 0 T + σm n F (x, σ)xt + m n (T )φ n T x T T ξ, w H(x, ξ)+σ m F (x, σ)xt 0 T + m(t )φ T x T Taking the supremum with respect to (ξ,σ) an using (52) then gives inf B = lim inf B n = lim inf K 1 n K 1 n + Bn (m n,w n ) B( m, w). Hence ( m, w) minimizes B. Moreover we note for later use that the above equalities entail that

28 1314 P. Caraliaguet et al. NoDEA T T F (x, m)xt = lim F n (x, m n )xt. 0 T n + 0 T In orer to prove the strong convergence of (m n,w n ), we rely on a stanar argument using Young measures. Let us efine the sequence of measures (μ n )on[0,t] T R by μ n (t, x, ρ) =txδ mn (t,x)(ρ). In view of (53), the sequence (μ n ) is tight an therefore converges weakly to some measure μ which can be isintegrate into μ(t, x, ρ) =μ t,x (ρ)tx. Then following Theorem 6.11 of [22], we have, as the F n are boune below: T 0 T F (x, m)xt = lim n + T 0 T T 0 R T F n (x, m n )xt F (x, ρ)μ t,x (ρ)xt, where, by convexity of F, T T F (x, ρ)μ t,x (ρ)xt F (x, ρμ t,x (ρ))xt 0 T R 0 T R T = F (x, m)xt. 0 T By strict convexity of F, this implies that μ t,x = δ m(t,x) a.e., which means that the sequence (m n ) actually strongly converges to m in L q (Proposition 6.12 of [22]). The strong convergence of the sequence (w n ) can be checke in the same way, by using the strict convexity of H. So we have checke that any converging subsequence of (m n,w n ) strongly converges to a minimizer ( m, w) of B on K 1. Since this minimizer is unique, the full sequence (m n,w n ) strongly converges to ( m, w). Next we turn to the convergence of (φ n,α n ). The growth conition (2) on f implies that the sequence (α n = f n (,m n )) converges in L p to ᾱ := f(, m). As (φ n ) is uniformly boune below, Theorem 3.3 implies that φ n L ((0,T ),L η (T )) + φ n Lγ ((0,T ) T ) C. The en of the proof follows closely the argument in Proposition 5.4: (Dφ n )is boune in L r, so that, up to a subsequence, (φ n ) converges weakly to some φ in L γ while Dφ n converges weakly to D φ in L r. Using the same argument as in Proposition 5.4, one can check that ( φ, ᾱ) belongs to K an is a minimizer of A over K. Theorem 6.4 then states that the pair ( φ, m) is a solution to the MFG problem (1). Acknowlegements We wish to thank the anonymous referee for useful comments. This work has been partially supporte by the Commission of the European Communities uner the 7th Framework Programme Marie Curie Initial Training Networks Project SADCO, FP7-PEOPLE-2010-ITN, No , by the French National Research Agency ANR-10-BLAN 0112 an ANR-12-BS an by the