A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES

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1 Manuscript submitted to Website: AIMS Journals Volume 00, Number 0, Xxxx XXXX pp A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES DIOGO A. GOMES, GABRIEL E. PIRES, HÉCTOR SÁNCHEZ-MORGADO. Abstract. In this paper we establish a new class of a-priori estimates for stationary mean-field games which have a quasi-variational structure. In particular we prove W 1,2 estimates for the value function u and that the players distribution m satisfies m W 1,2. We discuss further results for powerlike nonlinearities and prove higher regularity if the space dimension is 2. In particular we also obtain in this last case W 2,p estimates for u. D. Gomes was partially supported by CAMGSD/IST through FCT Program POCTI - FEDER and by grants PTDC/MAT/114397/2009, and by the CMU and UTAustin Portugal programs, through grants UTAustin/MAT/0057/2008, and UTA-CMU/MAT/0007/ Introduction. Mean field games is a recent area of research started in the engineering community by Peter Caines and his co-workers [21], [20], and, independently, in the context of partial differential equations and viscosity solutions by Pierre Louis Lions and Jean Michel Lasry [22, 23, 24, 25]. Mean field games attempt to understand the limiting behavior of systems involving very large numbers of rational agents which play dynamic games under partial information and symmetry assumptions. Inspired by ideas in statistical physics, these authors introduced a class of models in which the individual players contributions are encoded in a mean field that contains only statistical properties about the ensemble. In addition to mean-field type interactions, the agents are assumed to be identical and indistinguishable. Literature on mean field games and its applications is growing fast, for a recent survey see [27] and reference therein. Applications of mean field games arise in growth theory in economics [26] or environmental policy [2], for instance. We also believe that in the future, mean field games will play an important rôle in economics and population models. This is due to the fact that in many economics applications or population models there is a very large number of indistinguishable agents which behave in a rational but non-cooperative way. Understanding the behaviour of such systems as the number of agents tends to infinity is one of the most fundamental questions in these problems. Date: December 15, DEPARTAMENTO DE MATEMÁTICA AND CAMGSD, IST, LISBOA, PORTUGAL. E- MAIL: DGOMES@MATH.IST.UTL.PT DEPARTAMENTO DE MATEMÁTICA AND CAMGSD, IST, LISBOA, PORTUGAL. E- MAIL: GPIRES@MATH.IST.UTL.PT INSTITUTO DE MATEMÁTICAS, UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO. HECTOR@MATEM.UNAM.MX 1

2 2 DIOGO A. GOMES 1, GABRIEL E. PIRES 2 3, HÉCTOR SÁNCHEZ-MORGADO. There is also a growing interest in numerical methods for mean-field problems [2], [1], [3]. One author and his collaborators [14] have also considered the discrete time, finite state problem, the continuous time finite state problem [16], [15]. Such models are relevant in many problems where a large number of agents are faced with the problem of choosing between a finite number of states. Several problems have been worked out in detail in [19], [17], including applications to growth theory and the quadratic case. Mean field games are frequently formulated as Hamilton-Jacobi type equation coupled with a transport equation, the adjoint of the linearization of the Hamilton- Jacobi equation. This class of problems, in the case the Hamilton-Jacobi equation does not depend on the solution of the transport equation was introduced in [9]. These methods were applied to study the vanishing viscosity problem [9], the differentiability of solutions of the infinity laplacian [6], Aubry-Mather theory in the non-convex setting [5] and systems of Hamilton-Jacobi equations and obstacle type problems [4], just to mention a few. Given H : R d R, the Evans-Aronsson problem [7], [8], consists in minimizing inf e H(x,Dφ) dx. φ The stochastic Evans-Aronsson problem, which is a generalization of the previous problem and was introduced in [30], consists in minimizing inf e ǫ φ+h(x,dφ) dx. (1) φ While studying this last problem two of the authors observed in [18] a new connection between mean-field games and a class of calculus of variations problems. In fact, he Euler-Lagrange for (1) can be written as { ǫ u+h(x,du) = lnm+ H ǫ m div(d p Hm) = 0, which is the canonical example of a stationary mean field game. Here the constant H is due to an additional normalization requirement for m, i.e. m = 1. In particular in [18] we obtained several a-priori estimates as well as the existence of smooth solutions when H(p,x) is quadratic in p or in dimension 2. In this paper we establish new a-priori estimates for a class of stationary meanfield games. We consider mean-field games of the form { u+h(x,du,m) = H (2) m div(d p Hm) = 0, where H : R d P( ) R (here P( ) is the set of Borel probability measures on )whichsatisfiessuitablehypothesis. Werequiremtobeaprobabilitymeasure absolutely continuous with respect to Lebesgue measure. The constant H is chosen so that this normalization condition holds (we do not assume, however, uniqueness of H or solutions). We do not assume as in [22] that the dependence of H in m is continuous with respect to the weak topology in P( ). For instance we allow local dependence on m (assuming the probability measure m has a density). More precisely our setting is the following: we assume that H is quasi-variational, this means that there exists H 0 (x,p) : R d R and g : R R, with g concave

3 A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES 3 increasing such that H(x,p,m) H 0 (x,p)+g(m(x)) C. (3) This means that H has a local component g(m(x)) but does not imply that H is a local operator, that is, H could depend on m through convolution operators, difference operators or other non-local dependence. Note that if g is concave increasing then g 1 is convex increasing. To explain the name quasi-variational, note that if we consider the variational problem inf F( φ+h 0 (x,dφ) H)dx, φ the corresponding Euler-Lagrange equation can be written as { u+h 0 (x,du) H = (F ) 1 (m) m div(d p H 0 (x,du)m) = 0. By defining g = (F ) 1, we get H(x,p,m) = H 0 (x,p) g(m). (4) Additionally, if we assume that F is convex increasing, then g is concave increasing. Thus we can think of quasi-variational mean-field games as perturbations of meanfield games with a variational structure. IfH isasin(4)thentheuniquenesstechniquebylionsandlasry[22]canbeused to establish uniqueness. Note that, in this paper we do not require monotonicity in H. We should also observe that the setting we work here is different from the one in [22] as we do not require continuity in m with respect to weak convergence. In fact this fails even for Hamiltonians of the form (4). In this paper, to simplify the exposition we consider the periodic setting. In this way the discussion of boundary conditions is avoided. We believe, however, that the same ideas can be applied in a variety of problems which include boundary conditions of various types. The hypothesis under which the our results hold are discussed in 2. Then in the next section we prove various a-priori bounds, and is structured as follows: 1. a-priori bounds in H, proposition 2; 2. W 1,2 integrability for u, Corollary 1; 3. regularity for m, m W 1,2, proposition 2; 4. higher integrability for m, if g(m) = m γ, Corollary 3; 5. D(lnm) L 2, proposition 5; 6. W 2,2 and W 2,p estimates for u, in dimension 2, propositions 6 and 7. The first two bounds, namely a-priori bounds for H and W 1,2 estimates for u immediate for variational problems in the form (4) as one can use the variational formulation to extract bounds on any possiblem minimizer. However for quasivariational problems we cannot use this technique and these are not obvious at all. These a-priori bounds, combined with additional regularity hypothesis on H can be used to prove a-priori bounds for higher norms of u and m, at least if d = 2. These in turn can be combined with continuation methods to establish existence of smooth solutions. This will be the subject of a future publication.

4 4 DIOGO A. GOMES 4, GABRIEL E. PIRES 5 6, HÉCTOR SÁNCHEZ-MORGADO. Another class of problems that may be possible to address with similar methods are time-dependent mean-field games. These take the form { u t +H(x,Du,m) = u m t div(d p Hm) = m, subjected to initial-terminal conditions u(x,t) = ψ(x), m(x,0) = m 0 (x), forsuitableψ andm 0. Thekeydifficultyinthevariationalsettingisthatfunctionals of the form T F( u t +H 0 (x,du) u)dx 0 are not coercive. Nevertheless several a-priori bounds can still be established. 2. Main assumptions. We now describe the main assumptions, in addition to quasi-variationality, that we will be working with. We suppose that H 0 satisfies p 2 C +CH 0 (x,p). (5) This is a natural coercivity condition. One could of course work with another coercivity conditions such as p β C +CH 0 (x,p), (6) which would give yield analogous results. Given (x,p,m) R d P( ) we define the Lagrangian as L(x,p,m) = H(x,p,m)+pD p H(x,p,m). (7) This is the natural definition of Lagrangian if one recalls that in classical mechanics the Lagrangian corresponding to a Hamiltonian H(x, p) is given by and the supremum is achieved for L(x,v) = supv p H(p,x), p v = D p H(p,x). Thus (7) corresponds to the Lagrangian function written in terms of the positionmomentum (x, p) coordinates rather than position-velocity (x, v) as it is customary. We suppose that L(x,p,m) ch 0 (x,p) C. (8) In view of (5) this is a coercivity hypothesis on L and implies L(x,p,m) c p 2 C. From (5) and (8) it follows that there exists C such that for every function ϕ : R d, we have L(x,ϕ(x),m)dm C, (9) note that this last fact can hold independently of (8), for instance if L(x,v,m) = v 2 2 +lnm then, because mlnm 1 we obtain (9). We assume further that H is uniformly convex in p, D 2 pph(x,p,m) γ > 0. (10)

5 A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES 5 We assume the following bound D p H(x,p,m) 2 C +CH 0 (x,p), (11) wich can can be relaxed to the more general bound D p H(x,p,m) 2 C +CH 0 (x,p,m)+cm δ, (12) for 0 < δ < 1 2 2, where 2 is the Sobolev conjugated exponent 1 2 = d. Set Ĥxx(x,p,m) = D xx (H(x,p,m)+g(m)), Ĥ px = D x (D p H(x,p,m)), then we suppose the following estimates hold: Ĥxx C +CH 0, and Ĥpx 2 C +CH 0 ; (13) As we will see in section 3, remark 3, this bound can sometimes be relaxed to Finally we suppose Ĥxx, Ĥpx 2 C +CH 0 C +CH 0 (x,p)+cm 2 /2. This growth condition could be replaced by H 0 (x,p) C +C p 2. (14) H 0 (x,p) C +C p β, if we were working under (6). As a canonical example we could consider H(x,p,m) = p 2 2 +V(x)+W(η m) mγ, where V and W are smooth functions, η is a smoothing kernel and 0 < γ < 1. In this case L(x,p,m) = p 2 2 V(x) W(η m)+mγ. All the previous hypothesis can easily be checked. This example represents a situation in which there is a local interaction term of the form m γ (x), which is in fact repulsive. This means that increasing local concentrations at a point x increase the cost for a reference player located at x. The non-local term W(η m) takes into account concentration effects in a neighborhood of the position of each player, and the potential V(x) encodes the different desirability of the various positions. 3. A-priori estimates. In this section we study several a-priori estimates for the solutions of the mean-field game equation (2). Whereas for variational mean-field games one has, from the variational principle, estimates for H and u W 1,2, see [18], these are not obvious, for general mean-field games, even with the quasi-variational structure. As the key objective of this section is to establish a-priori estimates we assume all solutions to be classical smooth solutions. Proposition 1. Let (u,m, H) solve (2), then L(x,Du,m)dm = H. (15)

6 6 DIOGO A. GOMES 7, GABRIEL E. PIRES 8 9, HÉCTOR SÁNCHEZ-MORGADO. Proof. H = u+h(x,du,m)dm T d = H(x,Du,m) D p H(x,Du,m)Dudm T d = L(x,Du,m)dm In Aubry-Mather theory (see [10, 11, 12, 13, 28, 29], for instance) one considers this first order case where H does not depend on m. If one looks for the unique value H for which the cell problem has a viscosity solution, then one has H(x,Du) = H L(x,Du)dm = H. A similar result also holds for the stochastic Mather problem [17]. Proposition 2. Let (u,m, H) solve (2). Assume (3), (5), and (9). Then there exists a constant C, independent of (u,m) such that H C. (16) Proof. By Proposition 1 and (9) H = L(x,Du,m)dm C. To prove the opposite inequality, observe that by the quasi-variationality hypothesis (3) we have H H 0 (x,du) g(m) C + u. Then, using (5), g(m) C H + u Because g is increasing, we have m g 1( C H + u ). Since g 1 is a convex function, by Jensen s inequality we have g 1( u C H ) ( ) dx g 1 u C H dx = g 1( C H ). It follows that and then 1 g 1( C H ) H C. Corollary 1. Let (u,m, H) solve (2). Assume (3), (5), and (9). Then H 0 (x,du)dx C, and so, Du L 2.

7 A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES 7 Proof. H 0 (x,du)dx u+h(x,du,m)+g(m)dx+c (17) H + g(m)dx+c (18) C, (19) where (17) comes from hypothesis (3), (18) from (2) and udx = 0, (19) from Jensen s inequality using the concavity of g, and estimate (16) for H. Proposition 3. Let (u,m, H) be a solution of (2). Assume (3), (5), and (8). Then H 0 (x,du)dm C. (20) Proof. By Proposition 1 and (8) H = L(x,Du,m)dm T d c H 0 (x,du)dm+c. This, together with (16) implies (20). The previous a-priori estimates are valid for determinist mean-field games of the form { H(x,Du,m) = H div(d p Hm) = 0, with similar proofs. However, in order to obtain further regularity the presence of the Laplacian is essential. Corollary 2. Let (u,m, H) be a solution of (2). Assume (3), (5), (8), and either (11) or (12). Then Proof. Multiply m W 1,2 C. (21) m div(d p Hm) = 0 by lnm and integrate by parts to obtain Td Dm 2 m dx = D p HDm 1 D p H 2 dm+ 1 2 T 2 d Td Dm 2 m dx. Assuming (11) we get 4 D m 2 dx D p H 2 dm C +C H 0 dm. (22) From Proposition 3 and m = 1 we get (21). Remark 1. In the proof of the last proposition we could replace the assumption (11) by the more general hypothesis (12).

8 8 DIOGO A. GOMES 10, GABRIEL E. PIRES 11 12, HÉCTOR SÁNCHEZ-MORGADO. Indeed, taking into account Hölder s inequality ( ) 1 ( )1 fg f p p g q q 1 ; p + 1 q = 1, let s consider so that This means that from which we deduce and get Since m = 1, we have m 1+δ f p = m ; g q = m 2, fg = m 1+δ. 1 p + 2 p 1 = 1+δ, 2 p 1 2δ = 1 p 2 2 ; 1 q = 2δ 2 2 ( m 1+δ ) 1 ( ) 2δ p m m ( m 2 ) 2δ 2 2 C ( D m 2 ) 2 δ 2 2. Thus instead of (22) we have 4 D m 2 dx D p H 2 dm C +C H 0 dm+c m 1+δ dx C +C ( H 0 dm+c D m 2 ) 2 δ 2 2 This inequality yields (21) if that is, 2 δ 2 2 < 1, 0 < δ < Remark 2. Under either conditions of Corollary 2, we have m W 1,2 from which it follows m L 2 and so m L 2 /2. Proposition 4. Let (u,m, H) be a solution of (2). Assume (3), (5), (8), (10), and (13). Then g (m) Dm 2 dx C, D 2 u 2 dm C (23)

9 A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES 9 Proof. Applying the Laplacian to first equation of (2) we get u+ĥx ix i (x,du,m)+2ĥp k x i (x,du,m)u xk x i +Tr(D 2 pph(x,du,m)(d 2 u) 2 ) +D p H(x,Du,m)D u div(g (m)dm) = 0. Integrating w.r.t. m g (m) Dm 2 dx T d + Tr(DppH(x,Du,m)(D 2 2 u) 2 )+Ĥx ix i (x,du,m)+2ĥp k x i (x,du,m)u xk x i dm = 0. By elementary estimates g (m) Dm 2 dx+ γ D 2 u 2 m Ĥxx(x,Du,m) +C Ĥpx(x,Du,m) 2 dm. T 2 d From (13) and Proposition 3 we have g (m) Dm 2 dx+ γ D 2 u 2 dm C +C H 0 (x,du)dm C. (24) T 2 d Hence g (m) Dm 2 dx and D 2 u 2 dm are bounded. Remark 3. In the previous proposition we could have replaced (13) by Ĥxx, Ĥxp 2 C +CH 0 +Cm 2 /2 1, and requiring in addition (11) or (12) as in Remark 1. In this situation (24) is replaced by g (m) Dm 2 dx+ γ D 2 u 2 dm C +C H 0 (x,du)dm+c m 2 /2. T 2 d This, combined with Corollary (2), would give a similar estimate, using Remark 2. Similarly, under the assumptions of Remark 1, using again Remark 2 we get the same estimate. Corollary 3. Let g(m) = m γ, with 0 < γ < 1, and (u,m, H) be a solution of (2). Assume (3), (5), (8), (10), and (13). Then 2 (γ+1) C (25) m 2 where 2 1 is the Sobolev conjugate exponent 2 = d. If in addition 2γ +1 2 (γ +1), 2 then Proof. Let f = m 1 2 (γ+1). Since γ < 1 we have 0 f 1. Du 4 dm C. (26) By Poincaré inequality f 2 ( 2 f) = ( f f ) 2 C Df 2.

10 10 DIOGO A. GOMES 13, GABRIEL E. PIRES 14 15, HÉCTOR SÁNCHEZ-MORGADO. Thus, using (23) f W 1,2 1+C Df 2 1+C g (m) Dm 2 C. By Sobolev inequality f 2 C f W 1,2 C, proving (25). Assume now then g(m) 2 m = By (3) and Proposition 2 2γ +1 p = 2 γ +1 2, f p C f p 2 C. (27) H 0 (x,du) C u+g(m). Then Du 4 C +CH 0 (x,du) 2 C +Cg(m) 2 +C D 2 u 2. Integrating w.r.t. m, and using (27), (23), we get estimate (26). Proposition 5. Let (u,m, H) be a solution of (2). Assume (3), (5), (9), and(11). Then Dm 2 m 2 C. (28) Proof. Multiply m div(d p Hm) = 0 by 1 m and integrate by parts to obtain Td Dm 2 m 2 C D p H(x,Du,m) 2 C +C H 0 (x,du). Corollary 1 gives the estimate. Remark 4. Since Dlnm = Dm m, the previous proposition states that Dlnm L2. In [18] using the variational structure we also have that lnm L 2. It is not clear, however, if the same holds in the generalized setting of the present paper. Remark 5. Note that (28) also holds under the conditions of Corollary 2 if (11) is replaced by (12). Proposition 6. Let (u,m, H) be a solution of (2). Suppose (3), (5), (8), (11) and (14) hold. Assume g(m) = m γ with 0 < γ < 1. Suppose d = 2. Then u W 2,2. Proof. Observe that ( u) 2 C g(m) 2 +C H 0 (x,du) 2 +C. T 2 T 2 T 2 Because m W 1,2 we have m 2γ C. T 2 As in [18] we have the interpolation inequality (using (14)) ( ) 3/2 ( ) 1/2 H 0 (x,du) 2 C C + H 0 (x,du) ( u) 2, T 2 T 2 T 2

11 A-PRIORI ESTIMATES FOR STATIONARY MEAN-FIELD GAMES 11 which then yields the result. Proposition 7. Let (u,m, H) be a solution of (2). Suppose (3), (5), (8), (11) and (14) hold. Assume g(m) = m γ with 0 < γ < 1. Suppose d = 2. Then u W 2,p, for all 1 < p <. Proof. We have u C + H 0 (x,du) + g(m). Since u W 2,2, by Proposition (6), Du L q for all q, by Sobolev s Theorem. From (14) it follows H 0 (x,du) L p, for all 1 < p <. Also, since m W 1,2, we have g(m) L p for all 1 < p <. Thus u L p for all 1 < p < from which it follows u W 2,p, by standard elliptic regularity. REFERENCES [1] Y. Achdou and I. C. Dolcetta. Mean field games: numerical methods. preprint, [2] Julien Salomon Aimée Lachapelle and Gabriel Turinici. Computation of mean field equilibria in economics. [3] F. Camilli, Y. Achdou, and I. C. Dolcetta. Mean field games: numerical methods for the planning problem. preprint, [4] F. Cagnetti, D. Gomes, and H. V. Tran. Adjoint methods for obstacle problems and weakly coupled systems of PDE. submitted. [5] F. Cagnetti, D. Gomes, and H. V. Tran. Aubry-Mather measures in the non convex setting. submitted. [6] Lawrence C. Evans and Charles K. Smart. Adjoint methods for the infinity Laplacian partial differential equation. Arch. Ration. Mech. Anal., 201(1):87 113, [7] Lawrence C. Evans. Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations, 17(2): , [8] Lawrence C. Evans. Further PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations, 35(4): , [9] L. C. Evans. Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration. Mech. Anal., 197(3): , [10] A. Fathi. Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 325(6): , [11] A. Fathi. Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math., 324(9): , [12] A. Fathi. Orbite hétéroclines et ensemble de Peierls. C. R. Acad. Sci. Paris Sér. I Math., 326: , [13] A. Fathi. Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math., 327: , [14] D. Gomes, J. Mohr, and R. R. Souza. Discrete time, finite state space mean field games. Journal de Mathématiques Pures et Appliquées, 93(2): , [15] D. Gomes, J. Mohr, and R. R. Souza. Continuous time, finite state space mean field games - ode and pde methods. preprint, [16] D. Gomes, J. Mohr, and R. R. Souza. Mean-field limit of a continuous time finite state game. preprint, [17] D. Gomes. A stochastic analogue of Aubry-Mather theory. Nonlinearity, 15(3): , [18] D. Gomes and H Sanchez-Morgado. On the stochastic Evans-Aronsson problem. preprint, [19] O. Gueant. Mean Field Games and Applications to Economics. Ph.D. Thesis. Université Paris Dauphine, Paris, [20] Minyi Huang, Peter E. Caines, and Roland P. Malhamé. Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized ǫ- Nash equilibria. IEEE Trans. Automat. Control, 52(9): , [21] Minyi Huang, Roland P. Malhamé, and Peter E. Caines. Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst., 6(3): , 2006.

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