On the long time behavior of the master equation in Mean Field Games

Transcription

1 On the long time behavior of the master equation in Mean Field Games P. Cardaliaguet (Paris-Dauphine) Joint work with A. Porretta (Roma Tor Vergata) Mean Field Games and Related Topics - 4 Rome - June 14-16, 2017 P. Cardaliaguet (Paris-Dauphine) Mean field games 1 / 36

2 The discounted MFG system Given a positive discount factor δ > 0, we consider the MFG system where (MFG δ) t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d u δ = u δ (t, x) and m δ = m δ (t, x) are the unknown, H = H(x, p) : T d R d R is a smooth, unif. convex in p, Hamiltonian, f, g : T d P(T d ) R are smooth" and monotone, (P(T d ) = the set of Borel probability measures on T d ) m 0 P(T d ) is a smooth positive density The MFG system has been introduced by Lasry-Lions and Huang-Caines-Malhamé to study optimal control problems with infinitely many controllers. P. Cardaliaguet (Paris-Dauphine) Mean field games 2 / 36

3 Interpretation. If (u δ, m δ ) solves the discounted MFG system, then u is the value function of a typical small player : [ + ] u(t, x) = inf E e δs L(X α s,α s)+f(x s, m δ (s)) ds 0 where dx s = α sds + 2dW s for s [t,+ ), X t = x and L is the Fenchel conjugate of H : L(x, α) := sup p R d α p H(x, p) and m δ is the distribution of the players when they play in an optimal way : m δ := L(Y s) with dy s = H p(y s, Du(s, Y s))ds + 2dW s, s [0,+ ), L(Y 0 ) = m 0. P. Cardaliaguet (Paris-Dauphine) Mean field games 3 / 36

4 The limit problem Let (u δ, m δ ) be the solution to (MFG δ) t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d Study the limit as δ 0 + of the pair (u δ, m δ ). Motivation : classical question in economics/game theory (players infinitely patient). In contrast with similar problem for HJ equation, forward-backward system. One expects that (u δ, m δ ) converges" to the solution of the ergodic MFG problem λ ū + H(x, Dū) = f(x, m) in T d (MFG erg) m div( mh p(x, Dū)) = 0 in T d m 0 in T d, m = 1 T d where now the unknown are λ, ū = ū(x) and m = m(x). P. Cardaliaguet (Paris-Dauphine) Mean field games 4 / 36

5 The limit problem Let (u δ, m δ ) be the solution to (MFG δ) t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d Study the limit as δ 0 + of the pair (u δ, m δ ). Motivation : classical question in economics/game theory (players infinitely patient). In contrast with similar problem for HJ equation, forward-backward system. One expects that (u δ, m δ ) converges" to the solution of the ergodic MFG problem λ ū + H(x, Dū) = f(x, m) in T d (MFG erg) m div( mh p(x, Dū)) = 0 in T d m 0 in T d, m = 1 T d where now the unknown are λ, ū = ū(x) and m = m(x). P. Cardaliaguet (Paris-Dauphine) Mean field games 4 / 36

6 Classical results for decoupled problems For the Fokker-Plank equation driven by a vector-field V : t m m div(mv(x)) = f(x) in (0, ) T d (exponential) convergence of m(t) to the ergodic measure is well-known. For HJ equations : Let v = v(t, x) and u δ = u δ (x) be the solution to and t v v + H(x, Dv) = f(x) in (0,+ ) T d, δu δ u δ + H(x, Du δ ) = 0 in T d u(0, ) = u 0 in T d Convergence of δu δ as δ 0 and v(t)/t as T + to the ergodic constant λ : Lions-Papanicolau-Varadhan,... (Weak-KAM theory) Limit of v(t) λt as T + to a corrector : Fathi, Roquejoffre, Fathi-Siconolfi, Barles-Souganidis,... Convergence of u δ λ/δ as δ 0 + to a corrector : Davini, Fathi, Iturriaga and Zavidovique,... P. Cardaliaguet (Paris-Dauphine) Mean field games 5 / 36

7 For MFG systems For the MFG time-dependent system, convergence of v T /T and m T are known : Lions (Cours in Collège de France) Gomes-Mohr-Souza (discrete setting) C.-Lasry-Lions-Porretta (viscous setting), C. (Hamilton-Jacobi) Turnpike property (Samuelson, Porretta-Zuazua, Trélat,...) Similar results for δu δ and m δ are not known, but expected. Long-time behavior of v(t, ) λt vs limit of u δ λ/δ not known so far. P. Cardaliaguet (Paris-Dauphine) Mean field games 6 / 36

8 General strategy of proof Let (u δ, m δ ) be the solution to t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d (MFG δ) t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d As (u δ, m δ ) = (u δ (t, x), m δ (t, x)), two possible limits : When δ 0 : difficult (no obvious limit, dependence in m 0 unclear), When t + : easier. Expected limit : the stationary discounted problem (MFG bar δ) { δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) in T d m δ div( m δ H p(x, Dū δ )) = 0 in T d P. Cardaliaguet (Paris-Dauphine) Mean field games 7 / 36

9 General strategy of proof (continued) Show that lim = λ and identification of the limit of ū δ λ/δ. δ 0 +δūδ Collect all the equations (MFG δ) into a single equation : for m 0 P(T d ), set U δ (x, m 0 ) := u δ (0, x) where (u δ, m δ ) solves (MFG δ) with m(0) = m 0. Then U δ solves the discounted master equation. get Lipschitz estimate on U δ by compactness arguments, prove that U δ λ/δ converges to a solution Ū of the ergodic master equation (as δ 0, up to subsequences). Put the previous steps together to derive the limit of u δ λ/δ. P. Cardaliaguet (Paris-Dauphine) Mean field games 8 / 36

10 Derivatives and assumptions Outline 1 Derivatives and assumptions 2 The classical uncoupled setting 3 Small discount behavior of ū δ 4 The discounted and ergodic master equations 5 Small discount behavior of u δ P. Cardaliaguet (Paris-Dauphine) Mean field games 9 / 36

11 Derivatives and assumptions Detour on derivatives in the space of measures We denote by P(T d ) the set of Borel probability measures on T d, endowed for the Monge-Kantorovich distance d 1 (m, m ) = sup φ(y) d(m m )(y), φ T d where the supremum is taken over all Lipschitz continuous maps φ : T d R with a Lipschitz constant bounded by 1. Given U : P(T d ) R, we consider 2 notions of derivatives : The directional derivative δu (m, y) δm (see, e.g., Mischler-Mouhot) The intrinsic derivative D mu(m, y) (see, e.g., Otto, Ambrosio-Gigli-Savaré, Lions) P. Cardaliaguet (Paris-Dauphine) Mean field games 10 / 36

12 Derivatives and assumptions Directional derivative A map U : P(T d ) R is C 1 if there exists a continuous map δu δm : P(Td ) T d R such that, for any m, m P(T d ), 1 U(m ) U(m) = 0 T d δu δm ((1 s)m + sm, y)d(m m)(y)ds. Note that δu is defined up to an additive constant. We adopt the normalization convention δm T d δu (m, y)dm(y) = 0. δm Intrinsic derivative If δu δm is of class C1 with respect to the second variable, the intrinsic derivative D mu : P(T d ) T d R d is defined by δu D mu(m, y) := D y (m, y) δm P. Cardaliaguet (Paris-Dauphine) Mean field games 11 / 36

13 Derivatives and assumptions For instance, if U(m) = D mu(m, y) = Dg(y). Remarks. T d g(x)dm(x), then δu δm The directional derivative is fruitful for computations. (m, y) = g(y) gdm while T d The intrinsic derivative encodes the variation of the map in P(T d ). For instance : D mu = Lip U P. Cardaliaguet (Paris-Dauphine) Mean field games 12 / 36

14 Derivatives and assumptions Standing assumptions H : T d R d R is smooth, with : C 1 I d D 2 pph(x, p) CI d for (x, p) T d R d. Moreover, there exists θ (0, 1) and C > 0 such that D xxh(x, p) C p 1+θ, D xph(x, p) C p θ, (x, p) T d R d. the maps f, g : T d P(T d ) R are monotone : for any m, m P(T d ), (f(x, m) f(x, m ))d(m m )(x) 0, (g(x, m) g(x, m ))d(m m )(x) 0 T d T d the maps f, g are C 1 in m : there exists α (0, 1) such that sup m P(T d ) and the same for g. ( f(, m) 3+α + δf(, m, ) δm ) (3+α,3+α) + Lip 3+α ( δf δm ) <. P. Cardaliaguet (Paris-Dauphine) Mean field games 13 / 36

15 Derivatives and assumptions Example. If f is of the form : where f(x, m) = Φ(z,(ρ m)(z))ρ(x z)dz, R d denotes the usual convolution product (in R d ), Φ = Φ(x, r) is a smooth map, nondecreasing w.r. to r, ρ : R d R is a smooth, even function with compact support. Then f satisfies our conditions with δf (x, m, z) = δm R d Φ ρ(y z k) (y,ρ m(y))ρ(x y)dy m k Z d P. Cardaliaguet (Paris-Dauphine) Mean field games 14 / 36

16 The classical uncoupled setting Outline 1 Derivatives and assumptions 2 The classical uncoupled setting 3 Small discount behavior of ū δ 4 The discounted and ergodic master equations 5 Small discount behavior of u δ P. Cardaliaguet (Paris-Dauphine) Mean field games 15 / 36

17 The classical uncoupled setting The classical ergodic theory (Lions-Papanicolau-Varadhan, Evans, Arisawa-Lions,...) For δ > 0, let u δ solve the uncoupled HJ equation δu δ u δ + H(x, Du δ ) = f(x) in T d. Then (δu δ ) is bounded (maximum principle), Du δ is bounded (growth condition on H or ellipticity) Thus, as δ 0 + and up to a subsequence, (δu δ ) and (u δ u δ (0)) converge to the ergodic constant λ and a corrector ū : λ ū + H(x, Dū) = f(x) in T d. Uniqueness of λ and of ū (up to constants) (strong maximum principle). P. Cardaliaguet (Paris-Dauphine) Mean field games 16 / 36

18 The classical uncoupled setting The small discount behavior For δ > 0, let u δ solve the uncoupled HJ equation δu δ u δ + H(x, Du δ ) = f(x) in T d. Then u δ δ 1 λ actually converges as δ 0 to the unique solution ū of the ergodic cell problem such that ū m = 0, where m solves T d Proved by λ ū + H(x, Dū) = f(x) in T d m div( mh p(x, Dū)) = 0 in T d, m 0, m = 1. T d Davini, Fathi, Iturriaga and Zavidovique for the first order problem, Mitake and Tran (see also Mitake and Tran - Ishii, Mitake and Tran) for the viscous case P. Cardaliaguet (Paris-Dauphine) Mean field games 17 / 36

19 Small discount behavior of ū δ Outline 1 Derivatives and assumptions 2 The classical uncoupled setting 3 Small discount behavior of ū δ 4 The discounted and ergodic master equations 5 Small discount behavior of u δ P. Cardaliaguet (Paris-Dauphine) Mean field games 18 / 36

20 Small discount behavior of ū δ The stationary discounted MFG system It takes the form (MFG bar δ) { δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) in T d m δ div( m δ H p(x, Dū δ )) = 0 in T d Proposition There exists δ 0 > 0 such that, if δ (0,δ 0 ), there is a unique solution (ū δ, m δ ) to (MFG bar δ). Moreover, for any δ (0,δ 0 ), δū δ λ + D(ū δ ū) L 2 + m δ m L 2 Cδ 1/2. for some constant C > 0, where ( λ, ū, m) solves the ergodic MFG system (MFG ergo) { λ ū + H(x, Dū) = f(x, m) in T d m div( mh p(x, Dū)) = 0 in T d P. Cardaliaguet (Paris-Dauphine) Mean field games 19 / 36

21 Small discount behavior of ū δ Link with the discounted MFG system The solution (ū δ, m δ ) of the (MFG bar δ) system can be obtained the limit of the solution (u δ, m δ ) of (MFG δ) t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d Theorem Under our standing assumptions, if δ (0,δ 0 ), then D(u δ (t) ū δ ) L Ce γt t 0 and m δ (t) m δ L Ce γt t 1, where γ,δ 0 > 0 and C > 0 are independent of m 0. P. Cardaliaguet (Paris-Dauphine) Mean field games 20 / 36

22 Small discount behavior of ū δ Towards a limit of ū δ λ/δ Plugg the ansatz : into the equation for (ū δ, m δ ) : One has : ū δ λ δ + ū + θ +δ v, m δ m +δ µ, δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) m δ div( m δ H p(x, Dū δ )) = 0 in T d in T d λ+δū +δ θ +δ 2 v (ū +δ v)+h(x, D(ū +δ v)) = f(x, m +δ µ) ( m +δ µ) div(( m +δ µ)h p(x, D(ū +δ v))) = 0 P. Cardaliaguet (Paris-Dauphine) Mean field games 21 / 36

23 Small discount behavior of ū δ Towards a limit of ū δ λ/δ Plugg the ansatz : into the equation for (ū δ, m δ ) : ū δ λ δ + ū + θ +δ v, We recognize the equation for (ū, m) : m δ m +δ µ, δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) m δ div( m δ H p(x, Dū δ )) = 0 in T d in T d λ+δū +δ θ +δ 2 v (ū +δ v)+h(x, D(ū +δ v)) = f(x, m +δ µ) ( m +δ µ) div(( m +δ µ)h p(x, D(ū +δ v))) = 0 P. Cardaliaguet (Paris-Dauphine) Mean field games 21 / 36

24 Small discount behavior of ū δ Towards a limit of ū δ λ/δ Plugg the ansatz : into the equation for (ū δ, m δ ) : Expending and simplifying : ū δ λ δ + ū + θ +δ v, m δ m +δ µ, δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) m δ div( m δ H p(x, Dū δ )) = 0 in T d in T d δū +δ θ +δ 2 v (δ v)+h p(x, Dū) (δ v) = δf (x, m)(δ µ) δm (δ µ) div((δ µ)h p(x, Dū)) div( mh pp(x, Dū)(δ v))) = 0 P. Cardaliaguet (Paris-Dauphine) Mean field games 21 / 36

25 Small discount behavior of ū δ Towards a limit of ū δ λ/δ Plugg the ansatz : into the equation for (ū δ, m δ ) : ū δ λ δ + ū + θ +δ v, m δ m +δ µ, δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) m δ div( m δ H p(x, Dū δ )) = 0 Dividing by δ and omitting the term of lower order : in T d in T d ū + θ v + H p(x, Dū) D v = δf (x, m)( µ) δm µ div( µh p(x, Dū)) div( mh pp(x, Dū) v)) = 0 P. Cardaliaguet (Paris-Dauphine) Mean field games 21 / 36

26 Small discount behavior of ū δ Proposition There exists a unique constant θ for which the following has a solution ( v, µ) : ū + θ v + H p(x, Dū).D v = δf (x, m)( µ) δm in Td µ div( µh p(x, Dū)) div( mh pp(x, Dū)D v) = 0 T d µ = T d v = 0 in T d We can identify the limit of ū δ λ/δ : Proposition Let ( λ, ū, m), (ū δ, m δ ) and ( θ, v, µ) be as above. Then lim λ δ 0 + ūδ δ ū θ + m δ m = 0. P. Cardaliaguet (Paris-Dauphine) Mean field games 22 / 36

27 Small discount behavior of ū δ Proposition There exists a unique constant θ for which the following has a solution ( v, µ) : ū + θ v + H p(x, Dū).D v = δf (x, m)( µ) δm in Td µ div( µh p(x, Dū)) div( mh pp(x, Dū)D v) = 0 T d µ = T d v = 0 in T d We can identify the limit of ū δ λ/δ : Proposition Let ( λ, ū, m), (ū δ, m δ ) and ( θ, v, µ) be as above. Then lim λ δ 0 + ūδ δ ū θ + m δ m = 0. P. Cardaliaguet (Paris-Dauphine) Mean field games 22 / 36

28 Small discount behavior of ū δ This shows that ū δ λ/δ converges as δ 0 + to ū + θ, where θ is the unique constant such that the system has a solution ( v, µ). ū + θ v + H p(x, Dū).D v = δf (x, m)( µ) δm in Td µ div( µh p(x, Dū)) div( mh pp(x, Dū)D v) = 0 T d µ = T d v = 0 In the uncoupled case (f = f(x)), we have because δf δm in T d (ū + θ) m = 0, T d = 0 and, if we multiply the equation for v by m and integrate, we get 0 = = = m(ū + θ v + H p(x, Dū).D v) T d m(ū + θ)+ v( m div( mh p(x, Dū))) T d T d m(ū + θ) T d So one recovers the condition of Davini, Fathi, Iturriaga and Zavidovique. P. Cardaliaguet (Paris-Dauphine) Mean field games 23 / 36

29 The discounted and ergodic master equations Outline 1 Derivatives and assumptions 2 The classical uncoupled setting 3 Small discount behavior of ū δ 4 The discounted and ergodic master equations 5 Small discount behavior of u δ P. Cardaliaguet (Paris-Dauphine) Mean field games 24 / 36

30 The discounted and ergodic master equations The discounted master equation In order to study the limit behavior of (u δ, m δ ), we use the discounted master equation : δu δ xu δ + H(x, D xu δ ) f(x, m) [ div y D mu δ] dm(y)+ D mu δ H p(y, D xu δ ) dm(y) = 0 T d T d in T d P(T d ) where U δ = U δ (x, m) : T d P(T d ) R. Theorem (C.-Delarue-Lasry-Lions, 2015) Under our assumptions, the discounted master equation has a unique classical solution U δ. Previous results in that direction : Lasry-Lions, Gangbo-Swiech, Chassagneux-Crisan-Delarue,... P. Cardaliaguet (Paris-Dauphine) Mean field games 25 / 36

31 The discounted and ergodic master equations Idea of proof : Let us set U δ (x, m 0 ) := u δ (0, x), where (u δ, m δ ) solves t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d (MFG δ) t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d Then one expects that U δ solves the master equation because : Taking the derivative in t = 0 : so that Td δu δ U δ (x, m δ (t)) = u δ (t, x) t 0. Td δu δ δm (x, m 0, y) t m δ (0, dy) = t u(0, x), δm (x, m 0, y)( m 0 + div(m 0 H p(y, Du δ (0))) = δu δ (0) u δ (0)+H(x, Du δ (0)) f(x, m 0 ). Integrating by parts gives the master equation. P. Cardaliaguet (Paris-Dauphine) Mean field games 26 / 36

32 The discounted and ergodic master equations The key Lipschitz estimate Let U δ be the solution of the discounted master equation δu δ xu δ + H(x, D xu δ ) f(x, m) [ div y D mu δ] dm(y)+ D mu δ H p(y, D xu δ ) dm(y) = 0 T d T d in T d P(T d ) Proposition There is a constant C, depending on the data only, such that D mu δ (, m, ) C. 2+α,1+α In particular, U δ (, ) is uniformly Lipschitz continuous. Difficulty : equation for U δ neither coercive nor elliptic in m. P. Cardaliaguet (Paris-Dauphine) Mean field games 27 / 36

33 The discounted and ergodic master equations Idea of proof Representation formulas. Fix m 0 P(T d ) a initial condition and (u δ, m δ ) the associated solution of the discounted MFG system : t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded. For any smooth map µ 0 with T d m 0 = 0, one can show that Td δu δ δm (x, m 0, y)µ 0 (y)dy = w(0, x), where (w,µ) is the unique solution to the linearized system t w +δw w + H p(x, Du δ ).Dw = δf δm (x, mδ (t))(µ(t)) t µ µ div(µh p(x, Du δ )) div(m δ H pp(x, Du δ )Dw) = 0 µ(0, ) = µ 0 in T d, w bounded. in (0,+ ) T d in (0,+ ) T d P. Cardaliaguet (Paris-Dauphine) Mean field games 28 / 36

34 The discounted and ergodic master equations Key step for the estimate : DmU δ (, m, ) 2+α,1+α C. Lemma There exist θ,δ 0 > 0 and a constant C > 0 such that, if δ (0,δ 0 ), then the solution (w,µ) to the linearized system with T d µ 0 = 0 satisfies Dw(t) L 2 C(1+t)e θt µ 0 L 2 t 0 and µ(t) L 2 C(1+t)e θt µ 0 L 2 t 1. As a consequence, for any α (0, 1), there is a constant C (independent of δ) such that sup w(t) C 2+α C µ 0 (C 2+α ). t 0 Relies on the monotonicity formula and exponential decay of some viscous transport equation. P. Cardaliaguet (Paris-Dauphine) Mean field games 29 / 36

35 The discounted and ergodic master equations The ergodic master equation As in the classical framework, we have (up to a subsequence) : δu δ converges to a constant λ, U δ U δ (, m) converges to a Lipschitz continuous map Ū. Proposition The constant λ and the limit Ū satisfy the master cell-problem : λ x Ū(x, m)+h(x, D x Ū(x, m)) div(d m Ū(x, m))dm T d + D m Ū(x, m).h p(x,, D x Ū(x, m))dm = f(x, m) in T d P(T d ) T d (in a weak sense). Moreover, if ( λ, ū, m) is the solution to the ergodic MFG system then λ = λ and D x Ū(x, m) = Dū(x) x T d. P. Cardaliaguet (Paris-Dauphine) Mean field games 30 / 36

36 The discounted and ergodic master equations The ergodic master equation As in the classical framework, we have (up to a subsequence) : δu δ converges to a constant λ, U δ U δ (, m) converges to a Lipschitz continuous map Ū. Proposition The constant λ and the limit Ū satisfy the master cell-problem : λ x Ū(x, m)+h(x, D x Ū(x, m)) div(d m Ū(x, m))dm T d + D m Ū(x, m).h p(x,, D x Ū(x, m))dm = f(x, m) in T d P(T d ) T d (in a weak sense). Moreover, if ( λ, ū, m) is the solution to the ergodic MFG system then λ = λ and D x Ū(x, m) = Dū(x) x T d. P. Cardaliaguet (Paris-Dauphine) Mean field games 30 / 36

37 The discounted and ergodic master equations Remarks. One also shows that Ū is unique up to a constant. So the limits, up to subsequences, of U δ U δ (, m) is determined only up to a constant. To fix this constant, we use the identification of the limit of ū δ λ/δ. P. Cardaliaguet (Paris-Dauphine) Mean field games 31 / 36

38 Small discount behavior of u δ Outline 1 Derivatives and assumptions 2 The classical uncoupled setting 3 Small discount behavior of ū δ 4 The discounted and ergodic master equations 5 Small discount behavior of u δ P. Cardaliaguet (Paris-Dauphine) Mean field games 32 / 36

39 Small discount behavior of u δ Link between U δ and ū δ Let U δ be the solution to the discounted master equation : δu δ xu δ + H(x, D xu δ ) div(d mu δ )dm T d + D mu δ.h p(x, D xu δ (x, m))dm = f(x, m) in T d P(T d ). T d and (ū δ, m δ ) be the solution to discounted stationary problem : (MFG bar δ) Then, by construction of U δ, { δū δ ū δ + H(x, Dū δ ) = f(x, m δ ) in T d m δ div( m δ H p(x, Dū δ )) = 0 in T d U δ (, m δ ) = ū δ, because (ū δ, m δ ) is a stationary solution of the discounted MFG system (MFG δ). P. Cardaliaguet (Paris-Dauphine) Mean field games 33 / 36

40 Small discount behavior of u δ The main result Let U δ be the solution to the discounted master equation : δu δ xu δ + H(x, D xu δ ) div(d mu δ )dm T d + D mu δ.h p(x, D xu δ (x, m))dm = f(x, m) in T d P(T d ). T d Theorem As δ 0 +, U δ λ/δ converges uniformly to the solution Ū to the master cell problem such that Ū(x, m) = ū(x)+ θ, where θ is the unique constant for which the following linearized ergodic problem has a solution ( v, µ) : ū + θ v + H p(x, Dū).D v = δf (x, m)( µ) δm µ div( µh p(x, Dū)) div( mh pp(x, Dū)D v) = 0 T d µ = T v d = 0 in Td in T d P. Cardaliaguet (Paris-Dauphine) Mean field games 34 / 36

41 Small discount behavior of u δ The small discount behavior of v δ Fix m 0 P(T d ) and let (u δ, m δ ) be the solution to the discounted MFG system : (MFG δ) t u δ +δu δ u δ + H(x, Du δ ) = f(x, m δ (t)) in (0,+ ) T d t m δ m δ div(m δ H p(x, Du δ )) = 0 in (0,+ ) T d m δ (0, ) = m 0 in T d, u δ bounded in (0,+ ) T d Corollary We have, for any t 0, lim δ 0 uδ (t, x) λ/δ = Ū(x, m(t)), uniformly with respect to x, where Ū is the solution of the ergodic cell problem given in the main Theorem and (m(t)) solves the McKean-Vlasov equation t m m div(mh p(x, DŪ(x, m))) = 0 in (0,+ ) Td, m(0) = m 0. P. Cardaliaguet (Paris-Dauphine) Mean field games 35 / 36

42 Small discount behavior of u δ Conclusion We have established the small discount behavior of the discounted MFG system/master equation. We also show in the paper the long time behavior of the time-dependent MFG system/master equation. Open problems : First order setting. Convergence in the non-monotone setting. P. Cardaliaguet (Paris-Dauphine) Mean field games 36 / 36