Mean field games and related models

Transcription

1 Mean field games and related models Fabio Camilli SBAI-Dipartimento di Scienze di Base e Applicate per l Ingegneria Facoltà di Ingegneria Civile ed Industriale Camilli@dmmm.uniroma1.it Web page: Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

2 Numerical schemes for the MFG systems In this lecture I will discuss some numerical schemes for MFG systems. I will consider Finite difference schemes for 2 nd -order MFG system Semi-Lagrangian schemes for 1 st -order MFG system Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

3 References for numerical approximation of MFG system Second order problems Y. Achdou and I. Capuzzo-Dolcetta, Mean field games: Numerical methods, SIAM J. Numer. Anal., 48 (2010), pp Y.Achdou, F.Camilli and I.Capuzzo Dolcetta, Mean field games: numerical methods for the planning problem, SIAM J. of Control & Optimization 50 (2012), Y.Achdou, F.Camilli e I.Capuzzo Dolcetta, Mean field games: convergence of a finite difference method, arxiv: , to appear on SIAM J. Numer. Anal. Lachapelle; Salomon; Turinici, Computation of mean field equilibria in economics. Math. Models Methods Appl. Sci. 20, 2010 Gueant, Mean field games equations with quadratic Hamiltonian: a specific approach, Math. Models Methods Appl. Sci. 22, 2012 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

4 First order problems Camilli, Silva, A semi-discrete in time approximation for a model first order-finite horizon mean field game problem, Network and Heterogeneous Media 7 (2012). Carlini, Silva, A fully-discrete Semi-Lagrangian scheme for a first order mean field game problem, preprint, Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

5 Finite difference approximation of MFG systems u ν u + H(x, u) = Φ[m] t ( m + ν m + div m H ) (x, u) t p u(x, 0) = Φ 0 [m(0)] m(t ) = m 0 in (0, T ] T d = 0 in [0, T ) T d ( ) where H(x, p) = sup α R d (p α L(x, γ)). Most of what follows holds with Neumann or Dirichlet conditions. Take d = 2 for simplicity. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

6 Notations Let T h be a uniform grid on the torus with mesh step h, and x ij be a generic point in T h Uniform time grid: t = T /N T, t n = n t The values of u and m at (x i,j, t n ) are approximated by u n i,j and m n i,j The discrete Laplace operator: ( h w) i,j = 1 h 2 (w i+1,j + w i 1,j + w i,j+1 + w i,j 1 4w i,j ) Right-sided finite difference formulas for w x 1 (x i,j ) and w x 2 (x i,j ) (D 1 w) i,j = w i+1,j w i,j, and (D 2 w) i,j = w i,j+1 w i,j h h The collection of the 4 first order finite difference formulas at x i,j } [D h w] i,j = {(D 1 w) i,j, (D 1 w) i 1,j, (D 2 w) i,j, (D 2 w) i,j 1 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

7 Approximation of HJB equation We consider the semi-implicit monotone scheme u n+1 i,j t u n i,j u t ν u + H(x, u) = Φ[m] ν( h u n+1 ) i,j + g(x i,j, [D h u n+1 ] i,j ) = (Φ h [m n ]) i,j where [D h u] i,j R 4 is the collection of the two first order finite difference formulas at x i,j for x u and for y u. g(x i,j, [D h u n+1 ] i,j ) ) =g (x i,j, (D 1 u n+1 ) i,j, (D 1 u n+1 ) i 1,j, (D 2 u n+1 ) i,j, (D 2 u n+1 ) i,j 1 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

8 Assumptions on the discrete Hamiltonian g Monotonicity: g is nonincreasing with respect to q1 and q 3 g is nondecreasing with respect to to q2 and q 4 Consistency: g (x, q 1, q 1, q 3, q 3 ) = H(x, q), x T, q = (q 1, q 3 ) R 2 Differentiability: g is of class C 1 Convexity (for uniqueness and stability): (q 1, q 2, q 3, q 4 ) g (x, q 1, q 2, q 3, q 4 ) is convex Well known discrete Hamiltonians g can be chosen. For example, if the Hamiltonian is of the form H(x, u) = ψ(x, u ), a possible choice is the upwind scheme: ( ) g(x, q 1, q 2, q 3, q 4 ) = ψ x, (q 1 )2 + (q +2 )2 + (q 3 )2 + (q +4 )2. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

9 Coupling Local operator: if Φ[m](x) = F(m(x)), take (Φ h [m]) i,j = F(m i,j ) If Φ is a nonlocal operator, choose a consistent discrete approximation. For example, if it is possible, (Φ h [m]) i,j = Φ[m h ](x i,j ), calling m h the piecewise constant function on T d taking the value m i,j in the square x x i,j h/2 For uniqueness and stability, the following assumption will be useful: (Φ h [m] Φ h [ m], m m) 2 0 Φ h [m] = Φ h [ m] Same thing for Φ 0,h Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

10 Approximation of the Fokker-Planck equation m t ( + ν m + div m H ) (x, v) p = 0. ( ) It is chosen so that each time step leads to a linear system for m with a matrix whose diagonal coefficients are negative whose off-diagonal coefficients are nonnegative in order to hopefully get a discrete maximum principle The argument for uniqueness should hold in the discrete case, so the discrete Hamiltonian g should be used for ( ) as well Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

11 Discretize ( div m H ) (x, u) w = m H (x, u) w T d p T d p by h 2 i,j T i,j(u, m)w i,j h 2 i,j m i,j qg(x i,j, [D h u] i,j ) [D h w] i,j Discrete version of div(mh p(x, u)): T i,j (u, m) = 1 h + g g m i,j (x i,j, [D h u] i,j ) m i 1,j (x i 1,j, [D h u] i 1,j ) q 1 q 1 g g +m i+1,j (x i+1,j, [D h u] i+1,j ) m i,j q 2 (x i,j, [D h u] i,j ) q 2 g g m i,j (x i,j, [D h u] i,j ) m i,j 1 (x i,j 1, [D h u] i,j 1 ) q 3 q 3 +m i,j+1 g q 4 (x i,j+1, [D h u] i,j+1 ) m i,j g q 4 (x i,j, [D h u] i,j ) Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

12 The scheme for the MFG system By the previous steps we get the semi-implicit monotone scheme u n+1 i,j t u n i,j ν( h u n+1 ) i,j + g(x i,j, [D h u n+1 ] i,j ) = (Φ h [m n ]) i,j m n+1 i,j t m n i,j + ν( h m n ) i,j + T i,j (u n+1, m n ) = 0 The operator m ν( h m) i,j + T i,j (u, m) is the adjoint of the linearized version of u ν( h u) i,j g(x i,j, [D h u] i,j ). The discrete MFG system has the same structure as the continuous one. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

13 Existence of a solution Define the set of discrete { probability densities K = (m i,j ) 0 i,j<n : h 2 } i,j m i,j = 1, m i,j 0. Proposition If g is of class C 1, and monotone w.r.t. q ( x, (q 1, q 2, q 3, q 4 )) C(1 + q1 + q 2 + q 3 + q 4 ). g x Φ h and Φ 0,h are continuous operators on K then the discrete problem has a solution such that m n K, n. Strategy of proof As in the continuous case we use a Brouwer fixed point theorem in K taking advantage of the structure of the system if Φ is a nonlocal smoothing operator, discrete Lipschitz bounds on Φ h [m] yield uniform estimates on the discrete Lipschitz norm of u. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

14 A key identity for uniqueness and stability Consider the perturbed system ũ n+1 i,j ũi,j n ν( h ũ n+1 ) i,j + g(x i,j, [D h ũ n+1 ] i,j ) = (Φ h [ m n ]) i,j + ai,j n t m n+1 i,j t m n i,j + ν( h m n ) i,j + T i,j (ũ n+1, m n ) = b n i,j Multiply the 2 discrete HJB equations by mi,j n m i,j n, sum on n, i, j, and subtract the results Multiply the 2 discrete FP equations by u n+1 i,j ũ n+1 i,j, sum on n, i, j, and subtract the results Add the 2 resulting identities Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

15 One gets where 1 ( m N T m N T, u N T ũ N ) T t ( m 0 m 0, u 0 ũ 0) t 2 N T 1 +E(m, u, ũ) + E( m, ũ, u) + (Φ h [m n ] Φ h [ m n ], m n m n ) 2 N T 1 = n=0 N T (a n, m n m n ( ) 2 + b n 1, u n ũ n) 2 n=0 E(m, u, ũ) = i,j,n n=1 ( m n 1 g(xi,j, [Dũ n ] i,j ) g(x i,j, [Du n ] i,j ) i,j g q (x i,j, [Du n ] i,j ) ([Dũ n ] i,j [Du n ] i,j ) ) Convexity of g E(m, u, ũ) 0 if m 0 If Φ h is monotone, (Φ h [m n ] Φ h [ m n ], m n m n ) 2 0 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

16 Uniqueness Proposition If g is convex Φ h is monotone, i.e. (Φ h [m] Φ h [ m], m m) 2 0 Φ h [m] = Φ h [ m] If u 0 = Φ 0,h [m 0 ] and ( Φ0,h [m] Φ 0,h [ m], m m ) 2 0 Φ 0,h[m] = Φ 0,h [ m] then the discrete version of the MFG system has a unique solution. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

17 A convergence result with local coupling Assumptions (1/3) ν > 0 d = 2 (only for example) periodicity (but everything would work with Neumann boundary conditions, or suitable Dirichlet conditions) u t=0 = u 0 and the data u 0 and m T are smooth 0 < m T m T (x) m T Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

18 Assumptions (2/3) The Hamiltonian is of the form H(x, u) = H(x) + u β where β > 1 and H is a smooth function The discrete Hamiltonian is of the form g(x i,j, [D h u] i,j ). The function g : T R 4 R is defined by ( β g(x, q) = H(x) + (q 1 )2 + (q + 2 )2 + (q 3 )2 + (q + 4 )2) 2 where r + = max(r, 0) and r = max( r, 0) Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

19 Assumptions (3/3) Local coupling: the cost term is Φ[m](x) = F(m(x)) where F is C 1 on R + There exist three constants c 1 > 0 and γ > 1 and c 2 0 s.t. mf(m) c 1 F(m) γ c 2 m There exist three positive constants c 3, η 1 and η 2 < 1 s.t. F (m) c 3 min(m η 1, m η 2 ) m Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

20 Convergence Assume that the MFG system of pdes has a unique smooth solution (u, m) s.t. m m > 0. Let u h (resp. m h ) be the piecewise trilinear function in C([0, T ] T ) obtained by interpolating the values ui,j n (resp mi,j n ) at the nodes of the space-time grid. Then ( ) u u h L β (0,T ;W 1,β (T d )) + m m h L 2 η 2 ((0,T ) T d ) = 0 lim h, t 0 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

21 Main steps of the proof 1 Obtain a priori bounds on the solution of the discrete problem, in particular on F(m h ) L γ ((0,T ) T d ) 2 Plug the solution of the system of pdes into the numerical scheme, take advantage of the stability of the scheme and prove that u u h L β ((0,T ) T d ) and m m h L 2 η 2 ((0,T ) T d ) converge to 0 3 To get the full convergence for u, one has to pass to the limit in the Bellman equation. To pass to the limit in the term F(m h ), use the equi-integrability of F (m h ) and Vitali s theorem Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

22 Remarks: A corresponding convergence result holds for the nonlocal case. Convergence results can be also proved for finite difference scheme for stationary MFG system To solve the discrete MFG system, marching in time is not possible due to the forward-backward structure. One has to solve the system for u and m as a whole. This leads to large systems of nonlinear equations. An efficient Newton method is described in Achdou, Perez, Iterative strategies for solving linearized discrete mean field games systems, Networks and heterogeneous media, 2012 Some problems: finite difference for first order MFG; Semi-Lagrangial schemes; convergence results for weak solution; error estimates. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

23 Some problems: Convergence results for weak solution of MFG system Error estimates Finite difference for first order MFG Semi-Lagrangian schemes Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

24 Numerical examples (by Y.Achdou) Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

25 A. Exit from a hall with obstacles Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 m t u t + ν u H(x, m, u) = F (m), in (0, T ) Ω ( ν m div m H ) (, m, u) = 0, in (0, T ) Ω p u n = m n = 0 on walls u = k, m = 0 at exits

26 A. Exit from a hall with obstacles m t u t + ν u H(x, m, u) = F (m), in (0, T ) Ω ( ν m div m H ) (, m, u) = 0, in (0, T ) Ω p u n = m n = 0 on walls u = k, m = 0 at exits Congestion H(x, m, p) = H(x) + p β (c 0 + c 1 m) γ with c 0 > 0, c 1 0, β > 1 and 0 γ < 4(β 1)/β. Existence and uniqueness was proven by P-L. Lions, and hold in the discrete case. Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51

27 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 A. Exit from a hall with obstacles T = 6 ν = u(t = T ) = 0 F (m) = m Hamiltonian H(x, m, p) = p 2 (1 + 4m) 1.5 exit exit

28 Evolution of the density Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / t= t= t= t= t= t=

29 Evolution of the density Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 t=0 t=0.15 t=0.30 t=0.6 t=3 t=4.5

30 Velocity Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 t=0.6

31 Who will reach the goal? Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 F 1 (m 1, m 2 ) = m 1 + m 2, F 2 (m 1, m 2 ) = 20 m 1 + m 2 Ω = (0, 1) 2 \ ([0.4, 0.6] [0, 0.55]) T = 4, ν = u 1 (t = T ) = u 2 (t = T ) = 0 Same Hamiltonian for the two populations: p 2 H 1 (x, m, p) = H 2 (x, m, p) = H(x) (1 + 4m) 1.3 H(x) = 10 1 x/ ([0.6,1] [0,0.2])

32 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 Evolution of the densities (bottom: the xenophobic pop.; top: the other pop.) t= t= t= t= t= t= t= t=

33 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 Evolution of the densities (bottom: the xenophobic pop.; top: the other pop.) t=0 t=0.2 t=0.4 t=0.8 t=1.2 t=2 t=2.8 t=0 t=0.2 t=0.4 t=0.8 t=1.2 t=2 t=2.8

34 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 Evolution of the velocities (bottom: the xenophobic pop.; top: the other pop.) t=0.2 t=0.4 t=1.2 t=2.4 t=0.2 t=0.4 t=1.2 t=2.4

35 Two populations cross each other F 1 (m 1, m 2 ) = m 1 + m 2, F 2 (m 1, m 2 ) = 20 m 1 + m 2. Ω = (0, 1) 2 T = 4, ν = u 1 (t = T ) = u 2 (t = T ) = 0 Hamiltonians: pop1 pop 2 p 2 H i (x, m, p) = H i (x) (1 + 4m) 1.3 H 1 (x) = 10 1 x/ ([0.7,1] [0,0.2]) H 2 (x) = 10 1 x/ ([0.7,1] [0.8,1]) The two populations pay the same cost for moving and have the same sensitivity to congestion effects, but they aim at different corners Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51

36 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 Finally, at time t = 0, the densities of the two populations are given by m 1 (x, t = 0) = 4 1 [0,0.2] [0.4,0.9] (x) m 2 (x, t = 0) = 4 1 [0,0.2] [0.1,0.6] (x) pop1 pop 2

37 Fabio Camilli ( Sapienza" Univ. di Roma) Y. Achdou Mean field games Minneapolis Roma, maggio / 51 Evolution of the densities (bottom: the xenophobic pop.; top: the other pop.) t=0 t=0.2 t=0.4 t=0.8 t=1.2 t=2 t=2.8 t=3.4 t=0 t=0.2 t=0.4 t=0.8 t=1.2 t=2 t=2.8 t=3.4

38 Evolution of the velocities (bottom: the xenophobic pop.; top: the other pop.) t=0.2 t=0.4 t=1.2 t=2.4 t=0.2 t=0.4 t=1.2 t=2.4 Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

39 Approximation of first order MFG system with quadratic Hamiltonian t v Dv(t, x) 2 = F(x, m(t)), t m + div ( Dv(x, t)m ) = 0, v(x, T ) = G(x, m(t )), m(0) = m 0. Interpretation: The HJ equation determine the expected value v(x, t) and the optimal control of the agents at position (x, t) taking into account the density m of the other players, while the continuity equation describes the evolution of the density along the characteristic flow given by the optimal control α(x, t) = Dv(x, t). The problem presents additional difficulties with respect to the second order problem, in particular in the interpretation of the second equation since the vector field Dv(x, t) is only L Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

40 A brief study of the first order system P 1 : the space of probability measures over R d endowed with the Kantorovich-Rubinstein distance: { } δ 1 (µ, ν) = sup ϕ(x)d[µ ν](x) ; ϕ : R d R R d is 1-Lipschitz. Assumptions: F and G are continuous over R d P 1 and satisfies F(, m) C 2 + G(, m) C 2 C. Monotonicity conditions: For m 1 m 2 R d [F(x, m 1 ) F(x, m 2 )] d[m 1 m 2 ](x) > 0, R d [G(x, m 1 ) G(x, m 2 )] d[m 1 m 2 ](x) > 0. m 0 L d with density still denoted by m 0 L and supp(m 0 ) B(0, C). Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

41 Definition (v, m) W 1, loc (Rd [0, T ]) C([0, T ], P 1 ) is a solution if the first equation is satisfied in the viscosity sense, while the second one in the sense of distributions. Theorem Under the previous assumptions the first order MFG system admits a unique solution. Moreover m(t) L d, t [0, T ] The proof technique is based on a fixed point argument for the existence and a monotonicity argument for the uniqueness (Lions, Collège de France, 2008 and P.Cardaliaguet, Lecture Notes). We will discuss in the following some technical issues concerning the second equation. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

42 Strategy for the approximation scheme Aim: to define a semi-lagrangian scheme and show the convergence of the scheme to the solution of the limit problem. 1 st step: Semi-discretization in time We will consider first a discrete-in-time semi-lagrangian scheme for the 1 st MFG system. The strategy for the approximation is Define a semi-lagrangian scheme for the Hamilton-Jacobi equation Define a "transport along the characteristics scheme for the continuity equation Put together the two previous schemes for the approximation of the MFG system. 2 nd step: Fully discretization in space The semi-discrete problem is discretized by using a finite element method for the space variable Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

43 Semi-discretization of the HJ equation The unique viscosity solution of t w(x, t) Dw(x, t) 2 = f (x, t), in R d (0, T ), w(x, T ) = g(x) in R d. is given by the value function w(x, t) := T [ ] 1 inf α A(t) t 2 α(s) 2 + f (X x,t [α](s), s) ds + g(x x,t [α](t )) where X x,t [α] is the solution of { Ẋ(s) = α(s) for s (t, T ), X(t) = x and α A(t) := L 2 ([t, T ]; R d ). Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

44 Let h > 0 (which is in fact t ), with Nh = T. Set w h (x, n) := inf α A n { h N 1 k=n [1 2 α k 2 + f (X k, kh) ] } + g(x N ). where { Xk+1 = X k hα k for k = n,..., N 1, X n = x, The the discrete value function w h satisfies Discrete Hamilton-Jacobi equation w h (x, n) := inf α R d for all n = 0,..., N 1, x R d (Falcone-Ferretti J.Comput.Phys. (2002)) {w h (x hα, n + 1) + 12 h α 2 } + hf (x, nh) Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

45 Lemma (Properties of the discrete value function) (i) w h (x, n) C(1 + T ) for x R d, n = 1,..., N 1. (ii) There exists C 1 > 0, independent of h, such that w h (x 1, n 1 ) w h (x 2, n 2 ) C 1 ( x 1 x 2 + h n 1 n 2 ). (iii) w h (, n) is semiconcave with constant independent on h. (iv) If ᾱ(x, n) is the discrete optimal control, then ᾱ(x, n) = Dw h (x hᾱ(x, n), n + 1) a.e. in x (for the continuous problem ᾱ(x, t) = Dw(x, t) a.e. in x). Theorem (Convergence result) w h converges locally uniformly in R d [0, T ] to w. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

46 Semi-discretization of the Continuity equation { t µ(x, t) + div( Dw(x, t)µ(x, t)) = 0, in R d (0, T ), µ(x, 0) = m 0 (x) in R d (recall that Dw(x, t) = α(x, t) is the optimal control). Consider the flow { Φ(x, t) = Dw(Φ(x, t), t) for t (0, T ) Φ(x, 0) = x. and define the push-forward µ of m 0 along the flow Φ by µ(t)(a) := [Φ(, t) m 0 ](A) = m 0 {x R d : Φ(x, t) A} for any Borel set A. Theorem µ(t) P 1 is the unique solution of the continuity equation. Moreover µ( ) L d, µ L and supp{µ( )} B(0, C). Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

47 We define a discrete flow {Φ h (x, k)} N k=0 by the implicit Euler scheme { Φh (x, k + 1) = Φ h (x, k) hdw h (Φ h (x, k + 1), k + 1), Φ h (x, 0) = x, (Dw h L, hence the Euler scheme is not a priori well posed) Proposition There exists a constant C 4 > 0 such that C 4 Φ h (x, k) Φ h (y, k) x y Thus, x Φ h (x, k) is invertible in Φ h (R d, k) and the inverse Υ h (, k) is Lipschitz continuous. The result is a consequence of the uniform semi-concavity of w h. It follows that we can define the the discrete push-forward µ h of m 0 along the flow Φ h by µ h (k)(a) := [Φ h (, k) m 0 ](A) = m 0 {x R d : Φ h (x, k) A} Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

48 Lemma There exists C > 0 (independent of h) such that: (i) For all k 1, k 2 {1,..., N}, we have that d(µ h (k 1 ), µ h (k 2 )) C 5 h k 1 k 2. (1) (ii) µ h (k) is absolutely continuous (with density still denoted by µ h (k)), has a support in B(0, C) and µ h (k) C. Proposition As h 0 we have that µ h µ in C([0, T ]; P 1 ). Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

49 The semi-discrete scheme for the first order MFG equation The semi-discrete scheme for the MFG system is given by Semi-discrete scheme t v(t, x) Dv(t, x) 2 = F(x, m(t)), t m + div ( Dv m ) = 0, m(0) = m 0, v(x, T ) = G(x, m(t )). { v h (x, k) = inf α R d vh (x hα, k + 1) h α 2} + hf(x, m h (k)), m h (k) = Φ h (, k) m 0 m h (0) = m 0, where Φ h is the discrete optimal flow v h (x, N) = G(x, m h (N)) Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

50 Theorem (Well posed-ness) Given h > 0 small enough, there exists a unique solution for the semidiscrete MFG system. Existence: Schauder s fixed point Theorem Uniqueness: Consequence of the monotonicity assumption. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51

51 Theorem As h 0, v h converges locally uniformly to v and m h converge to m in C([0, T ]; P 1 ). Idea of the proof: By the Ascoli theorem, m h converges in C([0, T ]; P 1 ), up to some subsequence, to some measure m. By the stability of viscosity solution, v h converges uniformly to the viscosity solution of { t w Dw(t, x) 2 = F (x, m(t)) w(x, T ) = G(x, m(t )), Since Dv h weakly converges to Dw, the discrete flow Φ h uniformly converges to the flow Φ defined by Dw. Hence m h (t) = Φ h (, t) m 0 converges to µ(t) = Φ(, t) m 0 Since m h m, then m(t) = Φ(, t) m 0 and, by uniqueness, w = v and m = m. Fabio Camilli ( Sapienza" Univ. di Roma) Mean field games Roma, maggio / 51