Mean Field Games: numerical methods

Y. Achdou and I. Capuzzo-Dolcetta

November 19, 2008

Introduction

• Considera very large numberN of rational agents(or players) with

a limited global information on the game. Nash equilibria.

• In symmetric situations, Lasry and Lions have made a theory allowing

for passing to the limit whenN → ∞.

They have introduced, justified and analyzed models of mean field

games. Analogy with statistical physics and mechanics.

• New systems of PDEs with uniqueness, stability...

• For the justification, P-L. Lions has proposed a new mathematical

framework including

– differential calculus for functions of probability measures

– viscosity solutions for PDEs on probability measures.

1

Applications

• Finance

– Self-consistent models for volatility

– Price formation

• Economics

– Economical equilibrium with anticipation.

– Economic models for the behaviour of customers in front of a

technological breakthrough; ex: heat insulation of housesand

appartments.

2

Content

• Mean field games: some aspects of the theory of Lasry and Lions:

– The stationary case (infinite horizon)

– The non stationary case (finite horizon)

• Monotone numerical schemes

• Theoretical results on the numerical schemes

– Proof of existence in the non stationary case

– Uniqueness

– Proof of existence in the stationary case

• Numerical experiments.

• Open questions

3

Mean field games: some aspects of the theory of Lasry and Lions

ConsiderN players whose dynamics are

dXit =

√2νidW i

t − αidt, Xi0 = xi ∈ R

d, for 1 ≤ i ≤ N,

• (W 1t , . . . ,W

Nt ) independent Brownian motions inRd,

• νi > 0,

• The control of the playeri, i.e.αi is a bounded processassumed to beadapted to(W i

t ).

For simplicity, we assume thatXit ∈ T (the unit torus ofRd), and all the

functions used below will be periodic with period1 in every direction.

4

Cost function of the player i

J i(α1, . . . , αN ) = lim infT→∞

1

TE

(∫ T

0

(Li(Xi

t , αit) + F i(X1

t , . . . , XNt ))dt

),

given the initial conditionsX = (x1, . . . , xN ).

• F i is Lipschitz inTN ,

• Li is Lipschitz inxi uniformly in αi bounded,

•lim

|αi|→∞infxi

Li(xi, αi)

|αi| = +∞.

5

Nash equilibrium, given the initial conditions (x1, . . . , xN )

(α1, . . . , αN ) is a Nash point if

αi = Argminαi

J i(α1, . . . , αi−1, αi, αi+1, . . . , αN ), ∀i = 1, . . . , N.

Introduce the Hamiltonians

Hi(x, p) = supα∈Rd

(p · α− Li(x, α)

), i = 1, . . . , N, x ∈ T, p ∈ R

d.

Assume thatHi is C1.

6

Existence of a Nash equilibrium (Lasry-Lions)If there existsθ ∈ (0, 1) such that for|p| large,

infx∈T

„∂Hi

∂x· p +

θ

d(Hi)2 > 0

«> 0, ∀i, 1 ≤ i ≤ N,

then

1. ∃λ1, . . . , λN ∈ R, v1, . . . , vN ∈ C2(T), m1, . . . ,mN ∈ W 1,p(T) ∀p < ∞,

such that, for alli, 1 ≤ i ≤ N ,

8>>>>>>>><>>>>>>>>:

− νi∆vi + H

i(x,∇vi) + λi =

Z

TN−1

Fi(x1

, . . . , xi−1

, x, xi+1

, . . . , xN )

Y

j 6=i

mj(xj)dx

j,

− νi∆mi − div

„mi

∂Hi

∂p(x,∇vi)

«= 0,

Z

T

vidx = 0,

Z

T

midx = 1 and mi > 0 in T.

7

2. For any solutionλ1, . . . , λn, v1, . . . , vN ,m1, . . . ,mN of this system,

•αi(·) =

∂Hi

∂p(·,∇vi(·))

defines a feedback which is a Nash point for all(x1, . . . , xN ) ∈ TN .

•λi = J i(α1, . . . , αN )

= limT→∞

1

TE

(∫ T

0

(Li(Xi

t , αit(X

it)) + F i(X1

t , . . . , XNt ))dt

),

whereXit is the process corresponding to the feedbackαi.

8

Sketch of the proof:1) The existence for the system of PDEs follows from

• A priori estimate onλi: by the maximum principle,

|λi| ≤ ‖F i‖∞ + supx

|Hi(x, 0)|.

• A priori estimate on|∇vi|: using the technical hypothesis andBernstein method (i.e. write the equation satisfied by|∇vi|2 and use amaximum principle).

2) The existence of a Nash equilibria follows from a verification argumentusing Ito’s formula and the ergodicity formula

0 = limT→∞

0BBBB@

E

„1

T

Z T

0

Fi(X1

t , . . . , Xi−1t , X

it , X

i+1t , . . . , X

Nt )dt

«

−E

0@ 1

T

Z T

0

dt

Z

TN−1

Fi(x1

, . . . , xi−1

, Xit , x

i+1, . . . , x

N )Y

j 6=i

mj(xj)dx

j

1A

1CCCCA

.

9

Particular casesIf Li(x, α) = µi|α|2/2, then the change the unknown

φi = e−

vi

2νiµi

(∫

T

e−

vi

νiµi

)−1/2

and mi = φ2i

leads to the generalized Hartree equation: findφi > 0 s.t.

−2(νi)2µi∆φi+φi

∫

TN−1

F i(x1, . . . , xi−1, x, xi+1, . . . , xN )∏

j 6=i

φ2j(x

j)dxj = λiφi.

In particular, if

F i = 1/2∑

j 6=i

V (xi − xj) + V0(xi), νi = 1/2, µi = 1,

then one gets the Hartree equation: findφi > 0 s.t.

−1

2∆φi + φi

∑

j 6=i

V ∗ φ2j + V0φi = λiφi.

10

In a totally symmetric situation, one can find symmetrical solutions

Theorem: Symmetric caseIf the players are identical, i.e.νi = ν,

Hi = H and if

F i(x1, . . . , xN ) = F j(x1, . . . , xi−1, xj , xi+1, . . . , xj−1, xi, xj+1, . . . , xN ),

then there exists a solution of the previous system of PDE s.t.

λ1 = · · · = λN , v1 = · · · = vN , m1 = · · · = mN .

Uniqueness of the Nash equilibrium is false in general, evenin totallysymmetric situations.

Even symmetrical Nash equilibria may not be unique.

11

The limit N → ∞

We consider situations where there is a very large number of players and

each player chooses his optimal strategy in view of a limitedglobal

information on the game.

Assumptions

• The players are identical, thusνi = ν andHi = H.

• F i depends onxi andon the empirical density of the other players,namely

1

N − 1

∑

j 6=i

δxj .

12

• More precisely, assume that there exists an operatorV from the spaceof probability measures onT into a bounded set of Lipschitzfunctions onT such that

F i(x1, . . . , xN ) = V

1

N − 1

∑

j 6=i

δxj

(xi).

i.e.

Ji(α1

, . . . , αN ) = lim inf

T→∞

1

TE

0@

Z T

0

0@L(Xi

t , αit) + V

24 1

N − 1

X

j 6=i

δX

jt

35 (Xi

t)

1A dt

1A .

•

V [mn] converges uniformly onT to V [m] if mn weakly converges tom.

Typical examples forV include nonlocal smoothing operators.

13

TheoremUnder the above conditions, any Nash equilibriaλN1 , . . . , λN

N ,

vN1 , . . . , vN

N , mN1 , . . . ,mN

N satisfies:

• (λNi )i,N is bounded inR, (vN

i )i,N is relatively compact inC2(T), (mNi )i,N is

relatively compact inW 1,p(T), for any1 ≤ p < ∞.

•

supi,j

“|λN

i − λNj | + ‖vN

i − vNj ‖∞ + ‖mN

i − mNj ‖∞

”→ 0.

• Any converging subsequence(λN′

1 , vN′

1 , mN′

1 ) in R × C2(T) × W 1,p(T)

converges to(λ, v, m) such that8>>>>><>>>>>:

− ν∆v + H(x,∇v) + λ = V [m],

− ν∆m − div

„m

∂H

∂p(x,∇v)

«= 0,

Z

T

vdx = 0,

Z

T

mdx = 1, and m > 0 in T.

(∗)

14

Uniqueness for the mean field problem

By contrast with the preceding systems of PDEs, the system (*) is well

posed, under some conditions:

Theorem If V is strictly monotone, i.e.∫

T

(V [m1] − V [m2])(m1 −m2)dx ≤ 0 ⇒ m1 = m2,

then the solution of the mean field system (*) is unique.

15

Proof

Consider two solutions of (*):(λ1, v1,m1) and(λ2, v2,m2):

• multiply the first equation bym1 −m2

∫

T

(ν∇(v1 − v2) · ∇(m1 −m2) + (H(x,∇v1) −H(x,∇v2))(m1 −m2)

)dx

=

∫

T

(V [m1] − V [m2])(m1 −m2)dx

• multiply the second equation byv1 − v2

0 =

∫

T

ν∇(v1 − v2) · ∇(m1 −m2)dx

+

∫

T

(m1

∂H

∂p(x,∇v1) −m2

∂H

∂p(x,∇v2)

)· ∇(v1 − v2)dx.

16

• subtract:

0 =

8>>>>><>>>>>:

Z

T

m1

„H(x,∇v1) − H(x,∇v2) −

∂H

∂p(x,∇v1) · ∇(v1 − v2)

«dx

+

Z

T

m2

„H(x,∇v2) − H(x,∇v1) −

∂H

∂p(x,∇v2) · ∇(v2 − v1)

«dx

+

Z

T

(V [m1] − V [m2])(m1 − m2)dx

SinceH is convex andV is monotone, the 3 terms vanish.

The strict monotonicity ofV implies thatm1 = m2.

The identitiesv1 = v2 andλ1 = λ2 come from the uniqueness for the

ergodic problem:

−ν∆v +H(x,∇v) + λ = f.

17

Finite horizon Nash equilibrium with N playersCost:

E

∫ T

0

Li(Xi

s, αis) + V

1

N − 1

∑

j 6=i

δXjs

ds+ V0

1

N − 1

∑

j 6=i

δXj

T

Lasry and Lions show (formally only in most cases) that the value functions

atT − t: vNi (x1, . . . , xN , t) satisfy

∂vNi

∂t− ν

∑

j

∆xjvNi

+H(xi,∇xivNi ) +

∑

j 6=i

∂H

∂p(xj ,∇xjvN

i ) · ∇xjvNi

= V

1

N − 1

∑

j 6=i

δxj

(xi).

18

and

vi|t=0 = V0

1

N − 1

∑

j 6=i

δxj

.

The densitymN (x1, . . . , xN , t) of players onTN atT − t, obeys the

Fokker-Planck equation onTN × (0, T ):

∂mN

∂t+ ν

∑

i

∆ximN +∑

i

divxi

(mN ∂H

∂p(xi,∇xivN

i )

)= 0,

mN (x1, . . . , xN , T ) =N∏

i=1

m0(xi),

assuming that theN players initial conditions are random, independent,

with the same probability distributionm0.

19

Formal passage to the limit whenN → ∞

• vNi is expected to depend very little onxj , j 6= i.

• Since the players are independent,

mN (x1

, . . . , xN

, t) ≈Y

i

m(xi, t).

• Consider

vNi (xi, t) =

∫

TN−1

vNi (x1, . . . , xN , t)

∏

j 6=i

m(xj , t)dxj .

This should be asymptotically independent ofi:

vNi (x, t) ≈ v(x, t).

20

Multiplying the pde forvNi by

∏j 6=im(xj , t) and integrating onTN−1,

L.L. obtain asymptotically

∂v

∂t− ν∆v +H(x,∇v) = V [m],

∂m

∂t+ ν∆m+ div

(m∂H

∂p(x,∇v)

)= 0,

∫

T

mdx = 1, and m > 0 in T,

v(t = 0) = V0[m(t = 0)], m(t = T ) = m◦,

(∗∗)

since, from the law of large numbers,

Z

TN−1

V

24 1

N − 1

X

j 6=i

δxj

35 (xi)

Y

j 6=i

m(xj, t)dx

j → V [m](xi).

21

Existence for (**)

TheoremIf

• same kind of assumptions onV andV0 as in the stationary case (V and

V0 are nonlocal smoothing operators).

• H is smooth onT × Rd and

∣∣∣∣∂H

∂x(x, p)

∣∣∣∣ ≤ C(1 + |p|), ∀x ∈ T, ∀p ∈ Rd,

then (**) has at least a smooth solution.

22

Uniqueness for (**)

TheoremIf the operatorsV andV0 are strictly monotone, i.e.

∫

T

(V [m] − V [m])(m− m) ≤ 0 ⇒ V [m] = V [m],

∫

T

(V0[m] − V0[m])(m− m) ≤ 0 ⇒ V0[m] = V0[m],

then (**) has a unique solution.

23

Numerical Methods: taked = 2 and approximate numerically8>>>>>>>>><>>>>>>>>>:

∂u

∂t− ν∆u + H(x,∇u) = V [m],

∂m

∂t+ ν∆m + div

„m

∂H

∂p(x,∇u)

«= 0,

Z

T

mdx = 1, m > 0 in T,

u(t = 0) = V0[m(t = 0)], m(t = T ) = m◦,

(∗∗)

Let Th be a uniform grid on the torus with mesh steph, andxij be a genericpoint inTh.

LetNT be a positive integer and∆t = T/NT , tn = n∆t.

The values ofu andm at (xi,j , tn) are respectively approximated byUni,j

andMni,j .

24

Goal: use a fully implicit scheme, prove existence and possiblyuniqueness

Notation:

• The discrete Laplace operator:

(∆hW )i,j = − 1

h2(4Wi,j −Wi+1,j −Wi−1,j −Wi,j+1 −Wi,j−1).

• Right-sided finite difference formulas for∂1w(xi,j) and∂2w(xi,j):

(D+1 W )i,j =

Wi+1,j −Wi,j

h, and (D+

2 W )i,j =Wi,j+1 −Wi,j

h.

• The set of 4 finite difference formulas atxi,j :

[DhW ]i,j =((D+

1 W )i,j , (D+1 W )i−1,j, (D

+2 W )i,j , (D

+2 W )i,j−1

).

25

The discrete version of

∂u

∂t− ν∆u+H(x,∇u) = V [m]

is chosen as

Un+1i,j − Un

i,j

∆t− ν(∆hU

n+1)i,j + g(xi,j , [DhUn+1]i,j) =

(Vh[Mn+1]

)i,j,

where

•g(xi,j , [DhU

n+1]i,j)

=g`xi,j , (D

+

1 Un+1)i,j , (D

+

1 Un+1)i−1,j , (D

+

2 Un+1)i,j , (D

+

2 Un+1)i,j−1

´,

• for instance,(Vh[M ])i,j = V [mh](xi,j),

callingmh the piecewise constant function onT taking the valueMi,j

in the square|x− xi,j |∞ ≤ h/2.

26

Classical assumptions on the discrete Hamiltoniang

(q1, q2, q3, q4) → g (x, q1, q2, q3, q4) .

• Monotonicity: g is nonincreasing with respect toq1 andq3 and

nondecreasing with respect to toq2 andq4.

• Consistency:

g (x, q1, q1, q3, q3) = H(x, q), ∀x ∈ T, ∀q = (q1, q3) ∈ R2.

• Differentiability: g is of classC1, and there exists a constantC such

that∣∣∣ ∂g∂x (x, (q1, q2, q3, q4))

∣∣∣ ≤ C(1 + |q1| + |q2| + |q3| + |q4|) for all

x ∈ T andq1, q2, q3, q4.

• Convexity: the function(q1, q2, q3, q4) → g (x, q1, q2, q3, q4) is convex.

27

The discrete version of

∂m

∂t+ ν∆m+ div

(m∂H

∂p(x,∇v)

)= 0.

It is chosen so each time step corresponds to a linear system with a matrix

• whose diagonal coefficients are negative,

• whose off-diagonal coefficients are nonnegative,

in order to hopefully use somediscrete maximum principle.

• The argument for uniqueness should hold in the discrete case,

sothe discrete Hamiltoniang should be used for the second pde as well.

28

Principle

Discretize

−∫

T

div

(m∂H

∂p(x,∇u)

)w =

∫

T

m∂H

∂p(x,∇u) · ∇w

by

h2∑

i,j

mi,j∇qg(xi,j , [DhU ]i,j) · [DhW ]i,j

29

This yields the following scheme:

0 =Mn+1

i,j − Mni,j

∆t+ ν(∆hM

n)i,j

+1

h

8>>>>>>>><>>>>>>>>:

Mni,j

∂g

∂q1

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

−Mni−1,j

∂g

∂q1

`xi−1,j , (D

+

1 Un)i−1,j , (D

+

1 Un)i−2,j , (D

+

2 Un)i−1,j , (D

+

2 Un)i−1,j−1

´

+Mni+1,j

∂g

∂q2

`xi+1,j , (D

+

1 Un)i+1,j , (D

+

1 Un)i,j , (D

+

2 Un)i+1,j , (D

+

2 Un)i+1,j−1

´

−Mni,j

∂g

∂q2

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

+1

h

8>>>>>>>><>>>>>>>>:

Mni,j

∂g

∂q3

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

−Mni,j−1

∂g

∂q3

`xi,j−1, (D

+

1 Un)i,j−1, (D

+

1 Un)i−1,j−1, (D

+

2 Un)i,j−1, (D

+

2 Un)i,j−2

´

+Mni,j+1

∂g

∂q4

`xi,j+1, (D

+

1 Un)i,j+1, (D

+

1 Un)i,j+1, (D

+

2 Un)i,j+1, (D

+

2 Un)i,j

´

−Mni,j

∂g

∂q4

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

30

Classical discrete Hamiltoniansg can be chosen.For example, assume that the Hamiltonian is of the form

H(x,∇u) = ψ(x, |∇u|).

A possible choice is the Godunov scheme

g(x, q1, q2, q3, q4) =

ψ(x,√

min(q1, 0)2 + max(q2, 0)2 + min(q3, 0)2 + max(q4, 0)2).

If ψ is convex and nondecreasing with respect to its second argument, theng is a convex function of(q1, q2, q3, q4); it is nonincreasing with respect toq1 andq3 and nondecreasing with respect toq2 andq4.

Finally, it can be proven that the global scheme is consistent if H andg aresmooth enough.

31

Kushner-Dupuis scheme

More generally, ifH is given by

H(x, p) = supα∈R2

(p · α− L(x, α)) , x ∈ T, p ∈ R2,

then one may chooseg as

g(x, q1, q2, q3, q4) = supα∈R2

(−α−

1 q1 + α+1 q2 − α−

2 q3 + α+2 q4 − L(x, α)

).

The numerical Hamiltoniang is clearly convex, nonincreasing with respect

to q1 andq3 and nondecreasing with respect toq2 andq4.

32

Existence for the discrete problem

Theorem Assume thatMNT ≥ 0 and that∑

i,j MNT

i,j = 1. Under theassumptions above onV , V0 andg, the discrete problem has a solution.

Strategy of proof

K =

(Mi,j)0≤i,j<N : h2

∑

i,j

Mi,j = 1,Mi,j ≥ 0

.

Apply Brouwer fixed point theorem to a mapping

KNT → KNT ,

(Mn)n → (Un)n → (Mn)n.

33

Proof: a fixed point method in the compact and convex setKNT ,

Step 1: a mapping(Mn)n → (Un)n.Given(U0

i,j) and(MNT

i,j ), define the mapΦ:

Φ :(Mn)0≤n<NT∈ KNT → (Un)1≤n≤NT

, s.t.

Un+1i,j − Un

i,j

∆t− ν(∆hU

n+1)i,j + g(xi,j , [DhUn+1]i,j) =

(Vh[Mn+1]

)i,j.

• Existence is classical: (Leray-Schauder fixed point theorem at each time

step, making use of the monotonicity ofg, the uniform boundedness

assumption onV and ofH(·, 0)).• Uniqueness stems from the monotonicity ofg.

34

Step 2: estimates• There exists a constantC independent of(Mn)n s.t.

‖Un‖∞ ≤ C(1 + T ).

• From the assumption on the continuity ofV and well known results on

continuous dependence on the data for monotone schemes, themapΦ is

continuous.

•∣∣∣ ∂g∂x (x, (q1, q2, q3, q4)

∣∣∣ ≤ C(1 + |q1| + |q2| + |q3| + |q4|) implies

‖DhUn‖∞ ≤ LT, ∀n,

whereL does not depend on(Mn)0≤n<NT∈ KNT andh.

35

Step 3: A map(Mn)n → (Mn)n

For (Un)n = Φ((Mn)n) andµ > 0 large enough,

consider thelinear backward parabolic problem:

find (Mn)0≤n≤NTsuch that

MNT = MNT

and

36

−µMni,j =

fMn+1

i,j − fMni,j

∆t+ ν(∆h

fMn)i,j − µfMni,j

+1

h

8>>>>>>>><>>>>>>>>:

fMni,j

∂g

∂q1

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

−fMni−1,j

∂g

∂q1

`xi−1,j , (D

+

1 Un)i−1,j , (D

+

1 Un)i−2,j , (D

+

2 Un)i−1,j , (D

+

2 Un)i−1,j−1

´

+fMni+1,j

∂g

∂q2

`xi+1,j , (D

+

1 Un)i+1,j , (D

+

1 Un)i,j , (D

+

2 Un)i+1,j , (D

+

2 Un)i+1,j−1

´

−fMni,j

∂g

∂q2

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

+1

h

8>>>>>>>><>>>>>>>>:

fMni,j

∂g

∂q3

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

−fMni,j−1

∂g

∂q3

`xi,j−1, (D

+

1 Un)i,j−1, (D

+

1 Un)i−1,j−1, (D

+

2 Un)i,j−1, (D

+

2 Un)i,j−2

´

+fMni,j+1

∂g

∂q4

`xi,j+1, (D

+

1 Un)i,j+1, (D

+

1 Un)i,j+1, (D

+

2 Un)i,j+1, (D

+

2 Un)i,j

´

−fMni,j

∂g

∂q4

`xi,j , (D

+

1 Un)i,j , (D

+

1 Un)i−1,j , (D

+

2 Un)i,j , (D

+

2 Un)i,j−1

´

37

This is adiscrete backward linear parabolic problem.

From the previous estimates on(Un)n, one can findµ large enough andindependent of(Mn)n such that the iteration matrix is the opposite of aM-matrix, thusthere is a discrete maximum principle.

Therefore, there exists a unique solution(Mn)n. Moreover,

Mn ≥ 0 ⇒ Mn ≥ 0, ∀n,h2∑

i,j Mn = 1 ⇒ h2

∑i,j M

n = 1, ∀n.

ThusMn ∈ K for all n. Define the map

χ : KNT 7→ KNT ,

(Mn)0≤n<NT→ (Mn)0≤n<NT

38

Step 4: existence of a fixed point ofχ

From the boundedness and continuity of the mappingΦ, and from the fact

thatg is C1, we obtain thatχ : KNT 7→ KNT is continuous.

From Brouwer fixed point theorem,χ has a fixed point, which yields a

solution of (**).

39

Discrete weak maximum principle-1The discrete version of

∂m

∂t+ ν∆m+ div

(m∂H

∂p(x,∇v)

)= 0

can be written

−Mn − ∆tAnMn = −Mn+1.

Introduce the semi-norm|||W |||:

|||W |||2 =∑

i,j

((D+

1 W )2i,j + (D+2 W )2i,j

).

From the estimates onUn, there existsγ ≥ 0 (independent ofh and∆t) s.t.

(AnW,W )2 ≥ ν

2|||W |||2 − γ‖W‖2

2, ∀W, ∀n.

This is a discrete Garding inequality.

40

Discrete weak maximum principle-2

One can also prove that

−(AnW,W−)2 ≥ ν

2|||W−|||2 − γ‖W−‖2

2, ∀W.

Taking(Mn)− as a test-function in the discrete equation yields

(1 − 2γ∆t)∥∥(Mn)−

∥∥2

2+ ν∆t|||(Mn)−|||2 ≤

∥∥(Mn+1)−∥∥2

2,

so there is a discrete weak maximum principle if2γ∆t < 1.

41

Uniqueness

Theorem We make the same assumptions as above onV , V0,H andg, and

we assume furthermore that the operatorsVh andV0,h are strictly

monotone, i.e.(Vh[M ] − Vh[M ],M − M

)2≤ 0 ⇒ Vh[M ] = Vh[M ],

(V0,h[M ] − V0,h[M ],M − M

)2≤ 0 ⇒ V0,h[M ] = V0,h[M ].

There exists a nonnegative constantγ such that if2γ∆t < 1, then the

discrete problem has a unique solution.

Proof The choice of the scheme makes it possible to mimic the proof used

in the continuous case: uses the convexity assumption ong and the discrete

weak maximum principle.

42

A finite difference scheme for the stationary problem8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

−ν(∆hU)i,j + g(xi,j , [DhU ]i,j) + λ = (Vh[M ])i,j ,0BBBBBBBBBBB@

−ν(∆hM)i,j

−1

h

0B@

Mi,j∂g

∂q1

(xi,j , [DhU ]i,j) − Mi−1,j∂g

∂q1

(xi−1,j , [DhU ]i−1,j)

+Mi+1,j∂g

∂q2

(xi+1,j , [DhU ]i+1,j) − Mi,j∂g

∂q2

(xi,j , [DhU ]i,j)

1CA

−1

h

0B@

Mi,j∂g

∂q3

(xi,j , [DhU ]i,j) − Mi,j−1

∂g

∂q3

(xi,j−1, [DhU ]i,j−1)

+Mi,j+1

∂g

∂q4

(xi,j+1, [DhU ]i,j+1) − Mi,j∂g

∂q4

(xi,j , [DhU ]i,j)

1CA

1CCCCCCCCCCCA

= 0,

Mi,j ≥ 0,

and

h2∑

i,j

Mi,j = 1, and∑

i,j

Ui,j = 0.

43

Existence for the discrete problem: strategy of proof

• Use Brouwer fixed point theorem in the set of discrete probabilitymeasures for a mappingM → U →M .

• The mapM → U consists of solving

−ν(∆hU)i,j + g(xi,j , [DhU ]i,j) + λ = (Vh[M ])i,j ,∑i,j Ui,j = 0

• Ergodic approximation: pass to the limit asρ→ 0:

−ν(∆hU(ρ))i,j + g(xi,j , [DhU

(ρ)]i,j) + ρU (ρ))i,j = (Vh[M ])i,j ,

We need estimates onU (ρ) − U(ρ)0,0 uniform inρ andh.

44

DifficultyThe proof of existence for the continuous problem used the estimate

‖∇u‖∞ ≤ C, which was proven using the Bernstein method and the

assumption: there existsθ ∈ (0, 1) such that for|p| large,

infx∈T

(∂H

∂x· p+

θ

2H2 > 0

)> 0.

In the discrete case, this argument seems difficult to reproduce, and we were

not able so far to obtain an estimate on‖Dhu‖∞ independent ofh, under

the same assumptions.

45

Assumptions on the Hamiltonian

H(x, p) = maxα∈A

(p · α− L(x, α)) ,

where

• A is a compact subset ofR2,

• L is aC0 function onT ×A,

• Kushner-Dupuis scheme:

g(x, q1, q2, q3, q4) = supα∈A

(−α−

1 q1 + α+1 q2 − α−

2 q3 + α+2 q4 − L(x, α)

).

46

Proposition (using Kuo-Trudinger(1992) and Camilli-Marchi(2008))

Consider a grid functionV and make the assumptions:

• L is a continuous function,A is compact,

• g aC1 function onT × R4 such that∇qg is bounded.

• ‖V ‖∞ is bounded uniformly w.r.th,

For any real numberρ > 0, there exists a unique grid functionUρ such that

ρUρi,j − ν(∆hU

ρ)i,j + g(xi,j , [DhUρ]i,j) = Vi,j ,

and there exist two constantsδ, δ ∈ (0, 1) andC, C > 0, uniform inh andρ

s.t.

|Uρ(ξ) − Uρ(ξ′)| ≤ C|ξ − ξ′|δ, ∀ξ, ξ′ ∈ Th.

47

Proposition (using Krylov(2007) and Camilli-Marchi(2008))

Same assumptions as before, and furthermore

• L is uniformly Lipschitz w.r.t.x,

• ‖DhV ‖∞ is bounded uniformly w.r.th,

For any real numberρ > 0, there exists a unique grid functionUρ such that

ρUρi,j − ν(∆hU

ρ)i,j + g(xi,j , [DhUρ]i,j) = Vi,j ,

and there exist a constantC, C > 0, uniform inh andρ s.t.

|Uρ(ξ) − Uρ(ξ′)| ≤ C|ξ − ξ′|, ∀ξ, ξ′ ∈ Th.

48

Proposition Under the first set of assumptions, there exists a unique grid

functionU and a real numberλ such that

−ν(∆hU)i,j + g(xi,j , [DhU ]i,j) + λ = (Vh[M ])i,j ,∑i,j Ui,j = 0,

and there exist two constantsδ, δ ∈ (0, 1) andC, C > 0, uniform inh s.t.

|U(ξ) − U(ξ′)| ≤ C|ξ − ξ′|δ, ∀ξ, ξ′ ∈ Th.

Under the second set of assumptions,

|U(ξ) − U(ξ′)| ≤ C|ξ − ξ′|, ∀ξ, ξ′ ∈ Th.

49

Existence and uniqueness for the stationary problem

Theorem Under the above assumptions onV andg, the discrete stationary

problem has at least a solution and we have either a uniform Holder or a

Lipschitz estimate onU , depending on the assumptions.

Uniqueness: Ok if(Vh[M ] − Vh[M ],M − M

)2≤ 0 ⇒M = M.

50

The case whenV is a local operatorAssume thatV is a local operator, i.e.V [m](x) = F (m(x), x), whereF is

aC1 function onR × T.

The above mentioned existence theorem in its weakest form contains the

case whenVh is a continuous map from discrete probability measures to

bounded grid functions. In particular,Vh can be a local operator.

Existence and Holder estimates.

51

Numerical method: Long time approximation of the stationaryproblem

∂u

∂t− ν∆u+H(x,∇u) = V [m],

∂m

∂t− ν∆m− div

(m∂H

∂p(x,∇u)

)= 0,

u(0, x) = u0(x), m(0, x) = m0(x),

with∫

Tm0 = 1 and m0 ≥ 0, and we expect that

limt→∞

(u(t, x) − λt) = u(x), limt→∞

m(t, x) = m(x),

Same thing at the discrete level.We use a semi-implicit linearized scheme. It requires the numerical solutionof a linearized problem. Linearizing must be done carefullyand is notalways possible. In such cases, an explicit method can be used.

52

ν = 1,

H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|2,F (x,m) = m2

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.23.89e-16 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 -2 -2.2 -2.4 -2.6 -2.8

sin(2πx2) + sin(2πx1) + cos(4πx1).

53

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

200 400 600 800 1000 1200 1400 1600 1800 2000

"mean_8_200"

Convergence1t∫

Tu(x, t)dx→ λ ast→ ∞

54

0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.0051.73e-18 -0.005 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035

1.08 1.07 1.06 1.05 1.04 1.03 1.02 1.01 1 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.9

left: u, rightm.

55

0

2e-05

4e-05

6e-05

8e-05

0.0001

0.00012

0.00014

0.00016

0.00018

0.0002

0.01 0.015 0.02 0.025 0.03 0.035 0.04

"erroru"0.005*x

5e-05

0.0001

0.00015

0.0002

0.00025

0.0003

0.01 0.015 0.02 0.025 0.03 0.035 0.04

"errorm"0.005*x

Convergence ash→ 0

56

ν = 0.01,

H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|2,F (x,m) = m2

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

200 400 600 800 1000 1200 1400 1600 1800 2000

"mean_8_200"

Convergence1t∫

Tu(x, t)dx→ λ ast→ ∞

57

0.15 0.14 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.01

1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

left: u, rightm.

Note that the supports of∇u and ofm tend to be disjoint asν → 0.

58

Deterministic limit

Theorem (Lasry-Lions)

If

• H(x, p) ≥ H(x, 0) = 0,

• V [m] = F (m) + f0(x) whereF ′ > 0,

then

limν→0

(λν ,mν) = (λ,m),

where

m(x) =(F−1(λ− f0(x))

)+and

∫

T

mdx = 1.

59

ν = 0.001,

H(x, p) = |p|2,

V [m](x) = 4 cos(4πx) + m(x)

"u.gp" "m.gp"

left: u, rightm.The supports of∇u and ofm tend to be disjoint.

m(x) ≈ (λ− 4 cos(4πx))+

60

A nonlocal operator V

0.3 0.25 0.2 0.15 0.1 0.05-1.39e-17 -0.05 -0.1 -0.15 -0.2

7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5

ν = 0.1,

H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + |p|3/2,

F (x, m) = 200(1 − ∆)−1(1 − ∆)−1m

left: u, right m.

61

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05-6.94e-17 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3

13 12 11 10 9 8 7 6 5 4 3 2 1

ν = 0.1,

H(x, p) = sin(2πx2) + sin(2πx1) + cos(4πx1) + (0.6 + 0.59 cos(2πx))|p|3/2,

F (x, m) = 200(1 − ∆)−1(1 − ∆)−1m

left: u, right m.

62

Perspectives

• Obtain an estimate on‖DhU‖∞ uniform inh with more general

structure assumptions in the stationary case (important for stability and

convergence).

• Prove that the long-time strategy converges to the solutionof the

stationary problem.

• When convergence is OK, prove error estimates?

• Tackle practical situations.

63