Homogenization of a mean field game system in the
small noise limit
Claudio Marchi
University of Padova
Joint work with
Annalisa Cesaroni (Padova), Nicolas Dirr (Cardiff)
Rennes, 31st May, 2016
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
1 / 31
Outline
1
Brief introduction on MFG systems
2
Homogenization problems for MFG
3
Analysis of the effective operators and effective limit system
4
A (partial) convergence result.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
2 / 31
A brief introduction to Mean Field Games
The Mean Field Games model (MFG) was proposed by Lasry-Lions,
and independently by Huang-Malhamé-Caines, in 2006.
Distinctive features of the model:
The MFG theory is a model to describe interactions among a very
large number of agents. Aim of MFG theory is to relate individual
actions to mass behavior (fashion trends, Mexican wave, financial
crisis, crowd dynamics,...).
The MFG model has some analogies with Statistical Mechanics,
where an external field (usually a statistics of some given physical
quantity) influences the behavior of the particles. But in MFG
theory the agent is not a black-box, since it can decide a strategy
based on a set of preferences.
The single agent by itself cannot influence the collective behavior,
it can only optimize its own strategy. The mean field is given by
the collective behavior of the population.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
3 / 31
Model example
Consider a game with N rational and indistinguishable players. The
i-th player’s dynamics is
√
dXti = −αti dt + 2dWti ,
X0i = x i ∈ Tn
where W i are independent Brownian motions, while αi is the control
chosen so to minimize the cost functional

 



Z t |αi |2
X
 s + V Xsi , 1
δX j  ds + u0 XTi
E
s

 0
2
N −1
j6=i
The Nash equilibria are characterized by a system of 2N equations.
BUT, as N → +∞, this system reduces to the following one:
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
4 / 31
MFG system

1
2
(t, x) ∈ (0, T ) × Tn

 −ut − ∆u + 2 |∇u| = V (x, m)
mt − ∆m + div (m∇u) = 0
(t, x) ∈ (0, T ) × Tn


u(T , x) = u0 (x),
m(0, x) = m0 (x) x ∈ Tn
R
where m0 is the initial distribution of players: m0 ≥ 0 Td m0 = 1.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
5 / 31
MFG system

1
2
(t, x) ∈ (0, T ) × Tn

 −ut − ∆u + 2 |∇u| = V (x, m)
mt − ∆m + div (m∇u) = 0
(t, x) ∈ (0, T ) × Tn


u(T , x) = u0 (x),
m(0, x) = m0 (x) x ∈ Tn
R
where m0 is the initial distribution of players: m0 ≥ 0 Td m0 = 1.
Features of MFG system
The first equation is a backward Hamilton-Jacobi-Bellman
equation describing the expected value for an average player.
the second equation is a forward Fokker-Planck equation
describing the density m of the players.
The operator in the FP equation is the adjoint of the linearized of
the operator in the HJB equation.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
5 / 31
Two possible regimes for the coupling
1) nonlocal coupling: V : Rn × P1 → R smoothing in the space P1 of
probability measures, e.g.: V (x, m) = V (x, m ? k ).
Uniqueness when V (y , ·) is monotone increasing:
Z
(V (x, m) − V (x, n))(m − n)dx > 0.
Existence of smooth solutions via Schauder theorem (Lasry, Lions).
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
6 / 31
Two possible regimes for the coupling
1) nonlocal coupling: V : Rn × P1 → R smoothing in the space P1 of
probability measures, e.g.: V (x, m) = V (x, m ? k ).
Uniqueness when V (y , ·) is monotone increasing:
Z
(V (x, m) − V (x, n))(m − n)dx > 0.
Existence of smooth solutions via Schauder theorem (Lasry, Lions).
2) local coupling: V (x, m(x, t)) .
In our case H = |p|2 , solutions are smooth (Cardaliaguet, Lasry,
Lions, Porretta)
solutions are still smooth under some regularity-growth
assumptions (Lasry, Lions, Gomes, Pimentel, Sanchez Morgado)
Weak solutions of MFG system: second order (Porretta), first
order (Cardaliaguet), second order degenerate (Cardaliaguet,
Graber, Porretta, Tonon)
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
6 / 31
Asymptotic problems on MFG : convergence to steady
states
(
−ut − ∆u + 12 |∇u|2 = V (x, m) , u(x, T ) = u0 (x)
(MFG)
mt − ∆m − div(m∇u) = 0
m(x, 0) = m0 (x).
Stationary ergodic system (steady states)
(
R
λ − ∆u + 12 |∇u|2 = V (x, m) , Tn u(y )dy = 0
R
−∆m − div(m∇u) = 0
Tn m(y )dy = 1
Theorem
u(·, tT )
→ λ(1−t)
T
Z
u(·, tT )−
m(x, tT ) → m̄(x)
u(x, tT )dx → ū
in L2 (Tn ×(0, 1))
in Lp (Tn × (0, 1)) p < n + 2/n.
(Cardaliaguet, Lasry, Lions, Porretta, both local and nonlocal coupling)
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
7 / 31
Basic References for MFG theory:
Lasry-Lions, C.R. Math. Acad. Sci. Paris 343 (2006), 619-625.
Lasry-Lions, C.R. Math. Acad. Sci. Paris 343 (2006), 679-684.
Lasry-Lions, Jpn. J. Math. 2 (2007), 229-260.
Huang-Malhamé-Caines, Commun. Inf. Syst. 6 (2006), 221-251.
Lions’ online course at College de France
www.college-de-france.fr
Cardaliaguet, Notes on MFG (from Lions’ lectures at College de
France), www.ceremade.dauphine.fr/∼cardalia/
Achdou, Lecture Notes in Math. 2074 (2013), 1-47.
Cardaliaguet-Lasry-Lions-Porretta, Netw. Heterog. Media 7
(2012), 279-301.
Cardaliaguet-Lasry-Lions-Porretta, SIAM J. Control Optim.51
(2013), 3558-3591.
Gomes-Sáude, Dyn. Games Appl. 4 (2014), 110-154.
Cardaliaguet-Delarue-Lasry-Lions,
arxiv.org/abs/1509.02505
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
8 / 31
Homogenization problems for MFG systems
We are interested in the following asymptotic problem: the limit as
ε→0

1
x
ε
ε
ε 2
ε

(t, x) ∈ (0, T ) × Tn
−ut − ε∆u + 2 |∇u | = V ε , m
mtε − ε∆mε − div(mε ∇u ε ) = 0
(t, x) ∈ (0, T ) × Tn

 ε
u (x, T ) = u0 (x)
mε (x, 0) = m0 (x) x ∈ Tn .
where we consider periodic boundary conditions: both V (·, m) and
V ( ε· , m) are both Zn -periodic (we consider ε−1 ∈ N).
Remark. More generally we can assume just that V (·, m) is
Zn -periodic and consider the system in a bounded domain with
Neumann boudary conditions.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
9 / 31
Main assumptions
V (y , m) ∈ C 1 (Tn × R) is bounded,
V (y , ·) is monotone increasing in m,
V (·, m) is Zn -periodic in y , ε−1 ∈ N,
u0 is smooth and periodic,
m0 ≥ 0, m0 is smooth, periodic and
Claudio Marchi (University of Padova)
R
Tn
m0 (y )dy = 1.
Homogenization of a mean field game system in the small noise limit
10 / 31
Main assumptions
V (y , m) ∈ C 1 (Tn × R) is bounded,
V (y , ·) is monotone increasing in m,
V (·, m) is Zn -periodic in y , ε−1 ∈ N,
u0 is smooth and periodic,
m0 ≥ 0, m0 is smooth, periodic and
R
Tn
m0 (y )dy = 1.
Proposition
∀ε > 0, there exists a unique smooth solution to

1
x
ε
ε
ε 2
ε

(t, x) ∈ (0, T ) × Tn
−ut − ε∆u + 2 |∇u | = V ε , m
mtε − ε∆mε − div(mε ∇u ε ) = 0
(t, x) ∈ (0, T ) × Tn

 ε
u (x, T ) = u0 (x)
mε (x, 0) = m0 (x) x ∈ Tn .
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
10 / 31
Two main problems
1) identification of the limit system:
formal 2 scale (additive-multiplicative) asymptotic expansion
(
u ε (x, t) = u 0 (x, t) + εu(x/ε)
mε (x, t) = m0 (t, x) m(x/ε) + εm2 (x/ε)
solution of an ergodic MFG system (so called cell problem) to
identify the limit operators.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
11 / 31
Two main problems
1) identification of the limit system:
formal 2 scale (additive-multiplicative) asymptotic expansion
(
u ε (x, t) = u 0 (x, t) + εu(x/ε)
mε (x, t) = m0 (t, x) m(x/ε) + εm2 (x/ε)
solution of an ergodic MFG system (so called cell problem) to
identify the limit operators.
2) Proof of convergence (in a suitable sense)
To make rigorous the formal asymptotic expansion
two scale convergence (Nguetseng 89, Allaire 92)
perturbed test function method (Evans 89, 92)
Only preliminary results, under strong assumptions (i.e. assuming that
the asymptotic expansion holds).
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
11 / 31
Two scale expansion
(
u ε (t, x) = u 0 (t, x) + εu(x/ε)
mε (t, x) = m0 (t, x) m(x/ε) + εm2 (x/ε)
and we get the following (let y = xε )
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
12 / 31
Two scale expansion
(
u ε (t, x) = u 0 (t, x) + εu(x/ε)
mε (t, x) = m0 (t, x) m(x/ε) + εm2 (x/ε)
and we get the following (let y = xε )
terms in ε−1 in the FP:
−∆y m − divy ((∇x u 0 + ∇y u)m) = 0
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
12 / 31
Two scale expansion
(
u ε (t, x) = u 0 (t, x) + εu(x/ε)
mε (t, x) = m0 (t, x) m(x/ε) + εm2 (x/ε)
and we get the following (let y = xε )
terms in ε−1 in the FP:
−∆y m − divy ((∇x u 0 + ∇y u)m) = 0
terms in ε0 in the HJ:
1
−ut0 − ∆y u + |∇x u 0 + ∇y u|2 − V (y , m0 m) = 0
2
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
12 / 31
Two scale expansion
(
u ε (t, x) = u 0 (t, x) + εu(x/ε)
mε (t, x) = m0 (t, x) m(x/ε) + εm2 (x/ε)
and we get the following (let y = xε )
terms in ε−1 in the FP:
−∆y m − divy ((∇x u 0 + ∇y u)m) = 0
terms in ε0 in the HJ:
1
−ut0 − ∆y u + |∇x u 0 + ∇y u|2 − V (y , m0 m) = 0
2
We freeze x, t: given P = ∇u 0 (x, t) ∈ Rn and α = m0 (x, t) ≥ 0, find the
unique constant H̄(P, α) for which there exists a periodic solution to
(
−∆u + 21 |∇u +P|2 −V (y , αm) = H̄(P, α) y ∈ Tn
−∆m − div (m (∇u + P)) = 0
y ∈ Tn .
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
12 / 31
Proposition (cell problem)
Given P ∈ Rn and α ≥ 0, there exists a unique constant H̄ such that

1
2
n

−∆u+ 2 |∇u +P| −V (y , αm) = H(P, α) y ∈ T
−∆m − div (m (∇u + P)) = 0
y ∈ Tn

R
R

Tn m = 1
Tn u = 0
admits a solution (u, m).
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
13 / 31
Proposition (cell problem)
Given P ∈ Rn and α ≥ 0, there exists a unique constant H̄ such that

1
2
n

−∆u+ 2 |∇u +P| −V (y , αm) = H(P, α) y ∈ T
−∆m − div (m (∇u + P)) = 0
y ∈ Tn

R
R

Tn m = 1
Tn u = 0
admits a solution (u, m). Moreover
this solution is unique; u ∈ C 2,γ , m ∈ W 1,p with m ≥ c > 0;
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
13 / 31
Proposition (cell problem)
Given P ∈ Rn and α ≥ 0, there exists a unique constant H̄ such that

1
2
n

−∆u+ 2 |∇u +P| −V (y , αm) = H(P, α) y ∈ T
−∆m − div (m (∇u + P)) = 0
y ∈ Tn

R
R

Tn m = 1
Tn u = 0
admits a solution (u, m). Moreover
this solution is unique; u ∈ C 2,γ , m ∈ W 1,p with m ≥ c > 0;
H̄ is decreasing in α and coercive in P:
|P|2
|P|2
− kV k∞ ≤ H̄(P, α) ≤
+ kV k∞ ;
2
2
for any γ ∈ (0, 1), p ∈ (1, +∞), the following maps are continuous
(P, α) → H̄(P, α) ∈ R,
(P, α) → (u, m) ∈ C 1,γ × W 1,p .
Proof: based on [Cardaliaguet-Lasry-Lions-Porretta], [Bardi-Felequi].
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
13 / 31
Going back to the two scale expansion
terms in ε0 in the FP:
m0 (−∆y m2 − divy (m2 (∇x u 0 + ∇y u))) =
−mmt0 + mdivx (m0 (∇x u 0 + ∇y u)) + 2∇x m0 · ∇y m.
So the solvability condition for the equation in m2 gives
Z
0
0
mt − divx m
m(P + ∇u)dy
=0
Tn
We denote with
Z
m(P + ∇u)dy .
b̄(P, α) =
Tn
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Homogenization of a mean field game system in the small noise limit
14 / 31
Expected limit system
At the limit, we expect the following system

0
0
0

(t, x) ∈ (0, T ) × Rn
−ut + H̄(∇u , m ) = 0,
mt0 − div(m0 b̄(∇u 0 , m0 )) = 0,
(t, x) ∈ (0, T ) × Rn

 0
u (x, T ) = u0 (x)
m0 (x, 0) = m0 (x) x ∈ Rn
where
H̄(P, α) and b̄(P, α) are locally Lipschitz continuous
H̄(P, α) is coercive in P with quadratic growth (unif. in α)
H̄(P, α) is monotone decreasing in α
Questions. Existence of (weak) solutions? Uniqueness?
Even monotonicity of the system is not clear.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
15 / 31
(Brief) discussion on other cases
Case of finite-not vanishing- noise
(
−utε − ∆u ε + 12 |∇u ε |2 = V xε , mε
mtε − ∆mε − div(mε ∇u ε ) = 0
(t, x) ∈ (0, T ) × Tn ,
(t, x) ∈ (0, T ) × Tn .
Formal 2 scale asymptotic expansion
u ε (x, t) = u 0 (x, t) + ε2 u(x/ε)
mε (x, t) = m0 (t, x) m(x/ε) + ε2 m2 (x/ε)
cell problem decouples;
∆m = 0, m periodic and mean 1; hence, m ≡ 1;
expected limit system has a MFG structure with a monotone
coupling:
(
R
−ut0 − ∆u 0 + 12 |∇u 0 |2 = Tn V (y , m0 (x, t))dy
mt0 − ∆m0 − div(m0 ∇u 0 ) = 0.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
16 / 31
A case of nonlocal coupling
(
−utε − ε∆u ε + 12 |Du ε |2 = V xε , L(mε (·, t))
mtε − ε∆mε − div(mε ∇u ε ) = 0
(t, x) ∈ (0, T ) × Tn
(t, x) ∈ (0, T ) × Tn ,
where w ε := Lmε is the periodic function such that −∆w ε = mε − 1 on
Tn , with w ε (0) = 0.
We add the ansatz that also w ε satisfies the asymptotic expansion
x w ε (x, t) = w 0 (x, t) + ε2 w
,
ε
where −∆w 0 = m0 − 1, where w is a periodic function with zero
average such that ∆w(y ) = m0 (x)(m(y ) − 1). Also in this case the cell
system decouples.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
17 / 31
Properties of the effective operators
Lemma
H̄(P, α)
1
=
2
2
|P|
|P|→+∞
lim
Claudio Marchi (University of Padova)
|b̄(P, α) − P|
=0
|P|
|P|→+∞
lim
unif. in α.
Homogenization of a mean field game system in the small noise limit
18 / 31
Properties of the effective operators
Lemma
H̄(P, α)
1
=
2
2
|P|
|P|→+∞
lim
|b̄(P, α) − P|
=0
|P|
|P|→+∞
lim
unif. in α.
Proof: For P ∈ Rn , α > 0 fixed, let (uP , mP ) be the solution to
(
R
(i) −∆uP + 12 |∇uP +P|2 −V (y , αmP ) = H(P, α)
Tn uP = 0
R
(ii) −∆mP − div (mP (∇uP + P)) = 0
Tn mP = 1.
multiply (i) by mP − 1, (ii) by uP , integrate and substract:
R |∇uP |2
2 (mP + 1) + V (y , αmP )(mP − 1)dy = 0;
TN
by monotonicity: V (y , αmP )(mP − 1) ≥ V (y , α)(mP − 1);
√
|∇uP |/|P|, mP |∇uP |/|P| → 0 in L2
we integrate (i) on the torus, divide by |P|2 and send |P| → +∞
1/2 R
1/2
R
R
.
|b̄ − P| ≤ Tn |∇uP |mP ≤ Tn |∇uP |2 mP
Tn mP
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
18 / 31
Theorem
The following formulas hold:
Z
i)
ii)
∇P H̄(P, α) = b̄(P, α) −
Vm (y , αm)αm(∇P m)dy
Tn
Z h
i
∂α H̄(P, α) = −
Vm (y , αm)(m + α∂α m)2 + αm|∇∂α m|2 dy
Tn
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
19 / 31
Theorem
The following formulas hold:
Z
i)
ii)
∇P H̄(P, α) = b̄(P, α) −
Vm (y , αm)αm(∇P m)dy
Tn
Z h
i
∂α H̄(P, α) = −
Vm (y , αm)(m + α∂α m)2 + αm|∇∂α m|2 dy
Tn
Heuristic derivation: Variational characterization of H̄.
Z
|∇u + P|2
EP,α (u, m) =
m + (∇u + P) · ∇m − Φα (y , m)dy
2
Tn
Rm
where Φα (y , m) = 0 V (y , αs)ds. We have
∂m EP,α (u, m) = 0 = ∂u EP,α (u, m) ⇐⇒ (u, m) solves the cell problem
Z
H̄(P, α) = EP,α (u, m) +
Φα (y , m) − V (y , αm)m.
Tn
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
19 / 31
Proof of (i).
Let (u, m) and (uδ , mδ ) be the sol. to cell problem with (P, α) and
resp. with (P + δei , α). Let wδ = (uδ − u)/δ, nδ = (mδ − m)/δ.
a priori L2 bounds on ∇wδ , nδ
wδ , nδ are weakly converging (in H 1 and L2 ) to ui , mi , solution to


−∆ui + ∇ui · (∇u + P) + (∇u + P) · ei − Vm (y , αm)αmi = ci
−∆mi − div (P + ∇u)mi = div(m(∇ui + ei ))

R
R
Tn mi = Tn ui = 0.
ci = b̄(P, α) · ei −
Claudio Marchi (University of Padova)
R
Tn
Vm (y , αm)αmmi dy
Homogenization of a mean field game system in the small noise limit
20 / 31
Is the limit system always a MFG?
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
21 / 31
Is the limit system always a MFG?
Answer
NO.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
21 / 31
Is the limit system always a MFG?
Answer
NO.
Heuristic explanation: MFG with local coupling can be interpreted as
an optimality condition between two problems in duality. One is
Z
T
Z
min
(µ,w)∈K1ε
0
Tn
|w(t, x)|2
+ F (x/ε, µ(t, x))dxdt +
µ(t, x)
Z
Tn
u0 (x)µ(T , x)dx
Rs
where F (y , s) := 0 V (y , r )dr while K1ε is formed by the couples
R
(µ, w) ∈ L1 × (L1 )n , µ ≥ 0 a.e., µ(t, x)dx = 1, µ(0, x) = m0 (x) and
µt − ε∆µ + div(w) = 0.
[Cardaliaguet, Graber, Porretta, Tonon]: there exists a unique
minimizer (µ̄, w̄); moreover: µ̄ = mε . The “dual” problem “gives” u ε .
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
21 / 31
An explicit counter-example. Assume n = 1 and
V (y , m) = v (y ) + m
(not in the same assumptions, but we can extend all the results).
The formula reads:
∂ H̄(P, α)
α ∂kmk22
= b̄(P, α) −
∂P
2 ∂P
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Homogenization of a mean field game system in the small noise limit
22 / 31
Lemma
∂kmk22
6≡ 0.
∂P
Proof. By definition kmk1 = 1 and m 6≡ 1. By Jensen:
kmk22 > 1 = kmk21 .
It is sufficient to prove that m → 1 strongly in L2 as |P| → +∞.
kmk2 is uniformly bounded in P (reasoning as above)
m → 1 weakly in L2 (using the equation)
√
m is uniformly bounded in P in H 1 (multipy the equation for m by
log m and integrate)
H 1 compactly embedded in C 0 ; hence m → 1 in L2 .
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Homogenization of a mean field game system in the small noise limit
23 / 31
1.4
1.35
1.3
1.25
1.2
1.15
1.1
1.05
1
0.95
5
10
15
20
25
30
35
40
45
50
(by courtesy of Cacace and Camilli)
here α = 1, v (y ) = 50(sin(2πy ) + cos(4πy )).
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
24 / 31
Convergence result
Fix P ∈ Rn and consider the mean field game system

1
x
ε
ε
ε 2
ε

(t, x) ∈ (0, T ) × Rn
−ut − ε∆u + 2 |∇u | = V ε , m
mtε − ε∆mε − div(mε ∇u ε ) = 0
(t, x) ∈ (0, T ) × Rn

 ε
u (x, T ) = P · x
mε (x, 0) ≡ 1
x ∈ Rn .
(1)
The data P · x is not periodic: we rotate the reference system.
The limit system reads
(
−ut + H̄(∇u, m) = 0
mt − div(b̄(∇u, m)m) = 0
u(x, T ) = P · x
m(x, 0) = 1;
it has the trivial solution
u 0 (x, t) = P · x + (t − T )H̄(P, 1)
Claudio Marchi (University of Padova)
m0 (x, t) ≡ 1.
Homogenization of a mean field game system in the small noise limit
25 / 31
Theorem
For every compact Q in Rn ,
u ε → P · x + (t − T )H̄(P, 1) in L2 ([0, T ] × Q)
mε → 1 weakly in Lp ([0, T ] × Q) for 1 ≤ p < (n + 2)/n if n ≥ 3
and p < 2 if n = 2.
Main tools: energy estimates
We borrow main ideas from Cardaliaguet-Lasry-Lions-Porretta for the
long time behavior.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
26 / 31
Sketch of proof
Define
(
v ε (y , t) = 1ε u ε (εy , t) − P · y − 1ε (t − T )H̄(P, 1) − u(y )
nε (y , t) = mε (εy , t) − m(y )
where (u, m) is the solution to the cell problem with (P, 1).
Step 1. For every 0 ≤ t1 ≤ t2 ≤ T , there holds
Z
−ε
t2
v n =
n
T
Z
t2
Z
=
t1
Tn
ε ε
t1
2m + nε
|∇v ε |2 + [V (y , m + nε ) − V (y , m)] nε dy .
2
Proof. We cross test the equations fulfilled by (v ε , nε ).
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
27 / 31
Step 2. There holds:
T
Z
0
Z
Tn
2m + nε
|∇v ε |2 + [V (y , m + nε ) − V (y , m)] nε dy ≤ εC.
2
Proof. We introduce the energy
E(v , n) =
|∇v |2
+ ∇v · (P + ∇u)) + ∇v · ∇(n + m) − Φ1 (y , n)]dy
2
n
T
Rn
where Φ1 (y , n) = 0 (V (y , s + m) − V (y , m))ds. We use Step-1 and
Z
=
(n + m)[(
dE ε
(v (·, t), nε (·, t)) = 0.
dt
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
28 / 31
Step 3. There holds
(i)
(ii)
lim ∇v ε = 0
ε→0
lim nε = 0
ε→0
in L2 ((0, T ) × Tn )
in Lp ((0, T ) × Tn ).
Proof of (i). Use Step-2 and 2m + nε ≥ c > 0.
Proof of (ii). We use
Monotonicity of V and Step-2 entail: limε→0 nε = 0 in L1 ;
mε (εx, t) are uniformly bounded in Lp̄ for any p̄ ≤ (n + 2)/n if
n ≥ 3, for any p̄ < 2 if n = 2.
mε (εx, t) weakly converge to m1 in Lp̄ for any p̄ ≤ (n + 2)/n if
n ≥ 3, for any p̄ < 2 if n = 2.
Step 4. Go back to (u ε , mε ) rescaling the results in Step-3.
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
29 / 31
Perspectives
Other properties of the effective operators;
does b̄ depend on α?
What about monotonicity of the limit problem?
When the MFG structure is preserved? This happens for
V = v (y ) + log m...work in progress.
Convergence for general system?
Rate of convergence?
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
30 / 31
Thank you for your attention!
Claudio Marchi (University of Padova)
Homogenization of a mean field game system in the small noise limit
31 / 31