Introduction
Stationary problems
Time dependent problems
Mean field games with congestion
Diogo Gomes
King Abdullah University of Science and Technology (KAUST),
CEMSE Division,
Saudi Arabia.
February 8, 2015
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Outline
1
Introduction
2
Stationary problems
Variational mean-field games
Congestion models
3
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Outline
1
Introduction
2
Stationary problems
Variational mean-field games
Congestion models
3
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Collaborators
H. Mitake (Hiroshima)
H. S. Morgado (UNAM, Mexico)
S. Patrizi (Berlin)
E. Pimentel
V. Voskanyan (KAUST)
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Standard MFG in reduced form
Time dependent MFG
(
−ut + H(Du, x) = ∆u + F (m)
mt − div(Dp Hm) = ∆m
with m(x, 0) and u(x, T ) given.
Stationary version
(
H(Du, x) = ∆u + F (m) + H
− div(Dp Hm) = ∆m
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Standard MFG in reduced form
Time dependent MFG
(
−ut + H(Du, x) = ∆u + F (m)
mt − div(Dp Hm) = ∆m
with m(x, 0) and u(x, T ) given.
Stationary version
(
H(Du, x) = ∆u + F (m) + H
− div(Dp Hm) = ∆m
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Typical non-linearity F :
Non-local: F (m) = G(η ∗ m).
Power-like: F (m) = mα .
Logarithm: F (m) = ln m.
Typical Hamiltonian: H(x, p) = a(x)(1 + |p|2 )γ/2 + V (x), a, V
periodic, smooth, a > 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Typical non-linearity F :
Non-local: F (m) = G(η ∗ m).
Power-like: F (m) = mα .
Logarithm: F (m) = ln m.
Typical Hamiltonian: H(x, p) = a(x)(1 + |p|2 )γ/2 + V (x), a, V
periodic, smooth, a > 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Typical non-linearity F :
Non-local: F (m) = G(η ∗ m).
Power-like: F (m) = mα .
Logarithm: F (m) = ln m.
Typical Hamiltonian: H(x, p) = a(x)(1 + |p|2 )γ/2 + V (x), a, V
periodic, smooth, a > 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Typical non-linearity F :
Non-local: F (m) = G(η ∗ m).
Power-like: F (m) = mα .
Logarithm: F (m) = ln m.
Typical Hamiltonian: H(x, p) = a(x)(1 + |p|2 )γ/2 + V (x), a, V
periodic, smooth, a > 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
A more general case
Then the value function u solves the Hamilton-Jacobi equation
−ut + H(Dx u, x, m) =
and m solves
mt − div(Dp Hm) =
Diogo Gomes
σ2
∆u
2
σ2
∆m.
2
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Boundary data
The value function u and the probability measure m satisfy
initial-terminal problem
u(x, T ) = ψ(x)
m(x, 0) = m0 (x).
To simplify the presentation we work in the spatially periodic
setting. That is x ∈ Td , the standard d-dimensional torus
identified with [0, 1]d .
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
The solution u to the Hamilton-Jacobi equation is a value
function:
Z T
u(x, t) = inf E
L(x, v, m(·, s))ds + ψ(x(T ), m(·, T )),
v
t
where the infimum is taken, over all progressively measurable
controls v (w.r.t. the Brownian filtration of Wt ), and
dx = vdt + σdWt .
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Congestion
At the level of the Lagrangian, two important congestion models
are:
Logarithmic non-linearity
L(x, v , m) = L0 (x, v ) + ln m,
which makes low density regions extremely desirable.
Standard congestion problem:
L(x, v , m) = mα L0 (x, v ),
which makes very expensive to move in high-density areas.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
These two Lagrangians give yield the systems
Logarithmic non-linearity
(
−ut + H(x, Du) = ln m + ∆u
mt − div (Dp Hm) = ∆m.
Standard congestion
(
2
−ut + |Du|
mα = ∆u
mt − div (m1−α Du) = ∆m.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Research goals
Uniqueness (solved under general conditions by Lions)
Existence
Regularity
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Outline
1
Introduction
2
Stationary problems
Variational mean-field games
Congestion models
3
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Consider the variational problem
Z
min
eH(Du,x) dx.
Td
The Euler-Lagrange equation is the Mean Field Game:
(
H(Du, x) = ln m + H
− div(Dp Hm) = 0,
where the constant H is chosen so that
Diogo Gomes
R
m = 1.
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Consider the variational problem
Z
min
eH(Du,x) dx.
Td
The Euler-Lagrange equation is the Mean Field Game:
(
H(Du, x) = ln m + H
− div(Dp Hm) = 0,
where the constant H is chosen so that
Diogo Gomes
R
m = 1.
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
As proved by L. C. Evans - first order MFG with
quadratic-growth Hamiltonians admit classical solutions.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The stochastic Evans-Aronsson problem is the problem
Z
min
e−∆u+H(Du,x) .
u
Td
The Euler-Lagrange equation is
(
−∆u + H(Du, x) = ln m
−∆m − div(Dp Hm) = 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The stochastic Evans-Aronsson problem is the problem
Z
min
e−∆u+H(Du,x) .
u
Td
The Euler-Lagrange equation is
(
−∆u + H(Du, x) = ln m
−∆m − div(Dp Hm) = 0.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
In low dimension d ≤ 3 under quadratic growth conditions
we (G. and Sanchez-Morgado) also obtained existence of
a smooth solution.
This result was subsequently improved by G. , Patrizi,
Voskanyan for arbitrary dimension.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Running costs such as
L(x, v , m) = mα (x)
|v |2
− V (x)
2
correspond to the congestion MFG:
(
2
u + V (x) + |Du|
2mα = ∆u
m − div(m1−α Du) = ∆m + 1
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Theorem (D. G., H. Mitake)
Under the previous hypothesis, there exists a classical solution
(u, m) with m bounded by below if 0 < α < 1.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The proof relies on an a-priori bound for
1
m
in L∞ .
This bound depends on an explicit cancellation.
It is not known if similar results hold for general models or
time-dependent problems.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The proof relies on an a-priori bound for
1
m
in L∞ .
This bound depends on an explicit cancellation.
It is not known if similar results hold for general models or
time-dependent problems.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The proof relies on an a-priori bound for
1
m
in L∞ .
This bound depends on an explicit cancellation.
It is not known if similar results hold for general models or
time-dependent problems.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
The a-priori bound
The key a-priori bound follows from the identity:
i 1
|Du|2
−
V
dx
·
2mα
rmr
Td
Z h
i
1
dx
−
m − ∆m − div m1−α Du ·
(r + 1 − α)mr +1−α
Td
Z
1
=−
dx.
r +1−α
Td (r + 1 − α)m
Z
h
u − ∆u +
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Variational mean-field games
Congestion models
Which after some surprising cancellations yields
Z
Td
Z
Z
1
|Du|2
|Dm|2
dx
+
dx
dx
+
r
+α
r +2−α
(r + 1 − α)mr +1−α
Td 4rm
Td m
Z
Z
C
C
≤
dx +
dx.
r
r −α
Td rm
Td (r − α)m
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Outline
1
Introduction
2
Stationary problems
Variational mean-field games
Congestion models
3
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Initial-terminal value problem
−ut + H(Dx u, x) =
σ2
∆u + ln m
2
σ2
∆m.
2
Together with smooth initial conditions for m > 0 and terminal
conditions for u.
mt − div(Dp Hm) =
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Initial-terminal value problem
−ut + H(Dx u, x) =
σ2
∆u + ln m
2
σ2
∆m.
2
Together with smooth initial conditions for m > 0 and terminal
conditions for u.
mt − div(Dp Hm) =
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Initial-terminal value problem
−ut + H(Dx u, x) =
σ2
∆u + ln m
2
σ2
∆m.
2
Together with smooth initial conditions for m > 0 and terminal
conditions for u.
mt − div(Dp Hm) =
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Theorem (D. G., E. Pimentel)
The initial-terminal value problem for the logarithmic
nonlinearity with Hamiltonians
γ
H(x, p) = (1 + |p|2 ) 2 + V (x),
with 1 < γ < 45 , admits smooth solutions.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
The proof in three lines
kuk ≤ C + Ck ln mk
k ln mk ≤ C + Ckukβ
"result" (kuk bounded) follows if β < 1.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
The proof in three lines
kuk ≤ C + Ck ln mk
k ln mk ≤ C + Ckukβ
"result" (kuk bounded) follows if β < 1.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
The proof in three lines
kuk ≤ C + Ck ln mk
k ln mk ≤ C + Ckukβ
"result" (kuk bounded) follows if β < 1.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Key estimate
The key estimate is the following
Z
Z
d
1
1
2
≤ Ck|Dp H| kL∞
.
dt Td m
Td m
This estimate makes it possible to prove a bound
Z
| ln m|p ≤ C + Ck|Dp H|2 kpL∞ .
Td
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Key estimate
The key estimate is the following
Z
Z
d
1
1
2
≤ Ck|Dp H| kL∞
.
dt Td m
Td m
This estimate makes it possible to prove a bound
Z
| ln m|p ≤ C + Ck|Dp H|2 kpL∞ .
Td
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
The short time problem
H ∼ |p|γ for large p, and
(
Du
−ut + mα H(x, m
α ) = ∆u
Du
mt − div(Dp H(x, m
α )m) = ∆m.
u(x, T ) and m(x, 0) given
T small.
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Some a-priori bounds
Theorem (D. G. - V. Voskanyan)
Suppose α <
for any r .
2
d−2
and γ < 2. If T is small enough then
Diogo Gomes
Mean field games with congestion
1
m
∈ Lr ,
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
The proof of the theorem relies on the following estimate:
2
, r > 1, then there exists q > r such that
Assume α < d−2
Z
Td
Z tZ
Z tZ |Du|γ
1
1 2
dxdt
+
D
dx
+
dxdt
mr /2 r +ᾱ
mr (x, t)
0 Td m
0 Td
Z tZ
1
dxdt.
≤C+C
q
0 Td m
Diogo Gomes
Mean field games with congestion
Introduction
Stationary problems
Time dependent problems
Logarithmic nonlinearity
Short-time existence for congestion problems
Open questions
General stationary congestion models
Time dependent logarithmic non-linearity γ ≥ 54 .
Long time congestion problem.
Diogo Gomes
Mean field games with congestion