Slide 11
Slide 11 text
Dynamics
A typical time-dependent MFG system (derived from (9)) has the form
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:
t
2 + H (x, Dx (x, t), ⇢(x, t), t) = 0
⇢t
2 ⇢ divx (⇢DpH (x, Dx (x, t), ⇢(x, t), t)) = 0
⇢ 0,
R
Td
⇢(x, t)dx = 1, t 2 [0, T]
⇢(x, T) = ⇢T
(x), (x, 0) = 0(x, ⇢(x, 0)), x 2 Td .
(10)
Given:
H : Td ⇥ Rd ⇥ X ⇥ R ! R is a periodic in space and convex in
momentum Hamiltonian, where X = R+ or X = L
1(Td ; R+) or
X = R+ ⇥ L
1(Td ; R+). H = L
⇤
is a di↵usion parameter
0 : Td ⇥ X ! R, ⇢T
: Td ! R given initial-terminal conditions
Unknowns:
, ⇢ : Td ⇥ [0, T] ! R
has the form
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