Slide 31
Slide 31 text
Remarks on continuity equations 31
Under these assumptions, given α0
∈ Pc
(Rd)
solving/approximating (15) is quite straightforward, standard
Cauchy Lipschitz framework. Indeed, rewrite (15) as the
fixed-point problem
α = Φα0
(α) with Φα0
(α)t
:= Xα
t #
α0
, t ∈ [0, T]
where Xα
t
is the (globally well-defined) flow of B[α]:
d
dt
Xα
t
(x) = B[αt
](Xα
t
(x)), Xα
0
(x) = x, (t, x) ∈ [0, T]×Rd. (19)
for well chosen λ > 0, Φα0
is a contraction for the distance
dist(α1, α2) = supt∈[0,T ]
e−λtW2
(α1
t
, α2
t
). Existence, uniqueness,
Lipschitz dependence wrt initial condition.
Remarks on continuity equations/3