Projection markovienne de processus stochastiques

tel-00766235, version 1 - 17 Dec 2012
Hence, the limit when ǫ → 0 of
1
ǫ [Uǫ −I]Uth
exists, implying that Uth belongs to the domain of A for any h ∈ D 0 .
Thus,
T
belongs to the domain of A and
T
ǫ
ǫ
due −u Uuh
due −u T
UuAh = A due
ǫ
−u Uuh.
Since U is a contraction semigroup and given the continuity property of
Ut on the space D0 , one may take ǫ → 0 and T → ∞ in (2.15), leading
to ∞
e −t ∞
Uthdt = U0 +A due −u Uuh.
Thus
0
I −A ∞
due −u Uuh(s,x) = U0h(s,x) = h(s,x),
0
yielding h ∈ Im(I − A). We have shown that (Ut,t ≥ 0) generates
a strongly continuous contraction on C0([0,∞[×R d ) with infinitesimal
generator A (see [40, Theorem 2.2, Chapter 4]). The Hille-Yosida theorem
[40, Proposition 2.6, Chapter 1] then implies that for all λ > 0
Im(λ−A) = C0([0,∞[×R d ).
5. Now let pt(x0,dy) be another solution of (2.7). First, considering equation
(3.13) for the particular function g(y) = 1, yields
∀t ≥ 0 pt(x0,dy) = 1,
and pt(x0,dy) has mass 1.
R d
34
0