2548 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
and so there exists a set of measure zero Nh such that for ω /∈ Nh, the following equationholds in V ′ω
1h(I− τh)(Buh (·,ω))+ Ā(ω)(uh (·,ω)) = f (·,ω)
Let h denote a sequence converging to 0 and let N be a set of measure zero which includes∪hNh.
Letting uh ∈ V be the above solution to 76.3.18, it also follows from the above esti-mates 76.3.14 - 76.3.15 that for ω off N, ∥uh (·,ω)∥Vω
is bounded independent of h. Thus,for such ω off this set, there exists a subsequence still called uh such that the followingconvergences hold.
uh ⇀ u in Vω
Ā(ω)uh ⇀ ξ in V ′ω
1h(I− τh)(Buh)⇀ ζ in V ′ω
First we need to identify ζ . Let φ ∈C∞ ([0,T ]) where φ = 0 near T and let w ∈V . Then⟨∫ T
0ζ φ ,w
⟩= lim
h→0
⟨∫ T
0
1h(I− τh)(Buh) ,wφ
⟩
= limh→0
⟨∫ T
0
Buh (t)h
φ (t)−∫ T
h
Buh (t−h)h
φ (t) ,w⟩
= limh→0
⟨∫ T
0
Buh (t)h
φ (t)−∫ T−h
0
Buh (t)h
φ (t +h) ,w⟩
= limh→0
(⟨∫ T−h
0Buh
φ (t)−φ (t +h)h
,w⟩+∫ T
T−h
Buh (t)h
φ (t))
=
⟨−∫ T
0Bu(t)φ
′ (t) ,w⟩
Since this holds for all φ ∈ C∞c (0,T ) , it follows that ζ = (Bu)′. Hence letting φ be an
arbitrary function in C∞ ([0,T ]) which equals zero near T, this implies from the above that⟨−∫ T
0ζ φ ,w
⟩=
⟨∫ T
0Bu(t)φ
′ (t) ,w⟩
=
⟨∫ T
0
(Bu(0)+
∫ t
0(Bu)′ (s)ds
)φ′ (t) ,w
⟩
=∫ T
0⟨Bu(0) ,w⟩φ ′ (t)dt +
⟨∫ T
0
∫ t
0(Bu)′ (s)dsφ
′ (t) ,w⟩
=−⟨Bu(0) ,w⟩φ (0)+⟨∫ T
0(Bu)′ (s)
∫ T
sφ′ (t)dtds,w
⟩=−⟨Bu(0) ,w⟩φ (0)−
⟨∫ T
0(Bu)′ (s)φ (s)ds,w
⟩