2560 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
Taking a further subsequence if needed, one can also have
P(
supt|Mk (t)−M (t)|> 1
k
)≤ 1
2k
and so by the Borel Cantelli lemma, there is a set of measure zero such that off this set,supt |Mk (t)−M (t)| converges to 0. Hence for such ω,M∗k (T ) is bounded independent ofk. Thus for ω off a set of measure zero, 76.4.31 implies that for such ω,
sups∈[0,T ]
⟨Bur,ur⟩(s)+∫ T
0∥ur (s)∥p
V ds≤C (ω)
where C (ω) does not depend on the index r, this for the subsequence just described whichwill be the sequence of interest in what follows. Using the boundedness assumption for A,one also obtains an estimate of the form
sups∈[0,T ]
⟨Bur,ur⟩(s)+∫ T
0∥ur (s)∥p
V ds+∫ T
0∥zr∥p′
V ′ ≤C (ω) (76.4.36)
Lemma 76.4.2 There is a subsequence, still indexed by n and a set of measure zero N,containing all the preceding sets of measure zero such that for ω /∈ N,
sups∈[0,T ]
⟨Bun,un⟩(s)+∫ T
0∥un∥p
V ds≤C (ω)< ∞
From the theory of the stochastic integral, there is a further subsequence of the abovesuch that ∫ t
0ΦndW →
∫ t
0ΦdW strongly in C ([0,T ] ,W )
for all ω off a set of measure zero. Enlarge the exceptional set N and only use subsequencesof this one so that both the above estimate in the lemma and the above convergence holdfor ω /∈ N. Recall the integral equation solved.
Bun (t)−Bu0n +∫ t
0A(s,un,ω)ds =
∫ t
0f (s,ω)ds+B
∫ t
0ΦndW (76.4.37)
Thus (Bun−B
∫ (·)
0ΦndW −Bu0n
)′+Aun = f
Then for ω /∈ N, a subsequence of the one for which the above lemma holds, stilldenoted as {un} yields the following convergences,
un→ u weakly in Vω (76.4.38)
Aun ⇀ ξ weakly in V ′ω (76.4.39)(Bun−B
∫ (·)
0ΦndW −Bu0n
)′⇀ ζ weakly in V ′ω (76.4.40)