76.4. THE GENERAL CASE 2557
Now take a subsequence such that if m> nk,Cnk,m < 4−k. Then the above inequality impliesthat
P(
sups∈[0,T ] ⟨B(un−um) ,un−um⟩(s)+∫ T
0
∥∥unk −unk+1
∥∥α
U ds≥ 2−k
)≤ 4−k
2−k = 2−k
and so, by the Borel Cantelli lemma, there is a set of measure zero N including all earlierexceptional sets of measure zero such that for ω /∈ N,
sups∈[0,T ]
⟨B(un−um) ,un−um⟩(s)+∫ T
0
∥∥unk −unk+1
∥∥α
U ds < 2−k
for all k large enough. We will denote this new subsequence by {un} . Thus for suchω, it follows that {Bun} is a Cauchy sequence in C
(NC
ω ,W′) for Nω an exceptional set of
measure zero where B(un−um)(t) ̸=B(un (t)−um (t)) and also {un} is a Cauchy sequencein Lα (0,T,U). It follows
Bun→ z strongly in C(NC
ω ,W′) with uniform norm (76.4.28)
limm,n→∞
sups∈[0,T ]
⟨B(un−um) ,un−um⟩(s) = 0 (76.4.29)
There exists u ∈ Lα (0,T,U) such that for ω /∈ N,
∥un−u∥Lα (0,T,U)→ 0, un (t,ω)→ u(t,ω) for a.e.t in U (76.4.30)
Of course a technical issue is the fact that B is a degenerate operator which might notbe invertible. In the above limit, we do not know that z = Bu for some u. We resolve thisissue by obtaining pointwise estimates for a given ω and then pass to a limit. After this,a time integration will give the desired result. There are easier ways to do this if B is notdegenerate.
From now on, this or a subsequence of this one will be the sequence of interest. Returnto 76.4.27 and use the Ito formula again. Thus using the estimates,
12⟨Bun,un⟩(t)−
12⟨Bu0n,u0n⟩+δ
∫ t
0∥un∥p
V ds−λ
∫ t
0⟨Bun,un⟩ds
=12
∫ t
0⟨BΦn,Φn⟩ds+
∫ t
0c(s,ω)ds+
∫ t
0⟨ f ,un⟩ds+Mn (t)
where Mn (t) is a local martingale whose quadratic variation satisfies
[Mn] (t)≤C∫ t
0∥Φn∥2
L2∥Bun∥2
W ds
Then adjusting the constants,
⟨Bun,un⟩(t)+∫ t
0∥un∥p
V ds≤C (u0n,Φn, f ,c)+CM∗n (t)
where the expectation of the first constant on the right is no larger than a constant C whichis independent of n. Since the right term is increasing in t,
sups∈[0,t]
⟨Bun,un⟩(s)+∫ t
0∥un∥p
V ds≤C (u0n,Φn, f ,c)+CM∗n (t) (76.4.31)