2512 CHAPTER 74. A MORE ATTRACTIVE VERSION
Lemma 74.5.1 There exists a subsequence still denoted with the subscript k and an en-larged set of measure zero N including the earlier one such that BX l
k (t) ,BX rk (t) also con-
verges pointwise a.e. t to BX (t) in W ′ and X lk (t) ,X
rk (t) converge pointwise a.e. in V to
X (t) for ω /∈ N as well as having convergence of X lk (·,ω) to X (·,ω) in Lp ([0,T ] ;V ) and
BX lk (·,ω) to BX (·,ω) in L2 ([0,T ] ;W ′).
Proof: To see that such a sequence exists, let nk be such that∫Ω
∫ T
0
∥∥BX rnk(t)−BX (t)
∥∥2W ′
dtdP+∫
Ω
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dtdP+
∫Ω
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dtdP+
∫Ω
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
VdtdP < 4−k.
Then
P(∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dt +
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
Vdt > 2−k
)≤ 2k
(4−k)= 2−k
and so by Borel Cantelli lemma, there is a set of measure zero N such that if ω /∈ N,∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥X rnk(t)−X (t)
∥∥pV
dt+
∫ T
0
∥∥∥BX lnk(t)−BX (t)
∥∥∥2
W ′dt +
∫ T
0
∥∥∥X lnk(t)−X (t)
∥∥∥p
Vdt ≤ 2−k
for all k large enough. By the usual proof of completeness of Lp, it follows that X lnk(t)→
X (t) for a.e. t, this for each ω /∈ N, a similar assertion holding for X rnk
. Also BX lnk(t)→
BX (t) for a.e. t, similar for BX rnk(t). Denote these subsequences as
{X r
k
}∞
k=1 ,{
X lk
}∞
k=1 .Define the following stopping time.
τ p ≡ inf
{t : ∑
i⟨BX (t) ,ei⟩2 > p
}
By Lemma 74.3.2 τ p = ∞ for all p large enough off some set of measure zero. Also,BX (t)(ω) = B(X (t,ω)) for a.e. t and so for a.e.t,⟨BX (t) ,X (t)⟩= ∑i ⟨BX (t) ,ei⟩2 and so∥BXτ p (t)∥W ′ ≤ ∥B∥
√p for a.e.t. Hence BXτ p ∈ L∞ ([0,T ]×Ω,W ′).
Lemma 74.5.2 The process∫ t
0⟨BX l
k ,dM⟩
converges in probability as k→∞ to the integral∫ t0 ⟨BX ,dM⟩ which is a local martingale. Also, there is a subsequence and an enlarged set
of measure zero N such that for ω not in this set, the convergence is uniform on [0,T ].