2508 CHAPTER 74. A MORE ATTRACTIVE VERSION

where C (· · ·) is increasing in each argument, continuous, and C (0) = 0. Thus, enlargingN, for ω /∈ N,

supt∈D⟨BX (t) ,X (t)⟩=C (ω)< ∞ (74.3.15)

where∫

ΩC (ω)dP < ∞. By Lemma 69.4.1, there exists a countable set {ei} of vectors in

V such that ⟨Bei,e j

⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=0⟨Bx,ei⟩2 , Bx =

∞

∑i=1⟨Bx,ei⟩Bei

Thus for t not in a set of measure zero off which BX (t) = B(X (t)) ,

⟨BX (t) ,X (t)⟩=∞

∑i=0⟨BX (t) ,ei⟩2 = sup

m

m

∑k=1⟨BX (t) ,ei⟩2

Now from the formula for BX (t) , it follows that BX is continuous into V ′. For any t /∈ N̂so that (BX)(t) = B(X (t)) in Lq′ (Ω;V ′) and letting tk → t where tk ∈ D, Fatou’s lemmaimplies

E (⟨BX (t) ,X (t)⟩) = ∑i

E(⟨BX (t) ,ei⟩2

)= ∑

ilim inf

k→∞E(⟨BX (tk) ,ei⟩2

)

≤ lim infk→∞

∑i

E(⟨BX (tk) ,ei⟩2

)= lim inf

k→∞E (⟨BX (tk) ,X (tk)⟩)

≤ C(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)In addition to this, for arbitrary t ∈ [0,T ] , and tk→ t from D,

∑i⟨BX (t) ,ei⟩2 ≤ lim inf

k→∞∑

i⟨BX (tk) ,ei⟩2 ≤ sup

s∈D⟨BX (s) ,X (s)⟩

Hence

supt∈[0,T ]

∑i⟨BX (t) ,ei⟩2 ≤ sup

s∈D⟨BX (s) ,X (s)⟩

= sups∈D

∑i⟨BX (s) ,ei⟩2 ≤ sup

t∈[0,T ]∑

i⟨BX (t) ,ei⟩2

It follows that supt∈[0,T ] ∑i ⟨BX (t) ,ei⟩2 is measurable and

E

(sup

t∈[0,T ]∑

i⟨BX (t) ,ei⟩2

)≤ E

(sups∈D⟨BX (s) ,X (s)⟩

)≤ C

(||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω)

)