2510 CHAPTER 74. A MORE ATTRACTIVE VERSION

74.4 A Simplification Of The FormulaThis estimate in Lemma 74.3.2 also provides a way to simplify one of the formulas derivedearlier in the case that X0 ∈ Lp (Ω,V ) so that X−X0 ∈ Lp ([0,T ]×Ω,V ). Refer to 74.3.11.One term there is

⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩

Also,⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩

≤ 2⟨B(X (t1)−X0) ,X (t1)−X0⟩+2⟨BM (t1) ,M (t1)⟩It was observed above that 2⟨BM (t1) ,M (t1)⟩ → 0 a.e. and also in L1 (Ω) as k→∞. Applythe above lemma to ⟨B(X (t1)−X0) ,X (t1)−X0⟩ using [0, t1] instead of [0,T ] . The new X0equals 0. Then from the estimate 74.3.8, it follows that

E (⟨B(X (t1)−X0) ,X (t1)−X0⟩)→ 0

as k→ ∞. Taking a subsequence, we could also assume that

⟨B(X (t1)−X0) ,X (t1)−X0⟩ → 0

a.e. ω as k→ ∞. Then, using this subsequence, it would follow from 74.3.11,

⟨BX (tm) ,X (tm)⟩−⟨BX0,X0⟩= e(k)+

2∫ tm

0⟨Y (u) ,X r

k (u)⟩du++2∫ tm

0

⟨BX l

k ,dM⟩

+m−1

∑j=0

⟨B(M(t j+1

)−M (t j)

),M(t j+1

)−M (t j)

⟩−

m−1

∑j=1

⟨B(∆X (t j)−∆M (t j)) ,∆X (t j)−∆M (t j)

⟩(74.4.17)

where e(k)→ 0 in L1 (Ω) and a.e. ω and

∆X (t j)≡ X(t j+1

)−X (t j)

∆M (t j) being defined similarly. Note how this eliminated the need to consider the term

⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩

in passing to a limit. This is a very desirable thing to be able to conclude.Can you obtain something similar even in case X0 is not assumed to be in Lp (Ω,V )?

Let X0k ∈ Lp (Ω,V )∩L2 (Ω,W ) ,X0k → X0 in L2 (Ω,W ) . Then from the usual argumentsinvolving the Cauchy Schwarz inequality,

⟨B(X (t1)−X0) ,X (t1)−X0⟩1/2 ≤ ⟨B(X (t1)−X0k) ,X (t1)−X0k⟩1/2

+⟨B(X0k−X0) ,X0k−X0⟩1/2