2630 CHAPTER 77. STOCHASTIC INCLUSIONS

Also12⟨Buε ,uε⟩(T )−

12⟨Bu0,u0⟩+ ⟨u∗ε ,uε⟩+ ε ⟨Fuε ,uε⟩= ⟨ f ,uε⟩

Assume T is such that

⟨Buε ,uε⟩(T ) = ⟨B(uε (T )) ,uε (T )⟩ , Buε (T ) = B(uε (T ))

for all ε in the sequence converging to 0 and also

Bu(T ) = B(u(T )) ,⟨Bu,u⟩(T ) = ⟨B(u(T )) ,u(T )⟩

If not, carry out the argument for T̂ close to T for which this condition does hold. We havethe integral equation

Buε (t)−Bu0 +∫ t

0u∗ε ds+

∫ t

0εFuε ds =

∫ t

0f ds

and so Buε (t) converges to Bu(t) in U ′ weakly. This follows right away from the conver-gence of (Buε)

′ in the above. Also from the above equation,

Bu(t)−Bu0 +∫ t

0u∗ds =

∫ t

0f ds

Thus

Bu(0) = Bu0

(Bu)′+u∗ = f in U ′

Since V ′ ⊆U ′,

12⟨Bu,u⟩(t)− 1

2⟨Bu0,u0⟩+

∫ t

0⟨u∗,u⟩V ′,V ds =

∫ t

0⟨ f ,u⟩ds

Also12⟨Buε ,uε⟩(t)−

12⟨Bu0,u0⟩+

∫ t

0⟨u∗ε ,uε⟩V ′,V ds

+∫ t

0⟨εFuε ,uε⟩ds =

∫ t

0⟨ f ,uε⟩ds (77.5.56)

Now let {ei} be the vectors of Lemma 77.3.4 where these are in U . Thus

⟨Buε ,uε⟩(T ) =∞

∑i=1⟨B(uε (T )) ,ei⟩2