2566 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

≤∫

Ω

supt∈[0,T ]

⟨Bun,un⟩1/2 (t)

((∫ T

0∥Φn−Φ∥2

L2dt)1/2

)dP

≤

(∫Ω

supt∈[0,T ]

⟨Bun,un⟩(t)

)1/2(∫Ω

∫ T

0∥Φn−Φ∥2

L2dt)1/2

From 76.4.32

≤C(∫

Ω

∫ T

0∥Φn−Φ∥2

L2dt)1/2

which converges to 0.Return now to the equation solved by un in 76.4.37. Apply the Ito formula to this one.

This yields for a.e. t,

12⟨Bun (t) ,un (t)⟩−

12⟨Bu0n,u0n⟩+

∫ t

0⟨A(ω)un,un⟩ds =

12

∫ t

0⟨BΦn,Φn⟩ds

+∫ t

0⟨ f ,un⟩ds+

∫ t

0

(Φn ◦ J−1)∗Bun ◦ JdW (76.4.45)

Assume without loss of generality that T is not in the exceptional set. If not, consider allT ′ close to T such that T ′ is not in the exceptional set.∫ T

0⟨(λB+A(ω))un,un⟩ds

=12⟨Bu0n,u0n⟩−

12⟨Bun (T ) ,un (T )⟩+

∫ T

0⟨ f ,un⟩ds

+∫ T

0

(Φn ◦ J−1)∗Bun ◦ JdW +

12

∫ T

0⟨BΦn,Φn⟩ds+

∫ T

0⟨λBun,un⟩ds

Now it follows from 76.4.42 applied to t = T and the above lemma that

lim supn→∞

∫ T

0⟨(λB+A(ω))un,un⟩ds

≤ 12⟨Bu0,u0⟩−

12⟨Bu(T ) ,u(T )⟩+

∫ T

0⟨ f ,u⟩ds

+∫ T

0

(Φ◦ J−1)∗Bu◦ JdW +

12

∫ T

0⟨BΦ,Φ⟩ds+

∫ T

0⟨λBu,u⟩ds

and from 76.4.43, the expression on the right equals∫ T

0 ⟨λBu+ξ ,u⟩ds. Hence

lim supn→∞

∫ T

0⟨(λB+A(ω))un,un⟩ds≤

∫ T

0⟨λBu+ξ ,u⟩ds

Then since λB+A(ω) is monotone and hemicontinuous, it is type M and so this requiresA(ω)u = ξ .