79.2. REPLACING Φ WITH σ (u) 2697
so that an equivalent norm on W is ⟨Bu,u⟩1/2. Assume the monotonicity assumption: forzi ∈ A(ui,ω) ,
⟨λBu1 + z1− (λBu2 + z2) ,u1−u2⟩ ≥ δ ∥u1−u2∥2W (79.2.32)
for all λ large enough. Suppose σ (t,ω,u)∈L2(Q1/2U,W
)and has the growth properties
∥σ (t,ω,u)∥W ≤C+C∥u∥W
∥σ (t,ω,u1)−σ (t,ω,u2)∥L2(Q1/2U,W) ≤ K ∥u1−u2∥W (79.2.33)
and (t,ω)→ σ (t,ω,u(t,ω)) is progressively measurable whenever (t,ω)→ u(t,ω) is.Then there exists a unique solution u to the integral equation
Bu(t)−Bu0 +∫ t
0zds =
∫ t
0f ds+B
∫ t
0σ (s, ·,u)dW
The case of most interest is the usual one where V ⊆W = W ′ ⊆ V ′, the case of aGelfand triple in which B is the identity. We also are assuming that A does not have memoryterms. We obtain (t,ω)→ σ (t,ω,u(t,ω)) progressively measurable if (t,ω)→ σ (t,ω,u)is progressively measurable for each u ∈W thanks to continuity in u which comes from79.2.33.
Proof: For given w ∈ L2 (Ω,C ([0,T ] ,W )) each w being progressively measurable, de-fine u = ψ (w) as the solution to the integral equation
Bu(t,ω)−Bu0 (ω)+∫ t
0z(s,ω)ds =
∫ t
0f (s,ω)ds+B
∫ t
0σ (w)dW
which exists by above assumptions and Corollary 79.1.8. Here we write σ (w) for shortinstead of σ (t,ω,w). From Theorem 73.7.2, ⟨Bu,u⟩ is continuous hence bounded and soBu is in L∞ ([0,T ] ,W ′) which implies u∈ L∞ ([0,T ] ,W ) . Since Bu is essentially bounded inW ′ and equals a continuous function in V ′, it follows from density considerations that Buican be re defined on a set of meausure zero to be weakly continuous into W ′ hence weaklycontinuous into V ′. This re definition must yield the integral equation because all otherterms than Bu are continuous. However, this implies that if we let u(t) = B−1 (Bu)(t) , thenu is weakly continuous into W. By continuity of ∥u∥2 ≡ ⟨Bu,u⟩ , this shows that in fact,u is continuous into W thanks to the uniform continuity of the Hilbert space norm. Thusu(·,ω) ∈C ([0,T ] ,W ) .
Then from the estimates,
⟨Bu,u⟩(t)−⟨Bu0,u0⟩+δ
∫ t
0∥u∥p
V ds = 2∫ t
0⟨ f ,u⟩ds+C (b3,b4,b5)
+λ
∫ t
0⟨Bu,u⟩ds+
∫ t
0⟨Bσ (w) ,σ (w)⟩L2
ds+2M∗ (t)
≤ 2∫ t
0⟨ f ,u⟩ds+C (b3,b4,b5)+λ
∫ t
0⟨Bu,u⟩ds+
∫ t
0
(C+C∥w∥2
W
)ds+2M∗ (t)