2702 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
Cα,T
∫ t
0
∫ s
0E(∥w1−w2∥2
V
)dτds
+Cα,T
∫Ω
(∫ t
0∥σ (v1)−σ2 (v2)∥2
L2|v1− v2|2W ds
)1/2
dP
Then the usual manipulations and the Lipschitz condition on σ yields an inequality of theform
12
E
(sup
s∈[0,t]|v1− v2|2H (s)
)+α
∫ t
0E(∥v1− v2∥2
V
)ds≤
Cα,T
∫ t
0
∫ s
0E(∥w1−w2∥2
V
)dτds
+Cα,T
∫ t
0E
(sup
τ∈[0,s]|v1− v2|2H (τ)
)ds
Thus, Gronwall’s inequality yields
E
(sup
s∈[0,t]|v1− v2|2H (s)
)+∫ t
0E(∥v1− v2∥2
V
)ds
≤Cα,T
∫ t
0
∫ s
0E(∥w1−w2∥2
V
)dτds (79.3.38)
Now ψ (wi) is defined as w0 +∫ t
0 vids and so ψ (wi) is in C ([0,T ] ;V ) and from the above,
∥ψ (w1)(s)−ψ (w2)(s)∥2V ≤CT
∫ s
0∥v1 (τ)− v2 (τ)∥2
V dτ
and so
sups∈[0,t]
∥ψ (w1)(s)−ψ (w2)(s)∥2V = ∥ψ (w1)−ψ (w2)∥2
C([0,t];V )
≤ CT
∫ t
0∥v1 (s)− v2 (s)∥2
V ds
Using 79.3.38,
E(∥ψ (w1)−ψ (w2)∥2
C([0,t];V )
)≤CT
∫ t
0E(∥v1 (s)− v2 (s)∥2
V
)ds
≤Cα,T
∫ t
0
∫ s
0E(∥w1−w2∥2
V
)dτds≤Cα,T
∫ t
0E(∥w1−w2∥2
C([0,s];V )
)ds
Iterating this inequality shows that for all n large enough,
∥ψn (w1)−ψn (w2)∥2
C([0,t];V ) ≤12∥w1−w2∥2
C([0,t];V )