79.3. AN EXAMPLE 2701
where σ will be of the form given in Theorem 79.2.1. To show existence of a solution tothe above, note that by Theorem 79.2.1, if w ∈ L2 (Ω,C ([0,T ] ;V )) with w progressivelymeasurable, then there exists a unique solution v to the above integral equation 79.3.36.Let ψ (w)(t) = w0 +
∫ t0 vds. Suppose now you have wi ∈ L2 (Ω,C ([0,T ] ;V )) with vi being
the solution corresponding to the fixed wi. Then
v1 (t)− v2 (t)+α
∫ t
0L(v1− v2)(s)ds+
∫ t
0L(w1−w2)ds
+∫ t
0(Nw1−Nw2)ds =
∫ t
0(σ (v1)−σ2 (v2))dW
From the Ito formula,
|v1− v2|2H +2α
∫ t
0∥v1− v2∥2
V ds+∫ t
0
⟨∫ s
0L(w1−w2)dτ,v1 (s)− v2 (s)
⟩ds+
∫ t
0
⟨∫ s
0N (w1)−N (w2)dτ,v1 (s)− v2 (s)
⟩ds = 2
∫ t
0∥σ (v1)−σ2 (v2)∥2
L2ds+M (t)
where the quadratic variation of M (t) is∫ t
0∥σ (v1)−σ2 (v2)∥2
L2|v1− v2|2H ds
Using the estimates and the Lipschitz condition on Q, and letting C be a generic constantdepending on the subscripts, routine manipulations yield an inequality of the form
|v1− v2|2H +2α
∫ t
0∥v1− v2∥2
V ds≤ ε
∫ t
0∥v1 (s)− v2 (s)∥2
V ds
+Cε
∫ t
0
∫ s
0∥w1−w2∥2
V dτds+C∫ t
0|v1− v2|2H ds+M∗ (t)
Letting ε ≤ α, and modifying the constants as needed, Gronwall’s inequality yields aninequality of the form
|v1− v2|2H +α
∫ t
0∥v1− v2∥2
V ds≤Cα,T
∫ t
0
∫ s
0∥w1−w2∥2
V dτds+Cα,T M∗ (t)
Modifying the constants again if necessary, one obtains
sups∈[0,t]
|v1− v2|2H (s)+α
∫ t
0∥v1− v2∥2
V ds≤Cα,T
∫ t
0
∫ s
0∥w1−w2∥2
V dτds+Cα,T M∗ (t)
and now, by the Burkholder Davis Gundy inequality,
E
(sup
s∈[0,t]|v1− v2|2H (s)
)+α
∫ t
0E(∥v1− v2∥2
V
)ds≤