79.1. THE CASE OF UNIQUENESS 2689
+C∫
Ω
(∫ t
0∥Bun∥2
W ∥Φn∥2L2
ds)1/2
dP
Now ∥Bw∥= sup∥v∥≤1 ⟨Bw,v⟩ ≤ ⟨Bw,w⟩1/2. Also∫ t
0 ∥Φn∥2L2
ds≤∫ T
0 ∥Φ∥2L2
ds and so theabove inequality implies
E
(sup
s∈[0,t]⟨Bun,un⟩(s)
)+E
(∫ t
0∥un (s)∥p
V ds)
≤ C ( f ,λ ,c,Φ)+C∫
Ω
sups∈[0,t]
⟨Bun,un⟩1/2 (s)(∫ t
0∥Φ∥2
L2
)1/2
dP
Then adjusting the constants yields
12
E
(sup
s∈[0,T ]⟨Bun,un⟩(s)
)+E
(∫ T
0∥un (s)∥p
V ds)
≤C+C∫
Ω
∫ T
0∥Φ∥2
L2dtdP≡C (79.1.15)
If needed, you could use a stopping time to be sure that E(
sups∈[0,T ] ⟨Bun,un⟩(s))< ∞
and then let it converge to ∞.From the integral equation,
Bun (t)−Bum (t)+∫ t
0zn− zmds = B
∫ t
0(Φn−Φm)dW
Then using the monotonicity assumption and the Ito formula,
12⟨Bun−Bum,un−um⟩(t)≤ λ
∫ t
0⟨Bun−Bum,un−um⟩dss
+∫ t
0⟨B(Φn−Φm) ,Φn−Φm⟩d +
∫ t
0
((Φn−Φm)◦ J−1)∗B(un−um)◦ JdW
and so, from Gronwall’s inequality, there is a constant C which is independent of m,n suchthat
⟨Bun−Bum,un−um⟩(t)≤CMnm (t)≤CM∗nm (T )+C∫ t
0∥Φn−Φm∥2
L2ds
where Mnm refers to that local martingale on the right. Thus also
supt∈[0,T ]
⟨Bun−Bum,un−um⟩(t)≤CMnm (t)≤CM∗nm (T )+C∫ T
0∥Φn−Φm∥2
L2ds
(79.1.16)Taking the expectation and using the Burkholder Davis Gundy inequality again in a similarmanner to the above,
E
(sup
t∈[0,T ]⟨Bun−Bum,un−um⟩(t)
)≤C
∫Ω
∫ T
0∥Φn−Φm∥2
L2dtdP