2558 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
Now using the Burkholder Davis Gundy inequality as before and taking the expectation,
E
(sup
s∈[0,t]⟨Bun,un⟩(s)
)+E
∫ t
0∥un∥p
V ds
≤ C+C∫
Ω
(∫ t
0∥Φn∥2
L2∥Bun∥2
W ds)1/2
dP
≤ C+C∫
Ω
(∫ t
0∥Φn∥2
L2⟨Bun,un⟩ds
)1/2
dP
≤ C+C∫
Ω
sups∈[0,t]
⟨Bun,un⟩1/2 (s)(∫ t
0∥Φn∥2
L2ds)1/2
dP
Then adjusting the constants and using the approximation properties of Φn given above,there is a constant C independent of n, t ≤ T such that
E
(sup
s∈[0,t]⟨Bun,un⟩(s)
)+E
∫ t
0∥un∥p
V ds≤C
In particular
E
(sup
s∈[0,T ]⟨Bun,un⟩(s)
)+E
∫ T
0∥un∥p
V ds≤C (76.4.32)
Next use monotonicity to obtain
12⟨Bur−Buq,ur−uq
⟩(t) ≤ 1
2
∫ t
0
((Φr−Φq)◦ J−1)∗B(ur−uq)◦ JdW
+Cλ
∫ t
0
⟨Bur−Buq,ur−uq
⟩ds+
∫ t
0
∥∥Φr−Φq∥∥2 ds
and so, from Gronwall’s inequality, there is a constant C which is independent of r,q suchthat ⟨
Bur−Buq,ur−uq⟩(t)≤CMrq (t)≤CM∗rq (T )+C
∫ t
0
∥∥Φr−Φq∥∥2 ds
where Mrq refers to that local martingale on the right. Thus also
supt∈[0,T ]
⟨Bur−Buq,ur−uq
⟩(t)≤CMrq (t)≤CM∗rq (T )+C
∫ T
0
∥∥Φr−Φq∥∥2 ds (76.4.33)
Taking the expectation and using the Burkholder Davis Gundy inequality again, and similarestimates to the above, using appropriate stopping times as needed, we obtain
E
(sup
t∈[0,T ]
⟨Bur−Buq,ur−uq
⟩(t)
)≤C
∫Ω
∫ T
0
∥∥Φr−Φq∥∥2 dtdP