2556 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS
Then from Gronwall’s inequality, and adjusting the constants,
⟨B(un−um) ,un−um⟩(t)+∫ t
0∥un−um∥α
U ds
≤ C (u0n−u0m,Φn−Φm)+C (T )M∗mn (t)
where the expectation of the first constant on the right converges to 0 as m,n→ ∞. Here
M∗nm (t) = sups∈[0,t]
|Mnm (t)|
Since M∗ is increasing, this implies that after adjusting constants,
sups∈[0,t]
(⟨B(un−um) ,un−um⟩(s)+
∫ t
0∥un−um∥α
U ds)
≤ C (u0n−u0m,Φn−Φm)+C (T )M∗mn (t)
Then taking expectations and using the Burkholder Davis Gundy inequality,
E
(sup
s∈[0,t]
(⟨B(un−um) ,un−um⟩(s)+
∫ t
0∥un−um∥α
U ds))
≤ C (u0n−u0m,Φn−Φm)+
C (T )∫
Ω
(∫ t
0∥Φn−Φm∥2
L2(Q1/2U,W) ⟨B(un−um) ,un−um⟩ds)1/2
dP
≤ Cn,m +2C∫
Ω
sups∈[0,t]
⟨B(un−um) ,un−um⟩1/2 (s) ·
(∫ t
0∥Φn−Φm∥2
L2(Q1/2U,W)
)1/2
dP
Then adjusting the constants,
E
(sup
s∈[0,t]
(⟨B(un−um) ,un−um⟩(s)+
∫ t
0∥un−um∥α
U ds))
≤Cn,m +C∫
Ω
∫ T
0∥Φn−Φm∥2
L2(Q1/2U,W) dtdP≡Cn,m
where Cn,m→ 0 as n,m→ ∞. In particular, it is true for t = T
E
(sup
s∈[0,T ]⟨B(un−um) ,un−um⟩(s)+
∫ T
0∥un−um∥α
U ds
)≤Cn,m
Then
P
(sup
s∈[0,T ]⟨B(un−um) ,un−um⟩(s)+
∫ T
0∥un−um∥α
U ds≥ λ
)≤
Cn,m
λ