2552 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

≥ e−2λ t(

δ

∥∥∥eλ tu∥∥∥p

V− c(t,ω)

)≥ e−2λ t

(δ

∥∥∥eλ tu∥∥∥p

V− eλ pte−λ ptc(t,ω)

)≥ e−2λ tepλ t

(δ ∥u∥p

V − e−λ ptc(t,ω))≥ δ ∥u∥p

V − e−λ ptc(t,ω)

which is of the right form.Similarly

∥λBw+Aλ (t,w,ω)∥V ′ ≤ ∥λBw∥V ′ +∥∥∥e−λ tA

(t,eλ tw,ω

)∥∥∥V ′

≤ λ ∥B∥∥w∥V + e−λ t∥∥∥A(

t,eλ tw,ω)∥∥∥

V ′

≤ λ ∥B∥∥w∥V + e−λ tk∥∥∥eλ tw

∥∥∥p−1+ e−λ tc1/p′ (t,ω)

Since p≥ 2, this is no larger than

≤ (λ ∥B∥)p/(p−p′) +∥w∥p−1V + e(p−1)λ te−λ tk∥w∥p−1

V + e−λ tc1/p′ (t,ω)

≤(

e(p−2)λT k+1)∥w∥p−1

V + e−λ tc1/p′ (t,ω)+(λ ∥B∥)p/(p−p′)

≡ k̄∥w∥p−1V + c̄(t,ω)1/p′

Now note that w is a solution to

B(

w− e−λ (·)q)′+λBw+ e−λ (·)A

(t,eλ (·)w,ω

)= e−λ (·) f (·,ω)+λBe−λ (·)q(·,ω) in Vω

B(

w− e−λ (·)q)(0) = Bu0

if and only if u(t)≡ eλ tw(t) is a solution to

(B(u−q))′+A(t,u,ω) = f (·,ω) , B(u−q)(0) = Bu0

Thus the necessary uniqueness condition holds for the initial value problem for w andhence it follows from Proposition 76.3.6 that there exists a unique progressively measurablesolution to the initial value problem for w and hence a unique progressively measurablesolution to the above one for u.

Now suppose the situation of the above corollary and let E be a separable Hilbert spacewhich is dense in V and let

Φ ∈ L2([0,T ]×Ω,L2

(Q1/2U,E

)), Φ being progressively measurable,

so that one can consider the stochastic integral∫ t

0 ΦdW. Let

τn ≡ inf{

t :∥∥∥∥∫ t

0ΦdW

∥∥∥∥E> 2n

}