79.1. THE CASE OF UNIQUENESS 2685

Limit condition

Let U be a Banach space dense in V and that if ui ⇀ u in VI and u∗i ∈A(ui) with u∗i ⇀ u∗

in V ′I and t→ Bui (t) is continuous and

supi

supt ̸=s

∥Bui (t)−Bui (s)∥U ′|t− s|α

≤C (79.1.5)

then iflim sup

i→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0 (79.1.6)

it follows that for all v ∈ VI , there exists u∗(v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩V ′I ,VI

≥ ⟨u∗ (v) ,u− v⟩V ′I ,VI(79.1.7)

As to B(ω) , it is k (ω)B where B ∈ L (W,W ′) and is self adjoint and nonnegativewhere k is F0 measurable.

Progressively measurable condition

Condition 79.1.2 For each t ≤ T, if ω → u(·,ω) is Ft measurable into V[0,t], then thereexists a Ft measurable selection of A(u(·,ω) ,ω) into V ′[0,t].

Then there is a theorem. It was Theorem 77.7.4 which gave existence and uniquenessof progressively measurable solutions u to the integral equation.

Theorem 79.1.3 Assume the above conditions, 79.1.1 - , 79.1.7 along with the progressivemeasurability condition 79.1.2. Let u0 be F0 measurable and ω→ B(ω) also F0 measur-able and (t,ω)→X[0,t] (t) f (t,ω) is B ([0, t])×Ft product measurable into V ′ for eacht.

B(ω) = k (ω)B, k (ω)≥ 0,k measurable.

Also let t→ q(t,ω) be continuous and q is progressively measurable into V. Suppose thereis at most one solution to

Bu(t,ω)+∫ t

0z(s,ω)ds =

∫ t

0f (s,ω)ds+Bu0 (ω)+Bq(t,ω) , (79.1.8)

for each ω . Then the solution to the above integral equation u is progressively measurable.Moreover, for each ω , both Bu(t,ω) = B(u(t,ω)) for a.e. t and z(t,ω) ∈ A(u(t,ω) ,ω)for a.e. t. Also, for each a ∈ [0,T ] ,

Bu(t,ω)+∫ t

az(s,ω)ds =

∫ t

af (s,ω)ds+Bu(a,ω)+Bq(t,ω)−Bq(a,ω)

Letting q(t) =∫ t

0 ΦndW defined above with the filtration also being the one obtainedfrom the Wiener process, this implies the following theorem. The σ algebra of progres-sively measurable sets will be denoted by P .