2676 CHAPTER 78. A DIFFERENT APPROACH

Theorem 78.3.2 Suppose the conditions on A,78.3 - 78.3. Also let u0 be measurableinto W and f measurable into V ′. Let B ∈ L (W,W ′) be nonnegative and self adjointas described above. Let σ > 0 be small. Then there exist functions u,u∗ measurable intoV[0,T−σ ]×V ′[0,T−σ ] such that u∗ (ω) ∈ A

(X[0,T−σ ]u(ω) ,ω

)for each ω and for t ≤ T −σ ,

for each ω,

Bu(t)−Bu0 +∫ t

0u∗ (s)ds =

∫ t

0f (s)ds

Note that if for a given ω there is a unique solution to the evolution equation, thenwe can obtain the solution on (0,T ). However, σ was totally arbitrary so it seems likethere is not much difference between the above and the obtimum solution. However, onecould also index the above solutions relative to σ , take an appropriate extension of eachon (T −σ ,T ) and get similar estimates and pass to a limit as above as σ → 0 and therebyobtain a measurable solution valid on (0,T ). This time, it will be clear that Lu,Luσ are bothin V ′ so a monotonicity condition will hold for L without the delicate argument given abovewhich caused a smaller interval to be considered. Thus the following corollary will hold ifenough additional details are considered. The issue does not seem sufficiently significantto justify the consideration of these details.

Corollary 78.3.3 In the situation of Theorem 78.3.2 there exists the same kind of measur-able solution valid on (0,T ). This time, u∗,u are measurable into V and V ′ respectively.

One can give a very interesting generalization of Theorem 78.3.2.

Theorem 78.3.4 In the context of Theorem 78.3.2,let q(t,ω) be a product measurablefunction into V such that t→ q(t,ω) is continuous, q(0,ω) = 0.

Then for each small σ , there exists a solution u of the integral equation

Bu(t,ω)+∫ t

0u∗ (s,ω)ds =

∫ t

0f (s,ω)ds+Bu0 (ω)+Bq(t,ω) , t ≤ T −σ

where (t,ω) → u(t,ω) is product measurable. Moreover, for each ω , it follows thatBu(t,ω) = B(u(t,ω)) for a.e. t and u∗ (·,ω) ∈ A(u(·,ω) ,ω) for a.e. t, u∗ is productmeasurable into V ′. Also, for each a ∈ [0,T −σ ] ,

Bu(t,ω)+∫ t

au∗ (s,ω)ds =

∫ t

af (s,ω)ds+Bu(a,ω)+Bq(t,ω)−Bq(a,ω)

Proof: Define a stopping time

τr (ω)≡ inf{t : |q(t,ω)|> r}

Then this is the first hitting time of an open set by a continuous random variable and so itis a valid stopping time. Then for each r, let

Ar (ω,w)≡ A(ω,w+qτr (·,ω)) ,