2562 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

The question at this point is whether u is progressively measurable. From the assumedestimates, the Ito formula, and 76.4.27, the same kind of estimates used earlier show thatthere exists an estimate of the form

∥un∥V ≤C

Therefore, there exists a further subsequence such that

un→ ū weakly in V

It follows from Lemma 76.3.4 that off an enlarged exceptional set of measure zero, stilldenoted as N,

ū(·,ω) = u(·,ω) in Vω

Hence we can assume that u is progressively measurable into V . It follows that Bu isprogressively measurable into W ′.

Thus also(t,ω)→

∫ t

0ξ (s,ω)ds

is progressively measurable into V ′.Of course the next task is to identify ξ . This is always a problem even in the non

stochastic case. Here it is especially difficult because in order to identify ξ we need to usethe implicit Ito formula which only holds if ξ is sufficiently measurable. However, we haveobtained ξ as a weak limit for fixed ω . Therefore, this is a significant issue. In stochasticevolution problems where B = I this is not as difficult because one gets ξ as a weak limitin V and then ξ is progressively measurable. We cannot do it this way and still get the bestresults in which there is a solution to the integral equation which holds for all t off a set ofmeasure zero because of the degenerate nature of the operator B. However, ξ is only anequivalence class of functions. We show in the next lemma that there exists a representativeof this equivalence class for each ω off an exceptional set of measure zero such that theresulting ξ is progressively measurable. This will enable us to use the implicit Ito formulaand indentify ξ .

The following lemma will allow the use of the Ito formula and eventually identify ξ .

Lemma 76.4.4 Enlarging the exceptional set, one can assume that ξ is also progressivelymeasurable. In fact, if

ξ n ≡∫ t

t−(1/n)ξ ds

is known to be progressively measurable, ξ (t,ω) ≡ 0 for t < 0, then there exists a set ofmeasure zero N such that for ω /∈ N,ξ (t,ω) = ξ̄ (t,ω) for all t off a set of measure zeroand ξ̄ is progressively measurable.

Proof: Defineξ n ≡ n

∫ t

t−(1/n)ξ ds

where ξ is defined to be zero for t ≤ 0. Then by what was just shown, this is progressivelymeasurable. Also, standard approximate identity arguments verify that for each ω,ξ n →