2594 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Then one can obtain an estimate and pass to a limit as ε → 0 obtaining the followingconvergences.

εvε → 0 strongly in V

Bvε → Bv weak ∗ in L∞(0,T,L2 (

Ω,W ′))

uε (t)→ u(t) weak ∗ in L∞(0,T,L2 (Ω,V )

)Then one can simply pass to a limit in the approximate integral equation and obtain, thanksto linearity considerations, that

Bv(t)−Bv0 +∫ t

0A(u)ds =

∫ t

0f ds+

∫ t

0ΦdW, u(t) = u0 +

∫ t

0v(s)ds (76.7.73)

the equation holding in V ′. Thus for a.e. ω, the above holds for a.e. t. It is possible towork harder and have the equation holding for all t. This involves using the other form ofthe Ito formula, estimating for a fixed ω as done above and then arguing that by uniquenessone can use a single subsequence which works independent of ω .

Example 76.7.3 Let u0 ∈ L2 (Ω,V ) where V is described above and let v0 ∈ L2 (Ω,W ) forW described above. Let both of these initial conditions be progressively measurable. Alsolet f ∈ V ′ and Φ ∈ L∞

((0,T )×Ω,L2

(Q1/2U,H

)). Then there exists a unique solution to

the the integral equation 76.7.73 which can be written in the form

But (t)−Bv0 +∫ t

0A(

u0 +∫ t

0ut (r)dr

)ds =

∫ t

0f ds+

∫ t

0ΦdW

Note that a more standard model involves no point mass at the tip of the beam. Thiswould be done the same way but it would not require the generalized Ito formula presentedearlier. A more standard version would work.

One can find many other examples where this generalized Ito formula is a useful toolto study various kinds of stochastic partial differential equations. We have presented fiveexamples above in which it was helpful to have the extra generality.