ABSTRACT
Then, by (H2), it follows (see [7] or [15]) that, Vfc 6 V (A ') , Vi € [0,T], for all u € (I such that u(-,w) G L2(0,T, 17):
(F(u)(t,u),A*h) = j \ v { s , UyiG*elt-) A’ A*h)d, < cT H - ,« ) | | t , (0iTil7) ||A|| (1.2)
We have therefore proved that, for almost all u> E 0 , F(u)(t,u>) E V(A) Vi E [0,T] and:
p .F (u)(t,w )|| < cT ||«(-,w)||L*(o,T,to VO < t < T (1.3) Moreover writing F(u)(t) as / #T eaAGu{t — a, u>)ds (where u(t) = u(t) if t > 0 u(i) = 0 if t < 0) and arguing as in relations (1.2) and (1.3), we prove that AF(u) has continuous paths and, since it is adapted, it is also predictable. In particular by (1.3) we get that t -* AF(u)(t) is continuous from [0,T] into L2(0,£,IP).
