I have had some previous knowledge on evolution equations in a Banach space of the form $$u'(t)=Au(t),$$ where $A$ generates some strongly continuous operator semigroup. Now I am looking at a problem where randomness and dependence upon time are introduced to $A$.

Let $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Let $X$ be a Banach space. For each $\omega$ in $\Omega$ and each time $t$, let $A(\omega, t)$ be a linear operator on $X$ that is the generator of a strongly continuous semigroup. For each $\omega$ in $\Omega$, we have a linear evolution equation $$u_{\omega}'(t)=A(\omega,t)u_\omega(t),\ u_\omega(0)=u_0$$ where $u_0$ in $X$ is deterministic. (And for the problem to make sense, let us assume that $u_0$ is in the domain $D(A(\omega, 0))$ for every $\omega$ in $\Omega$.)

Surely such problem have been studied. Does anyone know some good textbooks/lecture notes on this topic?

Secondly, I have a more specific question. Suppose there exists an interval $I$ such that for each $\omega$, the solution $u_\omega(t)$ exists and is unique for all time $t$ in $I$. Then I may view the solutions $u_\omega(t)$ as a function $u:\Omega \rightarrow C^1(I,X)$ that maps each $\omega$ to the $C^1$ path $(t\mapsto u_\omega(t))$ in $X$. With some suitable norm, and hence a Borel $\sigma$-algebra on $C^1(I,X)$, what can be said about the measurability of this function $u:\Omega \rightarrow C^1(I,X)$?