Space-time stochastic processes and their generalizations

In the Section 1.2.6 of the book: Weak and measure-valued solutions to evolutionary partial differentional equations – Malek, Necas, Rokyta, Ruzicka, 1996, it is written something like this:

Let’s say we have some deterministic real valued function $ u(t,x):[0,T] \times A \rightarrow \mathbb{R}$ where $ A\subset \mathbb{R}^d$ . We could think of $ u$ in a different way. For $ u(t,x)$ the map $ $ u(t):x \mapsto u(t,x) $ $

is an element of some function space (Sobolev, Lebesgue,…). Then the function $ $ t \mapsto u(t) $ $

maps the interval [0,T] into that function space…

The idea is to regard a function of time and space as a collection of functions of space that is parametrized by time.

So what happens when we add some randomness in the function $ u(t,x)$ ? There are a lot of generalizations of stochastic processes but I am interested in the two space-time generalizations given bellow.

a) Could we say that the space-time stochastic process $ u(t,x,\omega)$ maps $ [0,T] \times A \times \Omega \rightarrow \mathbb{R}$ where $ A\subset \mathbb{R}^d$ ? I work on the problems in the Stochastic evolutionary equations I do not see this type of generalization almost nowhere. The one I see usually is

b) Instead of using $ u(t,x,\omega)$ given above it is custom to use $ u(t,\omega)$ that maps $ [0,T] \times \Omega$ into some function space. Definitions like this could be found for example in Chapter 3 of the book Stochastic Equations in Infinite Dimensions – Da Prato, Zabczyk, 1992.

So instead of using real-valued $ u(x,t,\omega) $ we use Banach space-valued $ u(t,\omega) $ . For example, spaces like $ u(t,\omega) \in L^p(\Omega,L^q(0,T;L^{\infty}))$ or $ u(t,\omega) \in L^p(\Omega,C([0,T];W^{k,r}))$ or something similar.

My questions are:

  1. What are the advantages of using processes b) instead of the processes a)? I.E. why would somebody use b) instead of a) and vice versa? So it is the question of using real-valued or Banach space-valued processes .

  2. For the processes in a) what would be their trajectory i.e. do we now fix $ x$ and $ \omega$ instead of the just $ \omega$ like in the typical stochastic processes? In the processes given in b) trajectories usually belong to the spaces $ C([0,T];L^p(\mathbb{R}^d))$ or similar.

Thanks for the help. And if anyone needs some clarification of this question or more examples, please write it in the comments bellow.