Understanding the construction of stochastic processes I've seen stochastic processes modeled/constructed in the following way. 


  Consider the
  probability space $(\Omega, \mathcal F, Pr)$ and let $\mathbb S$ be the (measurable)
  transformation $\mathbb S: \Omega \rightarrow \Omega$
  that we use to model the evolution of the sample point $\omega$ over time.
  Also, let $X$ be the random vector $X: \Omega \rightarrow \mathbb R^n$.
  Then, the stochastic process $\{ X_t: t=0,1,...\}$ is used to model
  a sequence of observations via the formula
  $
X_t(\omega) = X[\mathbb S^t(\omega)]
$
  or
  $
X_t = X \circ \mathbb S^t.
$


How should I understand the sample points $\omega \in \Omega$ and
the transformation $\mathbb S$ in this construction? (Could $\omega$ be something
like a sequence of shocks in certain cases?) 

For more concreteness,
how would I write these two processes in this notation? 

Process 1:
$$
X_{t+1} = \rho X_t + \varepsilon_{t+1} \tag{1}
$$
where $X_0 = 0$.

Process 2:
$$
X_{t+1} = \varepsilon_{t+1} \tag{2}
$$
This construction you describe is not fully general. In fact it characterizes strictly stationary time series. You see that it's shift-invariant. This operator $S$ is essentially a shift operator.

For comparison, here's the usual definition of, let's say discrete-time, processes:

Definition A stochastic process is a sequence $\{ X_t \}$ of Borel measurable maps on a probability space $( \Omega, \mathcal{F}, \mu )$.  

Now for what you're describing, you have a fixed Borel measurable map $X: \Omega \rightarrow \mathbb{R}^n$. It's the underlying measure that is evolving according to $S$. The map $S$ induces a new "push-forward measure" (in measure-theoretic parlance) on $\Omega$ by just taking preimages: define a measure $\mu_S$ by

$$
A \in \mathcal{F} \stackrel{\mu_S}{\mapsto} Pr(S^{-1}(A)).
$$

So the random vector $X: ( \Omega, \mathcal{F}, \mu_S) \rightarrow \mathbb{R}^n$ is $X \circ S$ by construction. They induce the same push-forward measure on $\mathbb{R}^n$. Do this with $S^t$ for each $t$ and you have your time series.

As for your question about $\omega$, inspecting the proof for the other direction should clarify this---i.e. any strictly stationary time series must necessarily take this form for some $( \Omega, \mathcal{F}, Pr)$, $X$, and $S$. 

The basic point is that, from a general point of view, a stochastic process is a probability measure on the set of its possible realizations. This is seen in, for example, Wiener's construction of Brownian motion; he constructed a probability measure on $C[0, \infty)$. So in general, an $\omega$ is a sample path and $\Omega$ consists of all possible sample paths.  

For example, take the two processes you named above. They are strictly stationary, if let's say the innovations are Gaussian. (Any covariance-stationary time series driven by Gaussian innovations is strictly stationary.) The construction would then start by taking $\Omega$ to be the set of all sequences, $\mathcal{F}$ the $\sigma$-algebra generated by coordinate maps, and $Pr$ the appropriate measure. For the white noise process (2), $Pr$ is just a product measure on an infinite product. 

Reference This characterization/construction by shift of strictly stationary time series is mentioned in White's Asymptotic Theory for Econometricians.