Consider the following two similar frameworks (the first one in discrete time, the second one in continuous time) in which I am trying to apply the Krylov-Bogoliubov theorem.
1) (Discrete time.) Consider a Markov process $\{X_t:t=1,2,\dots\}$, where $X_t$ is a vector with two entries, $X_{t}^{1}$ and $X_{t}^{2}$. The first entry $X_{t}^{1}$ takes values in $A\subset\mathbb{R}$ and it depends only on $X_{t-1}^{1}$; there exists a unique ergodic distribution $G^1$ for the first entry. The second entry $X_{t}^{2}$ takes values in $[0,1]$ and it depends on both $X_{t-1}^{1}$ and $X_{t-1}^{2}$.
What further assumptions do I have to impose to invoke the Krylov-Bogoliubov theorem and make sure that there exists an ergodic distribution $G$ for $X_t$?
To guarantee existence, I think that no further assumption on $X_t$ or $A$ is required. Here is my reasoning.
When constructing a stochastic process $X_t$, one can define a (deterministic) transformation $S:\Omega\rightarrow\Omega$ that maps the state of the world $\omega\in\Omega$ today into the state of the world $S(\omega)\in\Omega$ tomorrow. The stochastic process $X_t$ is then defined as a function $X:\Omega\rightarrow A\times[0,1]$. Then it's enough to define some $\Omega$ and $S$ satisfying the assumptions of the Bogoliubov-Krylov theorem, and an invariant measure exists on $\Omega$.
2) (Continuous time.) The framework in continuous time is similar. $\{X_t:t\geq 0\}$ is a markov process with two entries, $X_{t}^{1}$ and $X_{t}^{2}$. The first entry $X_{t}^{1}$ takes values in $A\subset\mathbb{R}$ and it depends only on past values of the first entry; there exists a unique ergodic distribution $G^1$ for the first entry. The second entry $X_{t}^{2}$ takes values in $[0,1]$ and it depends on past values of both entries. Moreover, $X_t$ is almost surely continuous in $t$ almost everywhere (I allow for jumps).
As before, what further assumptions do I have to impose to invoke the Krylov-Bogoliubov theorem?