Let $\xi_n\in \Xi$ be a sequence of iid random variables with $n \in\mathbb N\cup\{0\}$, which we call a noise process. Construct a process $ Z_{n+1} = f(Z_n,\xi_n)\quad(\star) $ with $Z_0\in E$ and $f:E\times\Xi\to E$ is a measurable function of two variables. Clearly, $Z$ is a Markov process.
On the other hand, consider a Markov process $X$ on $E$ given by its transition kernel $P(x,A)$. Is it always possible to find a noise process (on some set $\Xi$) and a measurable function $f:E\times \Xi\to E$ such that given any initial condition (maybe random) $X_0\in E$ $ \operatorname{Law}(X_1) = \operatorname{Law}(f(X_0,\xi_0)) $
Briefly speaking, if any Markov process can be presented in the form $(\star)$?