A random dynamic system is defined in Wikipedia. Its definition, which is not included in this post for the sake of clarity, reminds me how similar a Markov process is to a random dynamic system just in my very superficial impression. Let
$T=\mathbb{R}$ or $\mathbb{Z}$ be the index set,
$(\Omega, \mathcal{F}, P)$ be the probability space,
$X$, a measurable space (or a complete separable metric space with its Borel sigma algebra, as in Wikipedia's definition for random dynamic system), be the state space.
Questions:
I wonder if any Markov process $f: T \times \Omega \to X$ can be seen as a random dynamic system, i.e. can induce a random dynamic system $\varphi: T \times \Omega \times X \to X$ corresponding to it?
If no, what kinds of Markov processes can induce random dynamic systems?
When it is yes, how are the random dynamic system $\varphi$ and its base flow $\vartheta: T \times \Omega \to \Omega$ constructed from the Markov process $f$?
Thanks and regards!
I finally am able to read and understand the linked question by Ilya and reply by Byron. Yes they are closely related, in that Byron pointed out a theorem that can rewrite a discrete time Markov process into a kind of "randomized dynamic system".
Let $X$ be a process on $\mathbb{Z}_+$ with values in a Borel space $S$. Then $X$ is Markov iff there exist some measurable functions $f_1,f_2,\dots:S\times[0,1]\to S$ and iid $U(0,1)$ random variables $\xi_n$ independent of $X_0$ such that $X_n=f_n(X_{n-1},\xi_n)$ almost surely for all $n\in\mathbb{N}$. Here we may choose $f_1=f_2=\cdots =f$ iff $X$ is time homogeneous.
However, the form of $f_n$'s is not exactly $\varphi$ in the definition of a random dynamic system in the Wikipedia article I linked. So how shall I see if they are equivalent?