12
$\begingroup$

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:

  1. 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?

  2. 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?

  • 0
    [should be related](http://math.stackexchange.com/questions/66047/markov-processes-driven-by-the-noise)2012-05-13
  • 0
    @Ilya:Thanks! I will take a look.2012-05-13
  • 0
    @Ilya: Thanks! I wonder how to see if the form given by Byron and the definition of a random dynamic system in the Wikipedia article I linked are equivalent?2012-11-18
  • 0
    I think, measure-preserving maps are related to the stationarity - more precisely, if the shift operator over the Markov process $$ \vartheta(\omega_0,\omega_1,\omega_2,\dots) = (\omega_1,\omega_2,\dots) $$ is measure $\mathsf P_\mu$ preserving, than $\mu$ is the stationary distribution of the Markov process. Not every Markov process allows for the stationary distribution - however, maybe $\vartheta$ in that case can be something different from the shift. I don't think that the linked questions is a duplicate of yours, so I vote to reopen.2012-11-18

1 Answers 1