Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
should be related –  Ilya May 13 '12 at 8:48
    
@Ilya:Thanks! I will take a look. –  Tim May 13 '12 at 13:02
    
@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? –  Tim Nov 18 '12 at 18:32
    
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. –  Ilya Nov 18 '12 at 19:11
add comment

1 Answer

A different point of view:

Although the dynamics of a particle in a random walk are indeed random, the dynamics of its probability distribution certainly are not. Indeed note the probability distributions $\{\nu^{\star k}\}_{k\in\mathbb{N}}$ evolve deterministically as $\{\delta^e P^k:k\in \mathbb{N}\}$. Thus the random walk has the structure of a dynamical system $\{M_p(G),P\}$ with fixed point attractor $\{\pi\}$. The two canonical categories of dynamical systems (for which there is an existing literature of powerful methods) are topological and measure preserving dynamical systems. Unfortunately at first remove $\{M_p(G),P\}$ appears too coarse and structureless to apply any of these powerful methods. Also the mapping function $P$ is not necessarily invertible and this poses further problems. Indeed in many examples of walks exhibiting cut-off, $P$ may be seen to be singular. Hence the assumption that needs to be made on $P$ to put a structure on $\{M_p(G),P\}$ sufficient for application of dynamical systems methods to the cut-off phenomenon is overly strict. A more fundamental problem occurs in trying to put the structure of a measure preserving dynamical system on the walk in that if a meaningful (a measure $\kappa$ wouldn't be very meaningful if $\kappa(M_p(G))=\kappa(\{\pi\})$) measure is put on $M_p(G)$, the fact that $(M_p(G))P^k\underset{k\rightarrow \infty}{\rightarrow} \{\pi\}$ would imply that $P$ is in fact not measure preserving.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.