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.
I prefer the first definition by far. I relate the question to ergodic theory, as seems appropriate, and assume that the chain hass finitely many possible values, so as to not bother with positive recurrence.
Let us consider a finite state space $A$, and denote all the possible sequences of element in $A$ by $X:=A^{\mathbb{N}}$. Let us define a transformation $\sigma$ on $X$ by $(\sigma x)_n = x_{n+1}$ on $X$. For $x \in X$, we have $x_n = (\sigma^n x)_0$. In other words, by applying the transformation $\sigma$, I can read the successive values of a given sequence.
Now, let us take some probability measure $\mu$ on $A$ with full support (so as to see everything), and a stochastic matrix $P$ (the transition kernel). Using $\mu$ as the distribution of $X_0$ and the matrix $P$ to define transitions, we get a Markov chain $(X_n)_{n \geq 0} = x = ((\sigma^n x)_0)_{n \geq 0}$, which is a stochastic process with values in $A$. The distribution of $(X_n)_{n \geq 0}$ is a measure $\overline{\mu}$ on $A^{\mathbb{N}}$ which satisfies the usual conditions on cylinders, and whose first marginal is $\mu$.
The construction may look a bit confusing. However, if you forget about $\sigma$, it is what is done more or less informally when one defines Markov chains (that is the construction may be hidden, but it is there).
Hence, we can consider a Markov chain as a dynamical system $(X, \sigma)$ together with a probability measure $\overline{\mu}$. We can use the definitions of ergodic theory, and what we get in the end is that:
- the system $(X, \sigma, \overline{\mu})$ is measure-preserving if and only if $\mu$ is stationnary for $P$;
- the system $(X, \sigma, \overline{\mu})$ is ergodic (in the sense of ergodic theory) if and only if the Markov chain is irreducible;
- the system $(X, \sigma, \overline{\mu})$ is mixing if and only if the Markov chain is irreducible and aperiodic.
So these are two very different conditions, and aperiodicity does not correspond to ergodicity. As a corollary, one can apply ergodic theorems to Markov chains with no need for aperiodicity.
Best Answer
The article talks about a (stationary) Markov chain ${(X_n)}_{n \in \mathbb{Z}}$ in discrete time with each $X_n$ taking its values in a finite set $E$. The canonical space of the Markov chain is the product set $E^{\mathbb{Z}}$. The trajectory $X=(\ldots, X_{-1}, X_{0}, X_1, \ldots)$ of the Markov chain is a random variable taking its values in $E^{\mathbb{Z}}$. Denoting by $\mu$ its distribution (which could be termed as the law of the Markov process) then $\mu$ is invariant under the classical shift operator $T \colon E^{\mathbb{Z}} \to E^{\mathbb{Z}}$. Then the Markov chain can be considered as the dynamical system $(T,\mu)$. In fact here we only use the fact that ${(X_n)}_{n \in \mathbb{Z}}$ is a stationary process. In the Markov case we can say in addition that the ergodicity of $T$ is equivalent to the irreducibility of ${(X_n)}_{n \in \mathbb{Z}}$.