I have some difficulties understanding basic definitions and notations of RWRE and its customary usage. In addition to this, I am also confused with what really assumed and what implicitly assumed in research papers in this particular framework.
From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a graph, and $N_v:= \{k \in V: (v,k) \in E\}$ is the neighborhood of $v \in V$. $\Omega = \prod_{v \in V} M_1(N_v)$ together with the $\sigma-$algebra $\mathcal{F}$ generated by cylinder functions forms a measurable space $(\Omega, \mathcal{F})$. We define a probability measure $P$ on $(\Omega, \mathcal{F})$ and a time homogeneous Markov chain $\{X_n\}_{n \ge 0}$ with transition probabilities \begin{equation} Q_{\omega} (X_{n+1} = k|X_n=v) =\omega_v (k) \end{equation} $Q_{\omega}$ is called the quenched law of the RW, and the marginal of $\mathbb{P}:=P\otimes Q_{\omega}$ on $V^{\mathbb{N}}$ is the annealed law of the RW. I intentionally used the letters $Q$ and $k$ above.
- Is this what I should understand from this definition:
$\omega$ is simply an element of the set $\Omega$, and $P$ is a (probability) measure on $(\Omega, \mathcal{F})$. Using $\Omega$ as an index set, $Q_{\omega}$ is the probability measure on $(E_{\omega}, \Sigma_{\omega})$ and $X_n=X_n(\omega, z)$ is a $V-$valued measurable function, i.e., $X_n: (E_{\omega}, \Sigma_{\omega}, Q_{\omega}) \to (V,\mathcal{S})$ for each $\omega \in \Omega$.
If so, $\Omega$ is taken as $[0,1]^{\mathbb{Z}}$ and $\mathcal{F} = \mathcal{B}([0,1]^{\mathbb{Z}})$ Borel $\sigma-$algebra of $[0,1]^{\mathbb{Z}}$ in many papers regarding nearest neighbor RWRE in 1D integer lattice $\mathbb{Z}$, which makes sense because each number $p \in [0,1]$ represents a probability measure. Here is where I crack: They consider that the environment $\omega$ is a sequence of "random elements" $\omega=(\omega_n)_{n \in \mathbb{Z}}$ in $(\Omega, \mathcal{F})$ and $P$ is a probability measure in the measurable space $(\Omega, \mathcal{F})$. Why? I think $P$ is not the original $P$, instead it is the distribution of $\omega$ and as we change $\omega$, $P$ has to change.
We need to assume the probability measure $P$ to be stationary and ergodic in many papers that investigate RWRE in the sense of large deviations. Ergodicity of the probability measure $P$ is defined with respect to the right shift operator $\theta$. This makes sense because our space elements are sequences of probability measures. Simply pick a probability measure $P$ on $(\Omega, \mathcal{F})$ so that $\theta$ is measure preserving and for any $\theta - $invariant $A$, $P(A) \in \{0,1\}$. However, stationarity is still a complete mystery in this concept.
Here is my way of looking at these things, a particular example nearest neighbor RWRE in 1D $\omega_i(i)=0$: Suppose we have a family of sequences of random elements on $(\Omega, \mathcal{F}, P)$ in $([0,1], \mathcal{B}([0,1]))$, say $\Phi$. When we say $P$ is stationary, we mean the collection $\Phi$ contains stationary elements only. When we say $P$ is ergodic, we mean $\Phi$ contains ergodic elements only. The problem of this way of thinking is that the later is implicitly a subset of the former.
Thanks for your time and comments in advance.