Basic Definition and Notations in RWRE

Question

From the definition of Zeitouni's lecture notes on RWRE: $(V, E)$ is a special 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.

1) 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 $\omega \in \Omega$.

2) The environment $\omega$ can be viewed as a sequence of random variables. How can I connect these two definitions?

Thanks for your time and comments in advance.

Answer 1

Unfortunately, the probabilistic argot can be sometimes not so easy to understand for outsiders. Also, I think that your notation is unnecessarily complicated.

So, what do we have here? The state space is a graph $V$ (no idea, why it has to be ``special''). We consider only the nearest neighbor random walks, i.e., transitions are allowed only to neighbors, and the space of environments $\Omega$ is the set of all such transition operators (or transition matrices). Once you have fixed a starting point (or, more generally, an initial distribution) on $V$, each environment $\omega\in\Omega$ produces the corresponding Markov measure $\mathbf P_\omega$ on the space of nearest neighbor sample paths in $V$ (probabilists like to call the measures $\mathbf P_\omega$ quenched laws). If you now average the measures $\mathbf P_\omega$ with respect to a distribution $P$ on the space of environments $\Omega$, the resulting measure $\mathbf P = \int \mathbf P_\omega\,dP(\omega)$ on the space of nearest neighbor sample paths in $V$ is called the annealed law.