# Basic Definition and Notations in RWRE

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 $$Q_{\omega} (X_{n+1} = k|X_n=v) =\omega_v (k)$$ $Q_{\omega}$ 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?

• I think this question should be reopened. Although it's mainly about definitions and notation, they are definitions and notation that are quite specific to the study of RWRE (random walk in random environment). This is a highly active research area, and its basic notions are not normally taught in standard graduate courses. I would say this question is clearly in scope for MO. Jun 29, 2016 at 19:06

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.