Given random dynamical system $(X, \mathcal{B}, (T_{\omega})_{\omega\in \Omega}, \mu)$ where $(\Omega, \mathbb{P})$ is probability space with ergodic transformation $\sigma: \Omega \to \Omega$. Define random composition $T_\omega^n:=T_{\sigma^{n-1}\omega} \circ \dots \circ T_\omega$. Assume we have quasi-invariant absolutely continuous probability $\mu_\omega:= h_\omega d\mu$ such that $(T_\omega)_{*} \mu_\omega=\mu_{\sigma \omega}$.
There are two kinds of mixing rate I know so far, but could not tell the reason how it comes from.
the first type of mixing rate is : a.s. $\omega \in \Omega$, and any function $\phi, \psi \in L^{\infty}(\Omega \times X)$ (denote $\phi_{\omega}, \psi_{\omega}: X \to \mathbb{R}$ as function restricted on fiber $\omega$),
$$ \left|\int \varphi_{\sigma^n \omega} \circ T^n_\omega \cdot \psi_\omega \, d\mu_\omega -\int \varphi_{\sigma^n \omega} \, d\mu_{\sigma^n \omega} \int \psi_\omega \, d\mu_\omega\right| \le C_{\omega} \cdot \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n}$$
the second type of mixing rate is : a.s. $\omega \in \Omega$, and any function $\phi, \psi \in L^{\infty}(\Omega \times X)$, $$ \left|\int \varphi_{\sigma^n \omega} \circ T^n_\omega \cdot \psi_\omega \, d\mu_\omega -\int \varphi_{\sigma^n \omega} \, d\mu_{\sigma^n \omega} \int \psi_\omega \, d\mu_\omega\right| \le \|\varphi\|_\infty \cdot C_\psi \cdot e^{-n}$$
The first mixing rate depends on the random environment $\omega$, while the second one doesn't.
what is the reason making $C_{\omega}$ show up in the mixing rate? what kind of system has homogenous mixing rate, independent of $\omega$, i.e. does not have $C_{\omega}$. can we find examples in the real world to explain such difference? Thanks in advanced!
