Random dynamical system

Remarks

Clause 2 is the cocycle property. When $\Omega$ has one point, $\varphi$ is an ordinary dynamical system. Often $X$ is a topological space and each $\varphi(t, \omega)$ is required to be continuous.

Example: let $f_0, f_1 \colon X \to X$ be measurable maps. Let $\Omega = \set{0, 1}^{\Z}$ with the product of the measures giving weight $\frac{1}{2}$ to $0$ and to $1$, and let $\theta_t$ be the $t$-th power of the left shift, for $t \in \N$. Put $\varphi(0, \omega) = \id_X$ and $\varphi(n, \omega) = f_{\omega_{n-1}} \circ \cdots \circ f_{\omega_1} \circ f_{\omega_0}$ for $n \ge 1$. Then $\varphi(m + n, \omega)$ applies $f_{\omega_0}, \ldots, f_{\omega_{m-1}}$ and then $f_{\omega_m}, \ldots, f_{\omega_{m+n-1}}$, and the second block is $\varphi(n, \theta_m \omega)$, so clause 2 holds.

Non-example: with the same data, $\psi(n, \omega) = f_{\omega_0}^{\,n}$ fails clause 2 when $f_0^{\,2} \ne f_1 \circ f_0$. Indeed, for $\omega_0 = 0$ and $\omega_1 = 1$, $\psi(2, \omega) = f_0 \circ f_0$, while $\psi(1, \theta_1 \omega) \circ \psi(1, \omega) = f_1 \circ f_0$.

Stochastic differential equations with suitable coefficients generate random dynamical systems with $\Omega$ the space of Brownian paths and $\theta_t$ the shift of paths.