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I am having trouble understanding this definition, [Arnold: Random Dynamical Systems]:

Definition: A measurable random dynamical system on the measurable space $(X,\mathcal{B})$ over a metric dynamical system $(\Omega,\mathcal{F},\mathbb{P},(\theta(t))_{t\in\mathbb{T}})$ as a map $\phi$ with time $\mathbb{T}$ is a mapping

$$\phi:\mathbb{T}\times\Omega\times X\to X,$$

where $\phi$ is measurable and satisfies the cocycle property. $\blacksquare$

I understand that a metric dynamical system is a family of iterative mappings on the space $(\Omega,\mathcal{F},\mathbb{P})$ that act as endomorphisms, but I don't understand how it all ties together. As far as I can tell, $\phi$ is a deterministic map.

I don't understand how randomness is described. Can anyone enlighten / inspire me?

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1 Answer 1

up vote 2 down vote accepted

For every $(t,\omega,x)$ in $\mathbb T\times\Omega\times X$, $\phi(t,\omega,x)$ is some point in $X$ hence, for every $(t,x)$ in $\mathbb T\times X$, $\phi(t,x):\omega\mapsto\phi(t,\omega,x)$ is a random variable with values in $X$.

Furthermore, one asks that $\phi(t+s,x)=\phi(t,\phi(s,x))$ for every $(t,s,x)$ in $\mathbb T\times\mathbb T\times X$, in the sense that $\phi(t+s,\omega,x)=\phi(t,\omega,\phi(s,\omega,x))$ for every $(t,s,\omega,x)$ in $\mathbb T\times\mathbb T\times\Omega\times X$. In other words, each $\phi(\ ,\omega,\ )$ is a flow on $\mathbb T\times X$.

Edit: Here is a dynamical system. Assume that $\mathbb T=\mathbb N_0$, $X=\mathbb R$, that $\Omega=\{0,1\}^\mathbb N$ is endowed with its cylindrical sigma-algebra, and consider the shifts $(\theta_n)_{n\geqslant0}$ defined by $\theta_n(\omega)=(\omega_{k+n})_{k\geqslant1}$ for every $n\geqslant0$ and $\omega=(\omega_k)_{k\geqslant1}$ in $\Omega$, for example, $\theta_6(\omega_1,\omega_2,\omega_3,\ldots)=(\omega_7,\omega_8,\omega_9,\ldots)$. Finally, define $\phi(1,\omega,x)=A_{\omega_1}(x)$ for every $x$ in $X$ and every $\omega=(\omega_k)_{k\geqslant1}$ in $\Omega$, where $A_0$ and $A_1$ are some given functions on $\mathbb R$. In words, $\phi(1,\ ,\ )$ applies $A_0$ or $A_1$ to $x$, the index $0$ or $1$ being chosen uniformly randomly.

Let me suggest that you check that, for every $n\geqslant0$, $$ \phi(n,\omega,x)=A_{\omega_n}\circ \cdots\circ A_{\omega_2}\circ A_{\omega_1}(x), $$ and that the cocycle property holds in this setting.

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Thanks, yeah this makes sense, however I'm still confused by the cocycle property. $\phi(t+s,\omega ) = \phi(t,\theta(s)\omega ) \circ \phi(s,\omega ).$ What does the movement of $\omega$ through the metric space imply / represent? –  Chadman Oct 17 '13 at 22:52
Sorry but $\omega$ moves through the probability space. The alea $\omega$ is applied to the particle starting at time $0$. The alea $\theta_s(\omega)$ is what starts to be applied to the particle at time $s$. –  Did Oct 18 '13 at 5:34
I guess this is what ultimately confuses me, why do we plot the trajectory of $\omega$ through $\Omega$? I mean, what does $\phi(s,\omega)$ represent. \\ Presumably it's the statement "given $\omega$ has occurred, $x$ will be sent to $\phi(s,\omega)x$ after time $s$ has elapsed" \\ But if the previous statement is true, doesn't that imply (from the cocycle property) that if $\omega$ has occured at time $0$ then we know $\theta(s)\omega$ will occur after time $s$? Do you understand my confusion? –  Chadman Oct 18 '13 at 9:47
See Edit. $ $ $ $ –  Did Oct 18 '13 at 14:56
Fantastic! Thank you ever so much –  Chadman Oct 18 '13 at 18:08

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