Base flow (random dynamical systems)

In the definition of a random dynamical system, one is given a family of maps ${\displaystyle \vartheta _{s}:\Omega \to \Omega }$ on a probability space ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$. The measure-preserving dynamical system ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} ,\vartheta )}$ is known as the base flow of the random dynamical system. The maps ${\displaystyle \vartheta _{s}}$ are often known as shift maps since they "shift" time. The base flow is often ergodic.

The parameter ${\displaystyle s}$ may be chosen to run over

• ${\displaystyle \mathbb {R} }$ (a two-sided continuous-time dynamical system);
• ${\displaystyle [0,+\infty )\subsetneq \mathbb {R} }$ (a one-sided continuous-time dynamical system);
• ${\displaystyle \mathbb {Z} }$ (a two-sided discrete-time dynamical system);
• ${\displaystyle \mathbb {N} \cup \{0\}}$ (a one-sided discrete-time dynamical system).

Each map ${\displaystyle \vartheta _{s}}$ is required

• to be a ${\displaystyle ({\mathcal {F}},{\mathcal {F}})}$-measurable function: for all ${\displaystyle E\in {\mathcal {F}}}$, ${\displaystyle \vartheta _{s}^{-1}(E)\in {\mathcal {F}}}$
• to preserve the measure ${\displaystyle \mathbb {P} }$: for all ${\displaystyle E\in {\mathcal {F}}}$, ${\displaystyle \mathbb {P} (\vartheta _{s}^{-1}(E))=\mathbb {P} (E)}$.

Furthermore, as a family, the maps ${\displaystyle \vartheta _{s}}$ satisfy the relations

• ${\displaystyle \vartheta _{0}=\mathrm {id} _{\Omega }:\Omega \to \Omega }$, the identity function on ${\displaystyle \Omega }$;
• ${\displaystyle \vartheta _{s}\circ \vartheta _{t}=\vartheta _{s+t}}$ for all ${\displaystyle s}$ and ${\displaystyle t}$ for which the three maps in this expression are defined. In particular, ${\displaystyle \vartheta _{s}^{-1}=\vartheta _{-s}}$ if ${\displaystyle -s}$ exists.

In other words, the maps ${\displaystyle \vartheta _{s}}$ form a commutative monoid (in the cases ${\displaystyle s\in \mathbb {N} \cup \{0\}}$ and ${\displaystyle s\in [0,+\infty )}$) or a commutative group (in the cases ${\displaystyle s\in \mathbb {Z} }$ and ${\displaystyle s\in \mathbb {R} }$).

In the case of random dynamical system driven by a Wiener process ${\displaystyle W:\mathbb {R} \times \Omega \to X}$, where ${\displaystyle (\Omega ,{\mathcal {F}},\mathbb {P} )}$ is the two-sided classical Wiener space, the base flow ${\displaystyle \vartheta _{s}:\Omega \to \Omega }$ would be given by

${\displaystyle W(t,\vartheta _{s}(\omega ))=W(t+s,\omega )-W(s,\omega )}$.

This can be read as saying that ${\displaystyle \vartheta _{s}}$ "starts the noise at time ${\displaystyle s}$ instead of time 0".