Base flow (random dynamical systems)

In mathematics, the base flow of a random dynamical system is the dynamical system defined on the "noise" probability space that describes how to "fast forward" or "rewind" the noise when one wishes to change the time at which one "starts" the random dynamical system.

Definition

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 ${\displaystyle s\in \mathbb {R} }$).

Example

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".

References