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 formula_1 on a probability space formula_2. The measure-preserving dynamical system formula_3 is known as the base flow of the random dynamical system. The maps formula_4 are often known as shift maps since they "shift" time. The base flow is often ergodic.
The parameter formula_5 may be chosen to run over 
Each map formula_4 is required
Furthermore, as a family, the maps formula_4 satisfy the relations
In other words, the maps formula_4 form a commutative monoid (in the cases formula_26 and formula_27) or a commutative group (in the cases formula_28 and formula_29).
Example.
In the case of random dynamical system driven by a Wiener process formula_30, where formula_2 is the two-sided classical Wiener space, the base flow formula_1 would be given by
This can be read as saying that formula_4 "starts the noise at time formula_5 instead of time 0".