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 on a probability space. The measure-preserving dynamical system is known as the base flow of the random dynamical system. The maps are often known as shift maps since they "shift" time. The base flow is often ergodic.
The parameter may be chosen to run over
Each map is required
Furthermore, as a family, the maps satisfy the relations
- , the identity function on ;
- for all and for which the three maps in this expression are defined. In particular, if exists.
In other words, the maps form a commutative monoid or a commutative group.Example
In the case of random dynamical system driven by a Wiener process, where is the two-sided classical Wiener space, the base flow would be given by
This can be read as saying that "starts the noise at time instead of time 0".
