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.
In the definition of a random dynamical system, one is given a family of maps
ϑ
on a probability space
The measure-preserving dynamical system
is known as the base flow of the random dynamical system.
are often known as shift maps since they "shift" time.
The base flow is often ergodic.
may be chosen to run over Each map
is required Furthermore, as a family, the maps
satisfy the relations In other words, the maps
form a commutative monoid (in the cases
) or a commutative group (in the cases
In the case of random dynamical system driven by a Wiener process
is the two-sided classical Wiener space, the base flow
"starts the noise at time