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
(a two-sided continuous-time dynamical system);
(a one-sided continuous-time dynamical system);
(a two-sided discrete-time dynamical system);
(a one-sided discrete-time dynamical system).
Each map
is required
- to be a
-measurable function: for all
, 
- to preserve the measure
: for all
,
.
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 (in the cases
and
) or a commutative group (in the cases
and
).
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".
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