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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.

DefinitionEdit

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  ).

ExampleEdit

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".