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| {{Unreferenced|date=December 2009}}
| | Hello. Allow me introduce the author. Her name is Refugia Shryock. Supervising is my occupation. Body building is what my family and I appreciate. Years ago we moved to North Dakota.<br><br>My page; at home std testing ([http://torontocartridge.com/uncategorized/the-ideal-way-to-battle-a-yeast-infection/ he has a good point]) |
| {{Confusing|date=May 2009}}
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| 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.
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| ==Definition==
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| In the definition of a random dynamical system, one is given a family of maps <math>\vartheta_{s} : \Omega \to \Omega</math> on a probability space <math>(\Omega, \mathcal{F}, \mathbb{P})</math>. The [[measure-preserving dynamical system]] <math>(\Omega, \mathcal{F}, \mathbb{P}, \vartheta)</math> is known as the '''base flow''' of the random dynamical system. The maps <math>\vartheta_{s}</math> are often known as '''shift maps''' since they "shift" time. The base flow is often [[ergodic]].
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| The parameter <math>s</math> may be chosen to run over
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| * <math>\mathbb{R}</math> (a two-sided continuous-time dynamical system);
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| * <math>[0, + \infty) \subsetneq \mathbb{R}</math> (a one-sided continuous-time dynamical system);
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| * <math>\mathbb{Z}</math> (a two-sided discrete-time dynamical system);
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| * <math>\mathbb{N} \cup \{ 0 \}</math> (a one-sided discrete-time dynamical system).
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| Each map <math>\vartheta_{s}</math> is required
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| * to be a <math>(\mathcal{F}, \mathcal{F})</math>-[[measurable function]]: for all <math>E \in \mathcal{F}</math>, <math>\vartheta_{s}^{-1} (E) \in \mathcal{F}</math>
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| * to preserve the measure <math>\mathbb{P}</math>: for all <math>E \in \mathcal{F}</math>, <math>\mathbb{P} (\vartheta_{s}^{-1} (E)) = \mathbb{P} (E)</math>.
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| Furthermore, as a family, the maps <math>\vartheta_{s}</math> satisfy the relations
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| * <math>\vartheta_{0} = \mathrm{id}_{\Omega} : \Omega \to \Omega</math>, the [[identity function]] on <math>\Omega</math>;
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| * <math>\vartheta_{s} \circ \vartheta_{t} = \vartheta_{s + t}</math> for all <math>s</math> and <math>t</math> for which the three maps in this expression are defined. In particular, <math>\vartheta_{s}^{-1} = \vartheta_{-s}</math> if <math>- s</math> exists.
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| In other words, the maps <math>\vartheta_{s}</math> form a [[commutative]] [[monoid]] (in the cases <math>s \in \mathbb{N} \cup \{ 0 \}</math> and <math>s \in [0, + \infty)</math>) or a commutative [[group (mathematics)|group]] (in the cases <math>s \in \mathbb{Z}</math> and <math>s \in \mathbb{R}</math>).
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| ==Example==
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| In the case of random dynamical system driven by a [[Wiener process]] <math>W : \mathbb{R} \times \Omega \to X</math>, where <math>(\Omega, \mathcal{F}, \mathbb{P})</math> is the two-sided [[classical Wiener space]], the base flow <math>\vartheta_{s} : \Omega \to \Omega</math> would be given by
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| :<math>W (t, \vartheta_{s} (\omega)) = W (t + s, \omega) - W(s, \omega)</math>.
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| This can be read as saying that <math>\vartheta_{s}</math> "starts the noise at time <math>s</math> instead of time 0".
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| {{DEFAULTSORT:Base Flow (Random Dynamical Systems)}}
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| [[Category:Random dynamical systems]]
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Hello. Allow me introduce the author. Her name is Refugia Shryock. Supervising is my occupation. Body building is what my family and I appreciate. Years ago we moved to North Dakota.
My page; at home std testing (he has a good point)