|
|
| Line 1: |
Line 1: |
| {{context|date=August 2011}}
| | Greetings! I am Myrtle Shroyer. Years in the past we moved to North Dakota and I love every day living here. One of the issues she enjoys most is to study comics and she'll be beginning something else alongside with it. My day job is a librarian.<br><br>Feel free to surf to my weblog; at home std test ([http://mirim.ir/index.php?do=/profile-699/info/ visit the up coming internet site]) |
| In the [[Mathematics|mathematical]] field of [[dynamical systems]], a '''random dynamical system''' is a dynamical system in which the [[equations of motion]] have an element of randomness to them. Random dynamical systems are characterized by a [[state space]] ''S'', a [[set (mathematics)|set]] of [[map (mathematics)|map]]s ''T'' from ''S'' into itself that that can be thought of as the set of all possible equations of motion, and a [[probability distribution]] ''Q'' on the set ''T'' that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state <math>X \in S</math> evolving according to a succession of maps randomly chosen according to the distribution ''Q''.<ref name=Bhattacharya2003>{{cite journal|last=Bhattacharya, Rabi, and Mukul Majumdar. "Random dynamical systems: a review." Economic Theory 23, no. 1 (2003): 13-38.|first=Rabi|coauthors=Mukul Majumdar|title=Random dynamical systems: a review|journal=Economic Theory|year=2003|volume=23|issue=1|pages=13–38|doi=10.1007/s00199-003-0357-4|url=http://link.springer.com/article/10.1007/s00199-003-0357-4|accessdate=26 June 2013}}</ref>
| |
| | |
| An example of a random dynamical system is a [[stochastic differential equation]]; in this case the distribution Q is typically determined by ''noise terms''. It consists of a [[base flow (random dynamical systems)|base flow]], the "noise", and a [[Oseledec theorem|cocycle]] dynamical system on the "physical" [[phase space]].
| |
| | |
| ==Motivation: solutions to a stochastic differential equation==
| |
| | |
| Let <math>f : \mathbb{R}^{d} \to \mathbb{R}^{d}</math> be a <math>d</math>-dimensional [[vector field]], and let <math>\varepsilon > 0</math>. Suppose that the solution <math>X(t, \omega; x_{0})</math> to the stochastic differential equation
| |
| | |
| :<math>\left\{ \begin{matrix} \mathrm{d} X = f(X) \, \mathrm{d} t + \varepsilon \, \mathrm{d} W (t); \\ X (0) = x_{0}; \end{matrix} \right.</math>
| |
| | |
| exists for all positive time and some (small) interval of negative time dependent upon <math>\omega \in \Omega</math>, where <math>W : \mathbb{R} \times \Omega \to \mathbb{R}^{d}</math> denotes a <math>d</math>-dimensional [[Wiener process]] ([[Brownian motion]]). Implicitly, this statement uses the [[classical Wiener space|classical Wiener]] [[probability space]]
| |
| | |
| :<math>(\Omega, \mathcal{F}, \mathbb{P}) := \left( C_{0} (\mathbb{R}; \mathbb{R}^{d}), \mathcal{B} (C_{0} (\mathbb{R}; \mathbb{R}^{d})), \gamma \right).</math>
| |
| | |
| In this context, the Wiener process is the coordinate process.
| |
| | |
| Now define a '''flow map''' or ('''solution operator''') <math>\varphi : \mathbb{R} \times \Omega \times \mathbb{R}^{d} \to \mathbb{R}^{d}</math> by
| |
| | |
| :<math>\varphi (t, \omega, x_{0}) := X(t, \omega; x_{0})</math>
| |
| | |
| (whenever the right hand side is [[well-defined]]). Then <math>\varphi</math> (or, more precisely, the pair <math>(\mathbb{R}^{d}, \varphi)</math>) is a (local, left-sided) random dynamical system. The process of generating a "flow" from the solution to a stochastic differential equation leads us to study suitably defined "flows" on their own. These "flows" are random dynamical systems.
| |
| | |
| ==Formal definition==
| |
| | |
| Formally, a '''random dynamical system''' consists of a base flow, the "noise", and a cocycle dynamical system on the "physical" phase space. In detail.
| |
| | |
| Let <math>(\Omega, \mathcal{F}, \mathbb{P})</math> be a [[probability space]], the '''noise''' space. Define the '''base flow''' <math>\vartheta : \mathbb{R} \times \Omega \to \Omega</math> as follows: for each "time" <math>s \in \mathbb{R}</math>, let <math>\vartheta_{s} : \Omega \to \Omega</math> be a measure-preserving [[measurable function]]:
| |
| | |
| :<math>\mathbb{P} (E) = \mathbb{P} (\vartheta_{s}^{-1} (E))</math> for all <math>E \in \mathcal{F}</math> and <math>s \in \mathbb{R}</math>;
| |
| | |
| Suppose also that
| |
| # <math>\vartheta_{0} = \mathrm{id}_{\Omega} : \Omega \to \Omega</math>, the [[identity function]] on <math>\Omega</math>;
| |
| # for all <math>s, t \in \mathbb{R}</math>, <math>\vartheta_{s} \circ \vartheta_{t} = \vartheta_{s + t}</math>.
| |
| | |
| That is, <math>\vartheta_{s}</math>, <math>s \in \mathbb{R}</math>, forms a [[group (mathematics)|group]] of measure-preserving transformation of the noise <math>(\Omega, \mathcal{F}, \mathbb{P})</math>. For one-sided random dynamical systems, one would consider only positive indices <math>s</math>; for discrete-time random dynamical systems, one would consider only integer-valued <math>s</math>; in these cases, the maps <math>\vartheta_{s}</math> would only form a [[commutative]] [[monoid]] instead of a group.
| |
| | |
| While true in most applications, it is not usually part of the formal definition of a random dynamical system to require that the [[measure-preserving dynamical system]] <math>(\Omega, \mathcal{F}, \mathbb{P}, \vartheta)</math> is [[ergodic]].
| |
| | |
| Now let <math>(X, d)</math> be a [[complete space|complete]] [[separable space|separable]] [[metric space]], the '''phase space'''. Let <math>\varphi : \mathbb{R} \times \Omega \times X \to X</math> be a <math>(\mathcal{B} (\mathbb{R}) \otimes \mathcal{F} \otimes \mathcal{B} (X), \mathcal{B} (X))</math>-measurable function such that
| |
| | |
| # for all <math>\omega \in \Omega</math>, <math>\varphi (0, \omega) = \mathrm{id}_{X} : X \to X</math>, the identity function on <math>X</math>;
| |
| # for (almost) all <math>\omega \in \Omega</math>, <math>(t, \omega, x) \mapsto \varphi (t, \omega,x) </math> is [[continuous function|continuous]] in both <math>t</math> and <math>x</math>;
| |
| # <math>\varphi</math> satisfies the (crude) '''cocycle property''': for [[almost all]] <math>\omega \in \Omega</math>,
| |
| ::<math>\varphi (t, \vartheta_{s} (\omega)) \circ \varphi (s, \omega) = \varphi (t + s, \omega).</math>
| |
| | |
| In the case of random dynamical systems driven by a Wiener process <math>W : \mathbb{R} \times \Omega \to X</math>, the base flow <math>\vartheta_{s} : \Omega \to \Omega</math> would be given by
| |
| | |
| :<math>W (t, \vartheta_{s} (\omega)) = W (t + s, \omega) - W(s, \omega)</math>.
| |
| | |
| This can be read as saying that <math>\vartheta_{s}</math> "starts the noise at time <math>s</math> instead of time 0". Thus, the cocycle property can be read as saying that evolving the initial condition <math>x_{0}</math> with some noise <math>\omega </math> for <math>s</math> seconds and then through <math>t</math> seconds with the same noise (as started from the <math>s</math> seconds mark) gives the same result as evolving <math>x_{0}</math> through <math>(t + s)</math> seconds with that same noise.
| |
| | |
| ==Attractors for random dynamical systems==
| |
| The notion of an [[attractor]] for a random dynamical system is not as straightforward to define as in the deterministic case. For technical reasons, it is necessary to "rewind time", as in the definition of a [[pullback attractor]]. Moreover, the attractor is dependent upon the realisation <math>\omega</math> of the noise.
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| * Crauel, H., Debussche, A., & Flandoli, F. (1997) Random attractors. ''Journal of Dynamics and Differential Equations''. '''9'''(2) 307—341.
| |
| | |
| {{Stochastic processes}}
| |
| | |
| [[Category:Random dynamical systems|*]]
| |
| [[Category:Stochastic differential equations]]
| |
| [[Category:Stochastic processes]]
| |
Greetings! I am Myrtle Shroyer. Years in the past we moved to North Dakota and I love every day living here. One of the issues she enjoys most is to study comics and she'll be beginning something else alongside with it. My day job is a librarian.
Feel free to surf to my weblog; at home std test (visit the up coming internet site)