Per Enflo - Revision history

en>John of Reading: /* Bombieri norm */Typo fixing, replaced: use use → use using AWB

2014-12-20T18:31:22Z

Bombieri norm: Typo fixing, replaced: use use → use using AWB

← Older revision		Revision as of 20:31, 20 December 2014
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	~~{{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]]~~

en>Monkbot: Fix CS1 deprecated date parameter errors

2014-01-28T08:14:27Z

Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 10:14, 28 January 2014
Line 1:		Line 1:
	~~Hi there~~, ~~I am Felicidad Oquendo~~. ~~Bookkeeping is how he supports his family~~ and ~~his salary has been car~~ [http://~~www~~.~~austarps~~.com.au/~~UserProfile~~/~~tabid~~/~~105~~/~~userId~~/~~2471~~/~~Default~~.~~aspx auto warranty~~] ~~really satisfying~~. ~~Her~~ [~~http~~://~~www~~.~~cars.com~~/go/~~advice/Story~~.~~jsp?section~~=~~buy&story~~=~~usedWarranty&subject~~=~~warranty friends~~] ~~say it~~'s ~~not good~~ for ~~her but what she enjoys doing is extended car warranty flower arranging~~ and ~~she is trying~~ to ~~make it a profession. Delaware extended car extended car warranty warranty is~~ the ~~location~~ [~~http:~~//~~www~~.~~Popularmechanics.com~~/~~cars~~/~~how~~-to/~~repair~~/~~how~~-~~to-get~~-a~~-used-car-warranty-and-~~not-~~get-screwed-6654348 I adore~~] ~~most but I need~~ to ~~move~~ for ~~my family members.~~<br><br>~~My web site ... [http~~://~~pupo.ch~~/~~mypupo~~/~~panda~~/~~panda2~~/~~index.php?mod~~=~~users&action=view&id=161 extended car warranty]~~, ~~[http~~://~~Crossroad~~-~~rto~~.ca/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/~~55470~~/~~userId~~/~~428152~~/~~Default~~.~~aspx just click~~ the ~~up coming website~~],		{{context\|date=August 2011}}
			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]]

en>Omnipaedista: markup fix

2012-08-16T10:31:30Z

markup fix

New page

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