en>JRSpriggs: /* More variant forms */ if you go beyond linear

2012-06-28T06:04:03Z

More variant forms: if you go beyond linear

New page

{{Context|date=February 2010}}
In [[mathematics]], an '''absorbing set''' for a [[random dynamical system]] is a [[subset]] of the [[phase space]] that eventually contains the image of any [[bounded set]] under the cocycle ("flow") of the random dynamical system. As with many concepts related to random dynamical systems, it is defined in the [[Pullback attractor|pullback]] sense.

==Definition==
Consider a random dynamical system ''φ'' on a [[complete space|complete]] [[separable space|separable]] [[metric space]] (''X'', ''d''), where the noise is chosen from a [[probability space]] (Ω, Σ, '''P''') with [[base flow (random dynamical systems)|base flow]] ''ϑ'' : '''R''' × Ω → Ω. A [[random compact set]] ''K'' : Ω → 2''X'' is said to be '''absorbing''' if, for all ''d''-bounded [[deterministic]] sets ''B'' ⊆ ''X'', there exists a ([[Wikt:finite|finite]]) [[random variable|random time]] ''τ''''B'' : Ω → [0, +∞) such that

:<math>t \geq \tau_{B} (\omega) \implies \varphi (t, \vartheta_{-t} \omega) B \subseteq K(\omega).</math>

This is a definition in the pullback sense, as indicated by the use of the negative time shift ''ϑ''−''t''.

==References==
* {{cite journal
|last = Robinson
| first = James C.
| coauthors = Tearne, Oliver M.
| title = Boundaries of attractors of omega limit sets
| journal = Stoch. Dyn.
| volume = 5
| year = 2005
| issue = 1
| pages = 97–109
| issn = 0219-4937
| doi = 10.1142/S0219493705001304
|mr=2118757}} (See footnote (e) on p. 104)

[[Category:Random dynamical systems]]

Harmonic coordinate condition - Revision history

en>JRSpriggs: /* More variant forms */ if you go beyond linear