Absorbing set (random dynamical systems)

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 sense.

Definition

Consider a random dynamical system φ on a complete separable metric space (X, d), where the noise is chosen from a probability space (Ω, Σ, P) with 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 (finite) random time τ_B : Ω → 0, +∞) such that

$t \geq \tau_{B} (\omega) \implies \varphi (t, \vartheta_{-t} \omega) B \subseteq K(\omega).$

This is a definition in the pullback sense, as indicated by the use of the negative time shift ϑ_−t.

References

• Robinson, James C.; Tearne, Oliver M. (2005). "Boundaries of attractors of omega limit sets". Stoch. Dyn. 5 (1): 97–109. doi:10.1142/S0219493705001304. ISSN 0219-4937. MR 2118757. (See footnote (e) on p. 104)