In the mathematical theory of random dynamical systems, an absorbing set is a subset of the phase space that exhibits a capturing property.
It acts like a gravitational center, with the property that all trajectories of the system eventually enter and remain within that set.
As with many concepts related to random dynamical systems, it is defined in the pullback sense, which means they are understood through their long-term behavior.
The existence and properties of absorbing sets are fundamental to establishing the existence of global attractors and understanding the asymptotic behavior of solutions.
A random compact set K : Ω → 2X is said to be absorbing if, for all d-bounded deterministic sets B ⊆ X, there exists a (finite) random time τB : Ω → 0, +∞) such that This is a definition in the pullback sense, as indicated by the use of the negative time shift θ−t.