\Omega_{B} (\omega) := \left\{ x \in X \left| \exists t_{n} \to + \infty, \exists b_{n} \in B(\vartheta_{-t_{n}} \omega) \mathrm{\,s.t.\,} \varphi (t_{n}, \vartheta_{-t_{n}} \omega) b_{n} \to x \mathrm{\,as\,} n \to \infty \right.
Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth.