Admissibility and Generalized Nonuniform Dichotomies for Nonautonomous Random Dynamical Systems

The paper proves that, under the lacunary growth condition on µ and a forward bound (15), the unique solvability of the nonhomogeneous problem in the pair of Banach spaces Y_Π^C(Ω̃) and Y_0(Ω̃) implies a µ-dichotomy with properties P1–P5; see Theorem 3.3 and the definition of µ-dichotomy (P1–P5) in Section 2 . The proof constructs a bounded inverse T = (ℒ)^{-1} (Lemma 3.4), builds the invariant splitting and measurable projection P, establishes invertibility on ker P and measurability of the backward inverse, and then derives the dichotomy estimates using Lemma 3.1 and the lacunary hypothesis . The candidate solution follows the standard Green–Perron admissibility scheme: it frames the hypothesis as bijectivity of the difference operator ℒ on (Y_Π^C, Y_0), defines P via an impulse forcing and T, sketches P1–P4 and obtains P5 using T’s boundedness, the forward bound, and the lacunary property. While the model omits some lemmas (e.g., measurability and idempotence details, the explicit backward inverse construction, and the precise use of Lemma 3.1), its outline is consistent with the paper’s argument and relies on the same admissibility→dichotomy mechanism; hence both are correct, although the approaches differ in construction details of P and in how estimates are organized.