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If the RDS (θ, ϕ) possesses a random attractor in the universe D, then any backward invariant random closed set from D lies in the attractor. In particular the attractor contains every equilibrium u(ω) with the property {u(ω)} ∈ D. Now we prove a theorem on the existence of random attractors. 1. Let (θ, ϕ) be an asymptotically compact RDS in the universe D with an attracting random compact set B0 ∈ D. Then this RDS possesses a unique random compact pull back attractor {A(ω)} in the universe D, and A(ω) ⊂ B0 (ω) for all ω ∈ Ω.

Now we consider asymptotically compact affine RDS. 2. Assume that D is a universe of subsets of X such that {0} ∈ D and for any D ∈ D and λ > 0 the set ω → λD(ω) := {x : xλ−1 ∈ D(ω)} belongs to D. 48) and with an attracting random compact set B0 ∈ D. 51) exists for all ω ∈ Ω and is an equilibrium for the RDS (θ, ϕ). e. 52) −t ω) for any D ∈ D. Moreover {u(ω)} ∈ D and the RDS (θ, ϕ) possesses a unique equilibrium with this property. 9 Dissipative Linear and Affine RDS 47 Proof. 49) we get ψ(τ, θ−τ ω) = Φ(t, θ−t ω)ψ(τ − t, θ−τ ω) + ψ(t, θ−t ω), τ >t≥0.

9 Dissipative Linear and Affine RDS 47 Proof. 49) we get ψ(τ, θ−τ ω) = Φ(t, θ−t ω)ψ(τ − t, θ−τ ω) + ψ(t, θ−t ω), τ >t≥0. 53) Since {0} ∈ D, we have that ψ(τ, θ−τ ω) = ϕ(τ, θ−τ ω)0 → B0 (ω) as τ → ∞. 54) Hence there exist τn = τn (ω) → ∞ and b ∈ B0 (ω) such that ψ(τn , θ−τn ω) → b as n→∞. Since ψ(τ − t, θ−τ ω) = ϕ(τ − t, θ−τ ω)0 → B0 (θ−t ω) as τ →∞, we can choose a subsequence {τnk } and an element b1 (t) ∈ B0 (θ−t ω) such that ψ(τnk − t, θ−τnk ω) → b1 (t) as n → ∞. 53) we have b = Φ(t, θ−t ω)b1 (t) + ψ(t, θ−t ω) .

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General Facts about Random Dynamical Systems

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