Measurability of Random Attractors for Quasi Strong-to-Weak Continuous Random Dynamical Systems

Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise

“…In fact, this measurability (in more regular spaces) can be deduced from some abstract results on the concept of quasi-continuity, which seems to be first introduced by Li and Guo [22]. While the relationship between measurability and quasi-continuity has been discussed recently by Cui, Langa and Li [14].…”

“…Let ω 0 ∈ Ω be fixed and suppose ω k ∈ Ω such that ρ(ω k , ω 0 ) → 0 as k → ∞. We denote by (14) and the linearity of A , we have…”

“…Since Y → X and X * → Y * densely, it follows from inheritability of the quasi-continuity (see [14,Prop.3]) that the mapping is quasi-continuous from (Ω, ) to Y . Then, by the measurability of a quasi-continuous mapping (see [14,Prop.5]), the solution mapping is (F, B(Y )) measurable as required. It is similar to show that the solution mapping is (F, B(Z)) measurable.…”

“…In the definition of a bi-spatial random attractor given above, we require the F-measurability holds true in both state spaces, while the bi-spatial attractor given in [21] is F-measurable in the initial space only, and the bi-spatial attractor given in [14] is F-measurable in the terminate space only. The above definition may be more reasonable.…”

“…We drop the superscript for convenience. Taking the inner product of (14) with A v 2 in (L 2 (O), · g ), we find that,…”