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Asymptotic autonomy of random attractors for BBM equations with Laplace-multiplier noise
Renhai Wang,Yangrong Li
Keywords:Random attractor, asymptotic autonomy, backward compactness, Benjamin-Bona-Mahony equation, Laplace-multiplier noise, backward tempered set, measurability.
      We study asymptotic autonomy of random attractors for possibly non-autonomous Benjamin-Bona-Mahony equations perturbed by Laplace-multiplier noise. We assume that the time-indexed force converges to the time-independent force as the time-parameter tends to negative infinity, and then show that the time-indexed force is backward tempered and backward tail-small. These properties allow us to show that the asymptotic compactness of the non-autonomous system is uniform in the past, and then obtain a backward compact random attractor when the attracted universe consists of all backward tempered sets. More importantly, we prove backward convergence from time-fibers of the non-autonomous attractor to the autonomous attractor. Measurability of solution mapping, absorbing set and attractor is rigorously proved by using Egoroff, Lusin and Riesz theorems.