Invariant forward attractors of non-autonomous random dynamical systems

Abstract

Forward attractors, especially invariant ones, of non-autonomous and random dynamical systems have not been as well studied as pullback attractors. This is mainly due to the different role that time plays in each case. Pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past, whereas forward attractors are concerned with what happens in the distant future. Moreover, if they exist, they need not be unique. In this paper, ideas introduced in Kloeden [20] for asymptotically invariant forward attracting sets of non-autonomous ODEs are generalized and extended to study strictly invariant forward attractors of non-autonomous random dynamical systems on metric spaces. These are formulated as processes that include the effects of both noise and deterministic time variation. It is shown that the random forward attractors, when they exist, can be constructed by a pullback argument inside a positively invariant set. However, this is only a necessary condition, so additional sufficiency conditions are also given. Since forward attractors need not exist in dissipative systems, an alternative object, specifically the closed union of the forward omega limit sets for each initial time of a forward absorbing set is investigated. It is shown to be asymptotically invariant under certain conditions, so looks more and more like an invariant attractor as conventionally understood. It resembles the Vishik uniform attractor, but need not attract uniformly in the initial times and does not even require the system to be defined in the distant past.