Minimal Random Attractors

TL;DR

This paper proves the existence of minimal pullback and weak random attractors for general random dynamical systems, highlighting their uniqueness among attractors that attract arbitrary families of sets, unlike forward attractors.

Contribution

It introduces the concept of minimal random attractors for pullback and weak attractors and demonstrates their existence under broad conditions.

Findings

Minimal pullback and weak attractors always exist for given families of sets.

Forward attractors may not have minimal elements, as shown by an example.

The attractor concept applies to very general families of sets.

Abstract

It is well-known that random attractors of a random dynamical system are generally not unique. We show that for general pullback attractors and weak attractors, there is always a minimal (in the sense of smallest) random attractor which attracts a given family of (possibly random) sets. We provide an example which shows that this property need not hold for forward attractors. We point out that our concept of a random attractor is very general: The family of sets which are attracted is allowed to be completely arbitrary.