Connectedness of random set attractors

TL;DR

This paper investigates the connectedness of random set attractors in continuous-time random dynamical systems, proving connectedness under weak conditions and providing counterexamples under stronger assumptions.

Contribution

It establishes the connectedness of pullback attractors on connected spaces under weak continuity conditions and introduces a lemma about nullsets in convergence, with a counterexample for stronger assumptions.

Findings

Connectedness of pullback attractors on connected spaces under weak conditions.

Counterexample of a non-connected attractor with stronger assumptions.

A lemma on nullsets in convergence independent of compact sets.

Abstract

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynamical system we prove connectedness of the pullback attractor on a connected space. Additionally, we provide an example of a weak random set attractor of a random dynamical system with even more restrictive continuity assumptions on an even path-connected space which even attracts all bounded sets and which is not connected. On the way to proving connectedness of a pullback attractor we prove a lemma which may be of independent interest and which holds without the assumption that the state space is connected. It states that even though pullback convergence to the attractor allows for exceptional nullsets which may depend on the compact set, these nullsets can be chosen independently of the compact set (which is clear for $\sigma$-compact spaces but not at all clear for spaces which are not $\sigma$-compact).