Connectedness of random set attractors

Topological properties of random attractors are essential to understand the asymptotic behavior of random dynamical systems. In the deterministic case, set attractors of continuous-time systems are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space.

We prove that random attractors of continuous-time systems, which attract compact sets almost surely, are connected. Moreover, we present an example where compact sets converge to the attractor in probability and the attractor is not connected.

This is joint work with M. Scheutzow.