Weak Random Attractors

We deene point attractors and set attractors for random dynamical systems via convergence in probability forward in time. This deenition of a set attractor is weaker than the usual one via almost sure pullback convergence. We derive basic properties of our weak random attractors such as uniqueness, support of invariant measures and invari-ance under (random) coordinate transformations. The notion of weak random attractors allows us to consider local and relative attractors. We discuss attractor{repeller decompositions and give a characterization of the basin of attraction. 1. Introduction Attractors play an important role in the theory of dynamical systems. A global attractor for an autonomous dynamical system given by a (semi{))ow or the iterates of a map is a compact invariant set which attracts all trajec-tories of points or compact resp. bounded sets as time tends to innnity. That