  We provide sufficient conditions for synchronization by noise, i.e. under
these conditions we prove that weak random attractors for random dynamical
systems consist of single random points. In the case of SDE with additive
noise, these conditions are also essentially necessary. In addition, we provide
sufficient conditions for the existence of a minimal weak point random
attractor consisting of a single random point. As a result, synchronization by
noise is proven for a large class of SDE with additive noise. In particular, we
prove that the random attractor for an SDE with drift given by a
(multidimensional) double-well potential and additive noise consists of a
single random point. All examples treated in [Tearne, PTRF; 2008] are also
included.
