  We provide sufficient conditions for weak synchronization by noise for
order-preserving random dynamical systems on Polish spaces. That is, under
these conditions we prove the existence of a weak point attractor consisting of
a single random point. This generalizes previous results in two directions:
First, we do not restrict to Banach spaces and second, we do not require the
partial order to be admissible nor normal. As a second main result and
application we prove weak synchronization by noise for stochastic porous media
equations with additive noise.
