We provide sufficient conditions for weak synchronization/stabilization 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.
"Synchronization by noise for order-preserving random dynamical systems." Ann. Probab. 45 (2) 1325 - 1350, March 2017. https://doi.org/10.1214/16-AOP1088