Whenever a deterministic system like an ODE or PDE does not possess an
asymptotically stable constant solution but if noise is added then there
exists a random attractor which consists of a single (random) point,
then we call this phenomenon "synchronization by noise".
We first provide some specific examples and then present sufficient
conditions for synchronization to occur. Our results can be applied to
a large class of SDEs and some SPDEs with additive noise and to rather
general order-preserving random dynamical systems.
This is joint work with Franco Flandoli (Pisa) and Benjamin Gess (Leipzig).