Additive noise destroys the random attractor close to bifurcation

Abstract

We provide an example for stabilization by noise. Due to the presence of higher order differential operators our approach does not rely on monotonicity arguments, i.e. the preserved order of solutions. Moreover, as the noise is highly degenerate mixing properties of the system might not be available. In our examples already a scalar additive noise destroys the complexity of a high-dimensional deterministic attractor of a PDE on an unbounded domain. The main result shows that by adding a certain amount of noise all trajectories converge to a single stationary solution. Close to bifurcation there is a lower bound on the amount of noise necessary for this stabilization, which depends on the distance to bifurcation, and the presence of small (but not arbitrarily small) noise already suffices. We focus on stochastic PDEs posed on unbounded domains without any decay condition at infinity. This setting allows for spatially constant or periodic solutions of arbitrary period. But we need to work in weighted spaces and establish the existence of random attractors in that setting first.