Additive noise destroys the random attractor close to bifurcation

TL;DR

This paper demonstrates how additive noise can stabilize a high-dimensional PDE system near bifurcation, causing all trajectories to converge to a single stationary solution, even with degenerate noise.

Contribution

It introduces a novel approach to stabilization by noise that does not depend on monotonicity, applicable to complex PDEs with degenerate noise.

Findings

Additive noise destroys the high-dimensional random attractor.

Near bifurcation, all trajectories converge to a stationary solution.

The method applies to PDEs with higher order operators on unbounded domains.

Abstract

We provide an example for stabilization by noise. Our approach does not rely on monotonicity arguments due to the presence of higher order differential operators or mixing properties of the system as the noise might be highly degenerate. In the examples a scalar additive noise destroys a high-dimensional random attractor of a PDE on an unbounded domain. In the presence of small noise close to bifurcation all trajectories converge to a single stationary solution.