Attractors for Ergodic and Monotone Random Dynamical Systems

Abstract

We relate ergodicity, monotonicity and attractors of a random dynamical system (rds). Our first result states that an rds which is both monotone and ergodic has a weak random attractor which consists of a single point. Then we show that ergodicity alone is insufficient for the existence of a weak random attractor. In particular we present an rds in ℝd, d ≥ 2 namely an isotropic Brownian flow with drift, whose single-point motion is an ergodic diffusion process and which does not have a weak attractor. It seems that this is the first example of this kind in the literature.