Order-preserving random dynamical systems: equilibria, attractors, applications

This paper is meant as a first step towards a systematic study of order-preserving (or monotone) random dynamical systems, in particular of their long-term behavior and their attractors. A series of examples (including random I stochastic cooperative systems and random I stochastic parabolic equations) gives ample proof of the usefulness of the subject. We show that, given a sub- and super-equilibrium, there is always an equilibrium between them. Also, the random attractor of an order-preserving random dynamical system is bounded below and above by equilibria. We finally show by way of an example that omega-limit sets can contain non-trivial totally ordered subsets.

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