On the structure of attractors and invariant measures for a class of monotone random systems

Under rather general conditions we show that any monotone random dynamical system on an (admissible) subset of a partially ordered Banach space V has a unique invariant measure. This measure is Dirac, i.e. it is generated by some stationary process. If the cone V + of non-negative elements of V is normal, then this stationary process is a global random attractor with respect to convergence in probability. As examples we consider one-dimensional ordinary and retarded stochastic differential equations, a stochastic model of a biochemical control circuit, a class of parabolic stochastic partial differential equations (PDEs) with additive noise and interacting particle systems.