A Limit Set Trichotomy for Order-Preserving Random Systems

We study the asymptotic behavior of order-preserving (or monotone) random systems which have an additional concavity property called sublinearity (or subhomogeneity), frequently encountered in applications. Sublinear random systems are contractive with respect to the part metric, hence random equilibria are unique and asymptotically stable in each part of the cone. Our main result is a random limit set trichotomy, stating that in a given part either (i) all orbits are unbounded, or (ii) all orbits are bounded but their closure reaches out to the boundary of the part, or (iii) there exists a unique, globally attracting equilibrium. Several examples, including affine and cooperative systems, are given.