Monotone random systems theory and applications
The analysis of asymptotic properties of random dynamical systems is complicated by the fact that random forcing terms of deterministic systems act nonlocally and therefore tend to destroy for example local equilibrium or bifurcation patterns. The assumption of monotonicity simplifies things a lot, as is shown in the book under review. Random dynamical systems, i.e. random flows on finite- or infinite-dimensional state spaces for which the flow evolution is compatible with the time evolution of the random source, have the monotonicity property if they preserve ordering on the state space. Important classes of random dynamical systems are generated by random or stochastic differential equations which can be mapped into each other by random order preserving diffeomorphisms. Systems generated by random differential equations of the form \[ d x_t(\omega) = f(\theta_t\omega, x_t(\omega)) dt \] are monotonous e.g. provided the generating vector field \(f\) possesses positivity properties. The main aim of the monograph is to investigate the long time behavior of monotonous random dynamical systems, which reduces to the study of equilibria, and attracting sets related to them. This is done for infinite dimensional systems in the general part. When it comes to concrete systems given by their generators such as random or stochastic differential equations, the setting is finite-dimensional. The results are illustrated by a number of examples from different application areas which are revisited along the whole presentation. After the introduction of the concepts of random dynamical systems, their generators and asymptotics in Chapters 1 and 2, the main general results of the book are presented in the two subsequent Chapters. In Chapter 3, notions of partial ordering on vector spaces are introduced and build the basis for the concept of order-preserving random dynamical system. Compactness type criteria for the existence of equilibria are given, and their relationship with random attractors studied. Typically for monotonous systems, equilibria are contained in an interval between two extreme ones. The introduction of the notions of sub-linearity and concavity creates richer structures in Chapter 4, and gives rise to the central theorem of the book, claiming a trichotomy for the limit sets: orbits are either unbounded, or bounded with a closure touching the boundary of the cone in which they live, or are attracted to a global equilibrium. The final Chapters 5 and 6 are devoted to interpreting the main results in the setting of the two important special cases of systems generated by random or stochastic differential equations. Special emphasis is given to one space dimension and to a series of examples.
scientific article; zbMATH DE number 1749033