Monotone random systems theory and applications
type: Item
instance of: scholarly article
title: Monotone random systems theory and applications
zbMATH Open document ID: 1023.37030
published in: Lecture Notes in Mathematics
publication date: 2002-06-04T00:00:00Z
full work available at URL: t1779.htm
review text: 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 rando…
reviewed by: P. Imkeller
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 34C12
Mathematics Subject Classification ID: 37C65
Mathematics Subject Classification ID: 82C05
Mathematics Subject Classification ID: 37G35
Mathematics Subject Classification ID: 34F05
zbMATH DE Number: 1749033
zbMATH Keywords: random dynamical system
zbMATH Keywords: monotone system
zbMATH Keywords: random differential equation
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: asymptotic property
zbMATH Keywords: random attractor
zbMATH Keywords: random equilibrium
author: I. D. Chueshov
MaRDI profile type: Publication
OpenAlex ID: W338436792
Wikidata QID: Q59757571
DOI: 10.1007/B83277
description: scientific article; zbMATH DE number 1749033