Ergodic properties of weak asymptotic pseudotrajectories for semiflows
type: Item
instance of: scholarly article
title: Ergodic properties of weak asymptotic pseudotrajectories for semiflows
zbMATH Open document ID: 0998.37013
DOI: 10.1023/A:1026463628355
author: Michel Benaïm
author: Sebastian J. Schreiber
published in: Journal of Dynamics and Differential Equations
publication date: 2001-07-05T00:00:00Z
review text: The first author, with \textit{M. W. Hirsch} [J. Dyn. Differ. Equ. 8, 141-176 (1996; Zbl 0878.58053)], studied the limiting behavior of asymptotic pseudotrajectories for a semiflow, which they defined as follows. If \(\Phi\) is a semiflow on a metric space, then \(\{ X(t):t\geq 0\}\) is an asymptotic pseudotrajectory for \(\Phi\) if for any \(T>0\), one has \[ \lim_{t\to\infty} \sup_{0\leq h\leq T} d(X(t+h),\Phi_h(X(t)))=0. \] This theory provides a general framework for the study of long-term behavior of a large class of nonautonomous systems of difference/differential equations. In this paper, the authors are concerned with the ergodic and statistical behavior of asymptotic trajectories, and in fact in a class of more general statistical processes \(X(t)\) on the metric space which they call weak asymptotic pseudotrajectories. The main results are: (1) the weak\(^*\) limit points of the empirical measures for \(X\) are almost surely \(\Phi\)-invariant measures; (2) for any semiflow \(\Phi\), there exists a weak asymptotic pseudotrajectory of \(\Phi\) such that the set of weak\(^*\) limit points of its empirical measures is almost surely equal to the set of ergodic measures for \(\Phi\); (3) given an asymptotic pseudotrajectory \(X\) for a semiflow \(\Phi\), then they derive conditions on \(\Phi\) that insure convergence of the empirical measures of \(X\).
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 37H99
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 37M25
Mathematics Subject Classification ID: 60G48
zbMATH DE Number: 1556551
zbMATH Keywords: asymptotic autonomous semiflows
zbMATH Keywords: stochastic algorithms
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: invariant measures
zbMATH Keywords: minimal center of attraction
MaRDI profile type: Publication
full work available at URL: a:1026463628355
OpenAlex ID: W2123597892
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description: scientific article; zbMATH DE number 1556551