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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. I. General theory


Authors: Janusz Mierczyński and Wenxian Shen
Journal: Trans. Amer. Math. Soc. 365 (2013), 5329-5365
MSC (2010): Primary 37H15, 37L55, 37A30; Secondary 15B52, 34F05, 35R60
Published electronically: March 26, 2013
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Abstract: This series of papers is concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part of the series focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming part(s) the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.


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Additional Information

Janusz Mierczyński
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland
Email: mierczyn@pwr.wroc.pl

Wenxian Shen
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama 36849
Email: wenxish@auburn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05814-X
PII: S 0002-9947(2013)05814-X
Keywords: Random dynamical system, ordered Banach space, principal Lyapunov exponent, principal Floquet subspaces, exponential separation, entire positive solution, multiplicative ergodic theorem, Hilbert projective metric, skew-product semiflow
Received by editor(s): February 16, 2011
Received by editor(s) in revised form: February 11, 2012
Published electronically: March 26, 2013
Additional Notes: The first author was supported by Resources for Science in years 2009-2012 as research project (grant MENII N N201 394537, Poland)
The second author was partially supported by NSF grant DMS-0907752
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.