Ergodicity in nonautonomous linear ordinary differential equations

The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to th...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2019-11, Vol.479 (2), p.1441-1455
Main Authors: Pituk, Mihály, Pötzsche, Christian
Format: Article
Language:English
Subjects:
Ergodicity
Hilbert's projective metric
Ordinary differential equation
Perron–Frobenius theory
Positivity
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Summary:The weak and strong ergodic properties of nonautonomous linear ordinary differential equations are considered. It is shown that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2019.07.005