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arXiv · 2005.08059

Positive irreducible semigroups and their long-time behaviour

Wolfgang Arendt·Jochen Glück
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Abstract

The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of applications. By a brief walk-through of the field and with many examples, we highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as $t \to \infty$. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large $t$, while being absent for smaller times.

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Wolfgang Arendt, Jochen Glück. 2020-10-06. Positive irreducible semigroups and their long-time behaviour. https://doi.org/10.1098/rsta.2019.0611

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