Positive irreducible semigroups and their long-time behaviour

TL;DR

This paper explores the long-term behavior of positive irreducible semigroups, focusing on how positivity influences convergence to equilibrium and the phenomenon of positivity emerging only after some time.

Contribution

It provides a comprehensive overview of Perron-Frobenius theory for semigroups, highlighting new insights into positivity's role in long-time dynamics and recent phenomena of delayed positivity.

Findings

Positivity can ensure convergence to equilibrium as time approaches infinity.

Positivity may only appear after a certain time, not initially.

The theory has numerous applications across different fields.

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.