Lower Bounds and the Asymptotic Behaviour of Positive Operator Semigroups

TL;DR

This paper extends classical results on the convergence of positive operator semigroups, providing new proofs and generalizations for semigroups with weaker assumptions, including individual lower bounds and Banach lattice settings.

Contribution

It introduces simplified proofs and broadens the scope of convergence theorems for positive semigroups, including non-Markov and individual lower bound cases.

Findings

Semigroups with a non-trivial lower bound are strongly convergent.

Bounded semigroups without Markov property also converge under similar conditions.

Results are applicable to general Banach lattices and include counterexamples for optimality.

Abstract

If $(T_t)$ is a semigroup of Markov operators on an $L^1$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t \to \infty$. In this article we generalise and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalise a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.