Uniform convergence of stochastic semigroups

TL;DR

This paper establishes that for irreducible stochastic semigroups on L^1-spaces, uniform convergence to equilibrium is equivalent to the semigroup being partially integral and uniformly mean ergodic, linking structural properties to convergence behavior.

Contribution

It proves a new equivalence between uniform convergence, partial integrality, and mean ergodicity for stochastic semigroups, extending previous results on strong convergence.

Findings

Uniform convergence is equivalent to partial integrality and mean ergodicity.

A new Tauberian theorem characterizes uniform convergence via partial integrality and dual irreducibility.

The proof combines a uniform Lasota-Yorke lower bound with Banach lattice techniques.

Abstract

For stochastic $C_0$-semigroups on $L^1$-spaces there is wealth of results that show strong convergence to an equilibrium as $t \to \infty$, given that the semigroup contains a partial integral operator. This has plenty of applications to transport equations and in mathematical biology. However, up to now partial integral operators do not play a prominent role in theorems which yield uniform convergence of the semigroup rather than only strong convergence. In this article we prove that, for irreducible stochastic semigroups, uniform convergence to an equilibrium is actually equivalent to being partially integral and uniformly mean ergodic. In addition to this Tauberian theorem, we also show that our semigroup is uniformly convergent if and only if it is partially integral and the dual semigroup satisfies a certain irreducibility condition. Our proof is based on a uniform version of a lower bound theorem of Lasota and Yorke, which we combine with various techniques from Banach lattice theory.