The e-property of asymptotically stable Markov semigroups

TL;DR

This paper explores the relationships between asymptotic stability and the e-property of Markov semigroups on metric spaces, establishing conditions under which these properties are equivalent or imply each other, with implications for numerical stability.

Contribution

It proves that asymptotic stability combined with certain support conditions implies the eventual e-property, and that strong stochastic continuity ensures the e-property if the eventual e-property holds.

Findings

Asymptotic stability with non-empty support interior implies the eventual e-property.

Strong stochastic continuity plus the eventual e-property implies the e-property.

Weak stochastic continuity does not guarantee the e-property unless the space is compact.

Abstract

The relations between asymptotic stability, the eventual e-property and the e-property of Markov semigroups, acting on measures defined on general (Polish) metric spaces, are studied. While usually much attention is paid to asymptotic stability (and the e-property has been for years verified only to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov-Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov-Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual e-property, also has the e-property. We also present an example highlighting that strong stochastic continuity cannot be replaced by its weak counterpart, unless a state space of a process corresponding to a Markov semigroup is a compact metric space.