The e-property of asymptotically stable Markov semigroups

TitleThe e-property of asymptotically stable Markov semigroups
Publication TypeJournal Article
Year of Publication2022
AuthorsKukulski R, Wojewódka-Ściążko H
Keywords37A30, 46E27, 46N30, 60B10, 60J25, FOS: Mathematics, Probability (math.PR)

The relations between the e-property and the asymptotic stability of Markov semigroups are studied. In particular, it is shown that any stochastically continuous and asymptotically stable Markov-Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the e-property. Moreover, it is proved that any Markov-Feller semigroup, which is stochastically continuous, and which possesses the eventual e-property, has the e-property as well. An example pointing out that such an implication does not have to hold without assuming stochastic continuity is provided.