Ergodic measures of Markov semigroups with the e-property (Q2908174)

The authors are concerned with the study of the set of ergodic measures for a Markov semigroup defined on the class of all bounded Borel measurable functions on a Polish state space. The main assumption on this semigroup is the ``e-property'', an equicontinuity condition for bounded Lipschitz functions. In the first part of the paper, the authors show new consequences of the e-property on the set of ergodic measures. They later study the conclusions, which are drawn from a ``weak concentrating condition'', a concept introduced by them, around some compact set \(K\). The main result states that every ergodic measure is obtained as Cesáro weak limit starting at some point from the given compact set \(K\). In fact, the authors are able to find a Borel subset \(K_0\) of the compact set \(K\) that maps bijectively to the set of ergodic measures. This interesting result allows them to determine how many ergodic measures really exist. Sufficient conditions ensuring the existence of finitely or countably many ergodic measures are also provided. At the end of the paper, it is shown that a quite general condition related to the weak concentrating condition ensures that, on a Markov semigroup with the e-property, for every probability measure the Cesáro weak limit exists and is an invariant measure. This result is followed by interesting corollaries that give necessary and sufficient conditions for a Markov semigroup to be weak\(^{\star}\) mean ergodic and asymptotically stable.