During the talk I will describe the relationship between e-property and asymptotic stability of Markov semigroups. In particular, I will show that every 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, I will show 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 will be presented as well. My talk will be based on the following two recent results:

1. R. Kukulski, H. Wojewódka-Ściążko: The e-property of asymptotically stable Markov semigroups, arXiv:2211.16424v1 [math.PR]
2. R. Kukulski, H. Wojewódka-Ściążko: The e-property of asymptotically stable Markov–Feller operators, Colloquium Mathematicum 165 (2021), 269-283.