2013/08/22 by Joanna Jaroszewska, Jaroszewska, Joanna
Mathematics · #37A30 #60J25 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1308.4967
openalex publication_date 2013/08/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
T. Szarek [Stud. Math. 189 (2008), § 4] have discussed the relationship between two important notions concerning Markov semigroups: the asymptotic strong Feller property and the e-property, asserting that the former property implies the latter one. In this very short note we rectify this issue exhibiting a simple example of a Markov-Feller semigroup enjoying the asymptotic strong Feller property, for which the e-property is not satisfied. (See also the comment on a connection between the asymptotic strong Feller property and the e-property by T. Szarek, D. Worm [ETDS 32 (2012), § 1]). Additionally we give a very simple example - in comparing with the one given by T. Szarek [Stud. Math. 189 (2008), § 4] - showing that also the converse implication does not hold.