Ergodic measures of Markov semigroups with the e-property

Abstract

AbstractWe study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property, an equicontinuity condition. We introduce a weak concentrating condition around a compact set K and show that this condition has several implications on the set of ergodic measures, one of them being the existence of a Borel subset K0 of K with a bijective map from K0 to the ergodic measures, by sending a point in K0 to the weak limit of the Cesàro averages of the Dirac measure on this point. We also give sufficient conditions for the set of ergodic measures to be countable and finite. Finally, we give a quite general condition under which the Cesàro averages of any measure converge to an invariant measure.