An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

Abstract

For a regular jointly measurable Markov semigroup on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this semigroup in terms of subsets of the state space. In this way we extend results by Costa and Dufour (J. Appl. Probab. 43:767-781, 2006). As a consequence we obtain an integral decomposition of every invariant probability measure in terms of the ergodic probability measures. Our approach is completely centered around the reduction to and relationship with the case of a single regular Markov operator associated to the Markov semigroup, the resolvent operator, which enables us to fully exploit results in that situation (Worm and Hille in Ergod. Theory Dyn. Syst. 31(2):571-597, 2011).

Ergodic measures of Markov semigroups with the e-property

“…Recently S.C. Hille and the second author started considering ergodic decompositions of general Markov semigroups on Polish spaces. It appeared that then a quite general Yosida-type decomposition of the state space holds [18]. Similar results were obtained by O. Costa and F. Dufour in [3] in the setting of locally compact separable metric spaces.…”

“…We shall write [x] to denote the equivalence class of x. In [17,18] it is shown that Γ t , Γ cp and Γ cpie and [x] are Borel sets and that µ(Γ t ) = µ(Γ cp ) = µ(Γ cpie ) for all invariant probability measures µ. Furthermore, P erg (S) = {ε x : x ∈ Γ cpie } and ε x ([x]) = 1 for all x ∈ Γ cpie .…”

Mean ergodic theorems on norming dual pairs

“…The literature on weak Cesàro-convergence of Markov semigroups is rather extensive. Let us mention [9,19,20,21]. However, a characterization of mean ergodicity in the spirit of Theorem 1.1 is still missing.…”

The ergodic decomposition defined by actions of amenable groups

“…Our goal here is to obtain an ergodic decomposition defined by fairly general left actions of amenable groups on locally compact separable metric spaces. The decomposition discussed here is an extension of the decomposition defined by transition probabilities (obtained in [22] and in Worm and Hille [16]) and of the decomposition defined by transition functions (obtained in Chapter 5 and Section 6.2 of [23] and in Worm and Hille [17]); see also Worm's thesis [15].…”