Ergodic decompositions associated with regular Markov operators on Polish spaces

Abstract

For any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

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

“…In [32] we obtained a Yosida-type ergodic decomposition for regular Markov operators on Polish spaces, extending results by Hernàndez-Lerma, Lasserre and Zaharopol [17,18,36,37] on locally compact separable metric spaces. In this paper we will show that analogous results hold for regular Markov semigroups.…”

“…Moreover, the Cesàro averages for the semigroup and the resolvent, have the same convergence properties (made precise in Theorem 3.7 and its corollaries). These results imply that the Yosida-type ergodic decomposition associated to R, from [32], actually works for the semigroup (P (t)) t≥0 as well. We also obtain a full Yosida-type ergodic decomposition in Sect.…”

“…Following [14,20,26,32], we will call a Markov operator P regular if it satisfies the conditions of Proposition 2.5. The map U : BM(S) → BM(S) associated to a regular Markov operator P is unique and we call it the dual of P .…”

“…4.1 we define ergodicity of invariant measures for Markov operators and Markov semigroups and give several equivalent characterisations. We prove analogues of the ergodic decomposition results from [32] in the setting of Markov semigroups in Sect. 4.2, and give a full Yosida-type ergodic decomposition in Sect.…”

“…In our approach we build on definitions and results from [19,20] and an ergodic decomposition associated to Markov operators from [32] that we will recall in Sect. 2 for convenience.…”