An ergodic decomposition defined by regular jointly measurable Markov semigroups on Polish spaces Dani¨ el T. H. Worm, Sander C. Hille Mathematical Institute, University Leiden P.O. Box 9512, 2300 RA Leiden, The Netherlands E-mail: dworm@math.leidenuniv.nl, shille@math.leidenuniv.nl == MI-2010-02 == February 17, 2010 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. Prob. 43, 767–781). 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, “Ergodic decompositions associated to regular Markov operators on Polish spaces”, accepted by Ergodic Theory and Dynamical Systems ). 1 Introduction Regular Markov semigroups appear naturally in the context of continuous-time Markov processes as transition operators. If X t is the state of the process at time t, i.e. a random variable that takes values in a measurable space S, and µ 0 is the law of X 0 , then the law of X t is given by P (t)µ 0 . Here each P (t) is a regular Markov operator: an additive and positively homogeneous map on the convex cone of positive finite measures on S, given by a transition kernel, that leaves the set of probability measures invariant. The family of operators (P (t)) t≥0 forms a one-parameter semigroup. Accordingly, the behaviour of the dynamical system in the set of probability measures defined by a Markov semigroup is of 1