An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces
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An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces Daniël T.H. Worm · Sander C. Hille
Received: 16 February 2010 / Accepted: 1 July 2011 / Published online: 6 August 2011 © Springer Science+Business Media B.V. 2011
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). Keywords Markov semigroups · Ergodic decomposition · Ergodic measures · Resolvent operator Mathematics Subject Classification (2000) 60J25 · 37A30 · 47A35 · 60J35 1 Introduction Regular Markov semigroups appear naturally in the context of continuous-time Markov processes as transition operators. If Xt is the state of the process at time t , i.e. a random variable that takes values in a measurable space S, and μ0 is the law of X0 , then the law of Xt is given by P (t)μ0 . 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 D.T.H. Worm () · S.C. Hille Mathematical Institute, University Leiden, P.O. Box 9512, 2300 RA, Leiden, The Netherlands e-mail: [email protected] S.C. Hille e-mail: [email protected]
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D.T.H. Worm, S.C. Hille
dynamical system in the set of probability measures defined by a Markov semigroup is of special interest, in particular the question of existence and characterisation of invariant and ergodic probability measures for this semigroup. It is well-established that an invariant measure μ can be decomposed into an integral of ergodic measures. That is, there exists a representing measure ρμ on the set of ergodic measures Perg (S) for the semigroup (P (t))t≥0 , such that μ(E) = ε(E) dρμ (ε), (1.1) Perg (S)
see e.g. [31], Chap. 6. Dynkin [10] developed a general (probabilistic) theoretical framework for such decompositions, revolving around so-called (H -) sufficient statistics and σ algebras. A second line of research concerning ergodic decompositions goes back to pioneering work by Krylov and Bogolioubov [22], Beboutov [5] and Yosida [35] (the ‘KBBYapproach’), continued by Hernàndez-Lerma and Laserre [17], Costa and Du
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