Dissertation

Semigroups on Spaces of Measures

Promotor: S.M. Verduyn Lunel, Co-promotor: S.C. Hille

Author: Daniël Worm
Date: 16 September 2010
Links: Thesis in Leiden Repository

This thesis deals with Markov operators and semigroups. A Markov operator is a positive linear operator on the space of finite measures on some state space that preserves mass. A Markov semigroup is a family of Markov operators parametrised by the positive real numbers, satisfying the semigroup property. These appear naturally in various places: deterministic dynamical systems, iterated function systems, structured population models and more generally Markov chains and Markov processes. We will study general Markov operators and semigroups in a functional analytic framework. Because the usual topology on the space of measures, given by the total variation norm, is often too strong for applications, we consider weaker topologies on the space of measures. and study continuity properties of Markov semigroups and their restriction to invariant subspaces. In the latter part of the thesis we provide ergodic decompositions, yielding, among other things, a characterisation of ergodic measures and an 'explicit' integral decomposition of invariant measures into ergodic measures. Under extra equicontinuity assumptions the ergodic decompositions have some nice properties, allowing us to find various characterisations for the existence, uniqueness, mean ergodicity and stability of invariant measures, and giving us extra information on the set of ergodic measures.