Invariant Probabilities of Markov-Feller Operators and Their Supports

This text is centred on discrete time homogeneous Markov-Feller processes and it investigates supports of corresponding invariant measures. These processes are employed in the study of stochastic difference equations, to be applied in financial mathematics, for example. The processes are dealt with in the framework of the associated transition probabilities and operators. As a principal outcome, the supports of ergodic probabilities and of ergodic Markov-Feller operators are characterized in terms of topological limits. As a setting for handling ergodic probabilities, an original extension of the standard Bogolioubov-Yosida ergodic decomposition is built up. The main parts of the book treat preliminaries on Markov-Feller operators, the Krylov-Bogolioubov-Beboutoff-Yosida decomposition, unique ergodicity and equicontinuity (and unique ergodicity). Even though the book presents new results and deals mostly with topics of intense contemporary research, aid is given for the beginner to help gain a deeper understanding of the theory of Feller processes and Markov processes in general. The book offers good material for an advanced PhD seminar.

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