Invariant Probabilities of Markov-Feller Operators and Their Supports (Frontiers in Mathematics)

In this book invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, certain time series, etc. Rather than dealing with the processes, the transition probabilities and the operators associated with these processes are studied. Main features: - an ergodic decomposition which is a reference system for dealing with ergodic measures - formulas for the supports of invariant probability measures, some of which can be used to obtain algorithms for the graphical display of these supports - helps to gain a better understanding of the structure of Markov-Feller operators, and, implicitly, of the discrete-time homogeneous Feller processes - special efforts to attract newcomers to the theory of Markov processes in general, and to the topics covered in particular - most of the results are new and deal with topics of intense research interest. Invariant Probabilities of Markov-Feller Operators and Their Supports......Page 3 Contents......Page 7 Introduction......Page 9 Acknowledgements......Page 13 1. Preliminaries on Markov - Feller Operators......Page 15 1.1 Markov-Feller Pairs and Transition Probabilities......Page 16 1.2 Invariant Probabilities......Page 31 1.3 Special Topics: Topological Limits, Banach Limits,the Separability of C0(X), Order in Vector Spaces,and Equicontinuity......Page 38 2. The Krylov -Bogolioubov - Beboutoff -Yosida (KBBY) Decomposition......Page 51 2.1 A Weak KBBY Decomposition......Page 52 2.2 Supports of Elementary Invariant and ErgodicMeasures......Page 58 2.3 Minimal Markov–Feller Pairs......Page 64 3. Unique Ergodicity of Markov - Feller Operators and Related Topics......Page 71 3.1 Supports of Invariant Probabilities of CertainMarkov–Feller Pairs......Page 72 3.2 Generic Points and Unique Ergodicity......Page 75 3.3 Generic Points and Ergodic Measures......Page 82 4. Equicontinuity......Page 89 4.1 Unique Ergodicity and Equicontinuity......Page 90 4.2 A Diagonalization Procedure: Technical Preliminaries for Mean Ergodic Theorems......Page 98 4.3 Mean Ergodic Theorems......Page 104 Bibliography......Page 115 Index......Page 120 In this book invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, certain time series, etc. Rather than dealing with the processes, the transition probabilities and the operators associated with these processes are studied. Main features: - an ergodic decomposition which is a "reference system" for dealing with ergodic measures - "formulas" for the supports of invariant probability measures, some of which can be used to obtain algorithms for the graphical display of these supports - helps to gain a better understanding of the structure of Markov-Feller operators, and, implicitly, of the discrete-time homogeneous Feller processes - special efforts to attract newcomers to the theory of Markov processes in general, and to the topics covered in particular - most of the results are new and deal with topics of intense research interest. This book covers invariant probabilities for a large class of discrete-time homogeneous Markov processes known as Feller processes. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, and certain time series. From the reviews: "A very useful reference for researchers wishing to enter the area of stationary Markov processes both from a probabilistic and a dynamical point of view." --MONATSHEFTE FÜR MATHEMATIK "In this book invariant probabilities for a large class of discrete time homogeneous Markov processes known as Feller processes are discussed. These Feller processes appear in the study of iterated function systems with probabilities, convolution operators, certain time series, etc. Rather than dealing with the processes, the transition probabilities and the operators associated with these processes are studied."--BOOK JACKET