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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.
List of contents
Introduction.- 1. Preliminaries on Markov-Feller Operators.- 2. The KBBY Decomposition.- 3. Unique Ergodicity.- 4. Equicontinuity.- Bibliography.- Index
Summary
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. Special efforts are made 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.
Additional text
From the reviews:
This short monograph is 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 and opens the way to an interesting and active research area.
Monatshefte für Mathematik
All proofs are detailed, many illuminating examples are included, and familiarity with only basics of measure theory, general topology of metric spaces and functional analysis is sufficient to follow the exposition. Therefore, potential readers of this monograph are not only those looking for information about supports of invariant measures, but everybody interested in Markov operators who may find the author's approach inspiring in many respects.
Applications od Mathematics
“The book is essentially self-contained and very well written, in a reader-friendly style. The work is addressed to researchers interested in Markov-Feller operators and related topics, but it should also be useful to people in several fields, for instance, Markov processes, ergodic theory, and dynamical systems, to name a few.”(MATHEMATICAL REVIEWS)
Report
From the reviews:
This short monograph is 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 and opens the way to an interesting and active research area.
Monatshefte für Mathematik
All proofs are detailed, many illuminating examples are included, and familiarity with only basics of measure theory, general topology of metric spaces and functional analysis is sufficient to follow the exposition. Therefore, potential readers of this monograph are not only those looking for information about supports of invariant measures, but everybody interested in Markov operators who may find the author's approach inspiring in many respects.
Applications od Mathematics
"The book is essentially self-contained and very well written, in a reader-friendly style. The work is addressed to researchers interested in Markov-Feller operators and related topics, but it should also be useful to people in several fields, for instance, Markov processes, ergodic theory, and dynamical systems, to name a few."(MATHEMATICAL REVIEWS)
