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Invariant Probabilities of Transition Functions: Probability and Its Applications

Autor Radu Zaharopol
en Limba Engleză Hardback – 16 iul 2014
The structure of the set of all the invariant probabilities and the structure of various types of individual invariant probabilities of a transition function are two topics of significant interest in the theory of transition functions, and are studied in this book. The results obtained are useful in ergodic theory and the theory of dynamical systems, which, in turn, can be applied in various other areas (like number theory). They are illustrated using transition functions defined by flows, semiflows, and one-parameter convolution semigroups of probability measures. In this book, all results on transition probabilities that have been published by the author between 2004 and 2008 are extended to transition functions. The proofs of the results obtained are new.
For transition functions that satisfy very general conditions the book describes an ergodic decomposition that provides relevant information on the structure of the corresponding set of invariant probabilities. Ergodic decomposition means a splitting of the state space, where the invariant ergodic probability measures play a significant role. Other topics covered include: characterizations of the supports of various types of invariant probability measures and the use of these to obtain criteria for unique ergodicity, and the proofs of two mean ergodic theorems for a certain type of transition functions.
The book will be of interest to mathematicians working in ergodic theory, dynamical systems, or the theory of Markov processes. Biologists, physicists and economists interested in interacting particle systems and rigorous mathematics will also find this book a valuable resource. Parts of it are suitable for advanced graduate courses. Prerequisites are basic notions and results on functional analysis, general topology, measure theory, the Bochner integral and some of its applications.
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Specificații

ISBN-13: 9783319057224
ISBN-10: 3319057227
Pagini: 408
Ilustrații: XVIII, 389 p.
Dimensiuni: 155 x 235 x 27 mm
Greutate: 0.74 kg
Ediția:2014
Editura: Springer International Publishing
Colecția Springer
Seria Probability and Its Applications

Locul publicării:Cham, Switzerland

Public țintă

Research

Cuprins

​Introduction.- 1.Transition Probabilities .- 2.Transition Functions .- 3.Vector Integrals and A.E. Convergence .- 4.Special Topics.- 5.The KBBY Ergodic Decomposition, Part I.- 6.The KBBY Ergodic Decomposition, Part II.- 7.Feller Transition Functions.- Appendices: A.Semiflows and Flows: Introduction.- B.Measures and Convolutions.- Bibliography.- Index.

Recenzii

“The book is written with a high precision regardingdefinitions, notation, symbols and proofs. … The book may serve specialists asa comprehensive review of the topic. On the other hand, because it is soclearly written and almost self-contained, it can be recommended as a perfectprimer for beginners. … The monograph is written for readers searching for anupdated guide to functional methods of Markov (discrete- or continuous-time)processes.” (Wojciech Bartoszek, Mathematical Reviews, November, 2015)
“The present book is designed to provide a useful and complete presentation of the ergodic decomposition for transition functions defined on locally compact separable metric spaces. … The book is of great interest not only to research workers in the field of dynamical systems and the theory of Markov processes but also to scientists such asbiologists, physicists, etc. who need relatively straightforward rigorous mathematical methods in their studies and research as well.” (Chryssoula Ganatsiou, zbMATH, Vol. 1302, 2015)

Textul de pe ultima copertă

The structure of the set of all the invariant probabilities and the structure of various types of individual invariant probabilities of a transition function are two topics of significant interest in the theory of transition functions, and are studied in this book. The results obtained are useful in ergodic theory and the theory of dynamical systems, which, in turn, can be applied in various other areas (like number theory). They are illustrated using transition functions defined by flows, semiflows, and one-parameter convolution semigroups of probability measures. In this book, all results on transition probabilities that have been published by the author between 2004 and 2008 are extended to transition functions. The proofs of the results obtained are new.
For transition functions that satisfy very general conditions the book describes an ergodic decomposition that provides relevant information on the structure of the corresponding set of invariant probabilities. Ergodic decomposition means a splitting of the state space, where the invariant ergodic probability measures play a significant role. Other topics covered include: characterizations of the supports of various types of invariant probability measures and the use of these to obtain criteria for unique ergodicity, and the proofs of two mean ergodic theorems for a certain type of transition functions.
The book will be of interest to mathematicians working in ergodic theory, dynamical systems, or the theory of Markov processes. Biologists, physicists and economists interested in interacting particle systems and rigorous mathematics will also find this book a valuable resource. Parts of it are suitable for advanced graduate courses. Prerequisites are basic notions and results on functional analysis, general topology, measure theory, the Bochner integral and some of its applications.

Caracteristici

Suitable for advanced graduate courses Includes proofs of results published for the first time Includes examples and informative appendices Includes supplementary material: sn.pub/extras