Cycle Representations of Markov Processes / Edition 2

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This book is a prototype providing new insight into Markovian dependence via the cycle decompositions. It presents a systematic account of a class of shastic processes known as cycle (or circuit) processes - so-called because they may be defined by directed cycles. These processes have special and important properties through the interaction between the geometric properties of the trajectories and the algebraic characterization of the Markov process. An important application of this approach is the insight it provides to electrical networks and the duality principle of networks. In particular, it provides an entirely new approach to infinite electrical networks and their applications in topics as diverse as random walks, the classification of Riemann surfaces, and to operator theory.

The second edition of this book adds new advances to many directions, which reveal wide-ranging interpretations of the cycle representations like homologic decompositions, orthogonality equations, Fourier series, semigroup equations, and disintegration of measures. The versatility of these interpretations is consequently motivated by the existence of algebraic-topological principles in the fundamentals of the cycle representations. This book contains chapter summaries as well as a number of detailed illustrations.

Review of the earlier edition:

"This is a very useful monograph which avoids ready ways and opens new research perspectives. It will certainly stimulate further work, especially on the interplay of algebraic and geometrical aspects of Markovian dependence and its generalizations."

Math Reviews.

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Editorial Reviews

From the Publisher

From the reviews of the second edition:

"The first edition of this book [S. Kalpazidou, Cycle representations of Markov processes, Springer, New York, 1995; MR1336140 (96g:60002)] has been reviewed by this reviewer. This second edition amplifies Part I of the first edition, which consisted of 7 chapters, by adding four chapters: Chapter 8. Cycloid Markov processes; Chapter 9. Markov processes on Banach spaces on cycles; Chapter 10. The cycle measures; Chapter 11. Wide-ranging interpretations of the cycle representations of Markov processes. Also, there is a new Section 3.6 devoted to induced circuit chains in Part I, and a new Section 1.4 on Derriennic recurrence criteria in terms of weighted circuits in Part II. Besides, improvements have been introduced at different places (without completely eliminating linguistic slips).

The reviewer is glad that this second edition confirms his expectations concerning the applicability potential of the cycle representation topic. The book under review is indeed mainly motivated by the many applications of cycle representations which occurred in different fields, after the publication of the first edition. " (M. Iosifescu, Mathematical Reviews)

"The main purpose of the second edition … is to give systematic and unified exposition of shastic processes of the Markovian type, homogeneous and with either discrete or continuous parameter, which, under an additional assumption concerning the existence of invariant measures, can be defined by directed cycles or circuits. … The book will be useful for experts in representation theory of Markov processes." (Anatoliy Swishchuk, Zentralblatt MATH, Vol. 1113 (15), 2007)

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Product Details

  • ISBN-13: 9780387291666
  • Publisher: Springer New York
  • Publication date: 5/24/2006
  • Series: Stochastic Modelling and Applied Probability Series, #28
  • Edition description: 2nd ed. 2006
  • Edition number: 2
  • Pages: 304
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.75 (d)

Table of Contents

I Fundamentals of the cycle representations of Markov processes
1 Directed circuits 3
2 Genesis of Markov chains by circuits : the circuit chains 17
3 Cycle representations of recurrent denumerable Markov chains 29
4 Circuit representations of finite recurrent Markov chains 55
5 Continuous parameter circuit processes with finite state space 73
6 Spectral theory of circuit processes 93
7 Higher-order circuit processes 101
8 Cycloid Markov processes 131
9 Markov processes on Banach spaces on cycles 145
10 The cycle measures 163
11 Wide-ranging interpretations of the cycle representations of Markov processes 187
II Applications of the cycle representations
1 Stochastic properties in terms of circuits 207
2 Levy's theorem concerning positiveness of transition probabilities 225
3 The rotational theory of Markov processes 231
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