Cycle Representations of Markov Processes / Edition 2

Cycle Representations of Markov Processes / Edition 2

by Sophia L. Kalpazidou
     
 

ISBN-10: 0387291660

ISBN-13: 9780387291666

Pub. Date: 05/24/2006

Publisher: Springer New York

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

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Overview

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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Product Details

ISBN-13:
9780387291666
Publisher:
Springer New York
Publication date:
05/24/2006
Series:
Stochastic Modelling and Applied Probability Series, #28
Edition description:
2nd ed. 2006
Pages:
304
Product dimensions:
6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

IFundamentals of the cycle representations of Markov processes
1Directed circuits3
2Genesis of Markov chains by circuits : the circuit chains17
3Cycle representations of recurrent denumerable Markov chains29
4Circuit representations of finite recurrent Markov chains55
5Continuous parameter circuit processes with finite state space73
6Spectral theory of circuit processes93
7Higher-order circuit processes101
8Cycloid Markov processes131
9Markov processes on Banach spaces on cycles145
10The cycle measures163
11Wide-ranging interpretations of the cycle representations of Markov processes187
IIApplications of the cycle representations
1Stochastic properties in terms of circuits207
2Levy's theorem concerning positiveness of transition probabilities225
3The rotational theory of Markov processes231

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