Cycle Representations of Markov Processes

This book provides new insight into Markovian dependence via the cycle decompositions. It presents a systematic account of a class of stochastic processes known as cycle (or circuit) processes - so-called because they may be defined by directed cycles. An important application of this approach is the insight it provides to electrical networks and the duality principle of networks. This expanded second edition adds new advances, which reveal wide-ranging interpretations of cycle representations such as homologic decompositions, orthogonality equations, Fourier series, semigroup equations, and disintegration of measures. The text includes chapter summaries as well as a number of detailed illustrations.

Excellent pedagogical methods Promotes original ideas and methods and developes intuition Chapter summaries as well as a number of detailed illustrations included

Klappentext

The cycle representations of Markov processes have been advanced after the publication of the ?rst edition to many directions. One main purpose of these advances was the revelation of wide-ranging interpretations of the - cle decompositions of Markov processes such as homologic decompositions, orthogonality equations, Fourier series, semigroup equations, disinteg- tions of measures, and so on, which altogether express a genuine law of real phenomena. The versatility of these interpretations is consequently motivated by the existence of algebraic topological principles in the fundamentals of the - clerepresentationsofMarkovprocesses,whicheliberatesthestandardview on the Markovian modelling to new intuitive and constructive approaches. For instance, the ruling role of the cycles to partition the ?nite-dimensional distributions of certain Markov processes updates Poincare s spirit to - scribing randomness in terms of the discrete partitions of the dynamical phase state; also, it allows the translation of the famous Minty s painting lemma (1966) in terms of the stochastic entities. Furthermore, the methods based on the cycle formula of Markov p- cesses are often characterized by minimal descriptions on cycles, which widelyexpressaphilosophicalanalogytotheKolmogoroveanentropicc- plexity. For instance, a deeper scrutiny on the induced Markov chains into smallersubsetsofstatesprovidessimplerdescriptionsoncyclesthanonthe stochastic matrices involved in the taboo probabilities. Also, the rec- rencecriteriaon cyclesimprovepreviousconditionsbased on thestochastic matrices, and provide plenty of examples.

Zusammenfassung

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 stochastic 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)

Inhalt
Fundamentals of the Cycle Representations of Markov Processes.- Directed Circuits.- Genesis of Markov Chains by Circuits: The Circuit Chains.- Cycle Representations of Recurrent Denumerable Markov Chains.- Circuit Representations of Finite Recurrent Markov Chains.- Continuous Parameter Circuit Processes with Finite State Space.- Spectral Theory of Circuit Processes.- Higher-Order Circuit Processes.- Cycloid Markov Processes.- Markov Processes on Banach Spaces on Cycles.- The Cycle Measures.- Wide-Ranging Interpretations of the Cycle Representations of Markov Processes.- Applications of the Cycle Representations.- Stochastic Properties in Terms of Circuits.- Lévy's Theorem Concerning Positiveness of Transition Probabilities.- The Rotational Theory of Markov Processes.