Cycle Representations of Markov Processes

Sophia L. Kalpazidou - Collection Stochastic Modelling and Applied Probability

302 pages, parution le 31/05/2006 (2^eme édition)

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Résumé

This book presents an original and systematic account of a class of stochastic 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 finite-dimensional distributions.

An important application of this approach is the new insight it provides into Markovian dependence and electrical networks. In particular, it provides an entirely new approach to Markov processes and infinite electrical networks, and their applications in topics as diverse as random walks, ergodic theory, dynamical systems, potential theory, theory of matrices, algebraic topology, complexity theory, the classification of Riemann surfaces, and operator theory.

The author surveys the three principal developments in cycle theory: the cycle-decomposition formula and its relation to the Markov process; entropy production and how it may be used to measure how far a process is from being reversible; and how a finite recurrent stochastic matrix may be defined by a rotation of the circle and a partition whose elements consist of finite unions of circle-arcs.

The cycle representations have been advanced after the publication of the first edition to many directions, which reveal wide-ranging interpretations like homologic decompositions, orthogonality equations, Fourier series, semigroup equations, disintegration of measures, and so on. The versatility of these interpretations is consequently motivated by the existence of algebraic-topological principles in the fundamentals of the cycle representations, which elaborates the standard view on the Markovian modelling to new intuitive and constructive approaches.

Sommaire

Preface to the Second Edition
Preface
Acknowledgments
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
- List of Notations
Bibliography
Index

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	PAPIER
Éditeur(s)	Springer
Auteur(s)	Sophia L. Kalpazidou
Collection	Stochastic Modelling and Applied Probability
Parution	31/05/2006
Édition	2^eme édition
Nb. de pages	302
Format	16 x 24
Couverture	Relié
Poids	603g
Intérieur	Noir et Blanc
EAN13	9780387291666
ISBN13	978-0-387-29166-6