This book is a prototype providing 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. 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.
General
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