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.