By Sophia L. Kalpazidou

This e-book offers an unique and systematic account of a category of stochastic procedures often called cycle (or circuit) methods, so known as simply because they're outlined by means of directed cycles. those methods have exact and demanding houses during the interplay among the geometric houses of the trajectories and the algebraic characterization of the finite-dimensional distributions. an incredible software of this process is the recent perception it offers into Markovian dependence and electric networks. specifically, it presents a wholly new method of Markov approaches and countless electric networks, and their purposes in subject matters as assorted as random walks, ergodic idea, dynamical platforms, capability thought, conception of matrices, algebraic topology, complexity conception, the class of Riemann surfaces, and operator theory.The writer surveys the 3 relevant advancements in cycle conception: the cycle-decomposition formulation and its relation to the Markov technique; entropy creation and the way it can be used to degree how some distance a strategy is from being reversible; and the way a finite recurrent stochastic matrix will be outlined by way of a rotation of the circle and a partition whose parts include finite unions of circle-arcs. The cycle representations were complicated after the booklet of the 1st variation to many instructions, which display wide-ranging interpretations like homologic decompositions, orthogonality equations, Fourier sequence, semigroup equations, disintegration of measures, and so forth. the flexibility of those interpretations is for that reason encouraged by way of the life of algebraic-topological ideas within the basics of the cycle representations,which elaborates the normal view at the Markovian modelling to new intuitive and confident methods.

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**Extra resources for Cycle Representations of Markov Processes**

**Example text**

One reason for choosing a deterministic algorithm is that the correspondence P → C becomes nearly one-to-one, that is, the class C approximates the probabilistic one. It is proved below that the class C may be the limit of an increasing sequence n C of finite classes of overlapping directed circuits. The one-to-one correspondences P → C are particularly important for plenty of problems arising in various fields. For example, we may refer here to the so-called coding problem arising in the context of dynamical systems, that in turn leads to the problem of mapping stochastic matrices into partitions.

Then w(i, j) = wc Jc (i, j), c∈C for all i, j ∈ S. By arguing analogously for w and choosing as representative class of circuits to be C = {c : c is the reversed circuit of c, c ∈ C }, we obtain wc = wc for any c ∈ C , and the decomposition of w by (C , wc ). The proof is complete. 5) are called, by Kalpazidou ((1993a), (1994a)), cycle generating equations. They can be used as an implicit definition of the directed circuits. Accordingly, we shall say that the functions w and w are respectively represented by (C, wc ) and (C , wc ).

We are now prepared to prove a deterministic circuit decomposition of P following S. Kalpazidou (1993c). As was already mentioned, we are interested in representing the chain ξ by a class (C, wc ) provided by a deterministic algorithm such that the correspondence P → C becomes nearly one-to-one, that is, C will approximate the collection of all the circuits occurring along almost all the sample paths. Then the trivial case of the class containing only the circuits of period two will be avoided.