Wide-ranging interpretations of the cycle representations of Markov processes (Q2769683)

Stochastic processes of Markovian type which are homogeneous (with discrete or continuous parameter) and admit invariant measures can be defined by directed cycles and are called cycle (Markov) processes. The corresponding equations that describe the finite-dimensional distribution are called cycle representations and they are proved to be finite or denumerable linear equations of directed cycles \(c\) with real coefficients \(w_c\) (cycle weights). The corresponding collection \(\{c,w_c\}\) of cycles and weights is of particular interest for the Kolmogorov's famous algorithmic approach to chaos, with its profound connections between probability, dynamics, information, and complexity.NEWLINENEWLINENEWLINEThe aim of the present paper is to reveal several interesting interpretations of the cycle representation of Markov processes, as they were recently studied by the author and co-authors. The homologic, the algebraic, the Banach space, the measure-theoretic, and the stochastic interpretations of the cycle representation of Markov processes are proved to express, each one and altogether, a genuine law of real phenomena. The versatility of these interpretations as orthogonality equations, as linear expressions on cycles, as Fourier series, as semigroup equations etc., is consequently motivated by the existence of algebraic-topological principles in the fundamentals of cycle representations of Markov processes.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].