Explore the dynamic advancements in Markov process cycle representations with the 2nd Edition of “Cycle Representations of Markov Processes.” This comprehensive guide unveils diverse interpretations, from homologic decompositions and orthogonality equations to Fourier series and measure disintegrations, revealing the profound connection between abstract principles and real-world phenomena. Discover how algebraic-topological foundations liberate Markovian modeling, paving the way for intuitive and constructive approaches. Delve into the ruling role of cycles in partitioning finite-dimensional distributions, modernizing Poincaré’s spirit of describing randomness. This edition offers minimal descriptions on cycles, echoing Kolmogorov’s entropic complexity, and provides improved recurrence criteria, making it an essential resource for researchers and practitioners in stochastic modeling and applied probability.