Investigating the Architecture of Coupled Markov Chains: From Foundational Traits to Unified Convergence

This paper systematically investigates the structural properties and convergence behavior of a class of composite Markov processes formed by coupling multiple Markov subprocesses. The state space is expressed as a direct product space, with each subprocess evolving within its own subspace, while a coupling mechanism integrates them into a global transition kernel. Key analytical focuses include: a) Irreducibility, where necessary and sufficient conditions are established based on the connectivity of subprocesses and the ergodicity of the coupling; b) Ergodicity and stationary distribution, proving the existence of a unique stationary distribution under suitable coupling conditions. Convergence rates are quantified using the spectral gap and the logarithmic Sobolev constant. The proposed framework offers a unified approach for analyzing complex systems such as biomolecular processes, interacting particle systems, and queueing networks.