Beyond Ergodicity: Persistent Dynamics in Non-Stationary Markov Chains

This paper investigates the behavior of non-stationary Markov chains, systems for which the foundational assumption of ergodicity does not hold. Traditional Markov chain analysis relies on the existence of a unique, time-invariant stationary distribution, implying that time averages converge to ensemble averages. However, in many real-world complex systems, such as those found in finance, climate science, and biology, transition probabilities are inherently time-dependent, precluding convergence to a single equilibrium. This work moves beyond the ergodic framework to explore the concept of persistent dynamics—recurrent, metastable, or quasi-stationary behaviors that characterize the long-term evolution of these non-stationary systems. We develop and apply a suite of analytical and computational tools, including time-windowed spectral analysis of transition operators and information-theoretic measures, to identify and quantify these persistent patterns. Through a series of simulated case studies involving periodic, slowly drifting, and abruptly shifting environments, we demonstrate how these methods can reveal underlying structures, such as limit cycles and evolving quasi-stationary distributions. The results show that even in the absence of a global equilibrium, non-stationary Markov chains exhibit rich, predictable, and meaningful dynamic behaviors. This research provides a robust methodological framework for analyzing complex systems that operate far from stationarity, offering deeper insights into their adaptability, resilience, and regime-shifting nature.