Self-Switching Markov Chains: Emerging dominance phenomena

The law of many dynamical systems changes with the evolution of the system. These changes are often associated with the occurrence of certain events whose time of occurrence depends on the trajectory of the system itself. Dynamics that take longer to change will be observed more frequently and may dominate in the long run (the only ones observed). This article proposes a Markov chain model, called Self-Switching Markov Chain, in which the emergence of dominant dynamics can be rigorously addressed. We present conditions and scaling under which we observe with probability one only the subset of dominant dynamics.