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Coarse resistance tree methods for stochastic stability analysis

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Abstract

Emergent behavior in natural and manmade systems can often be characterized by the limiting distribution of a class of Markov processes termed regular perturbed processes. Resistance trees have gained popularity as a computationally efficient way to characterize the support of the limiting distribution; however, there are three main limitations of this approach. First, it requires finding a minimum weight spanning tree for each state in a potentially large state space. Second, perturbations to transition probabilities must decay at an exponentially smooth rate. Lastly, the approach is shown to hold purely in the context of finite Markov chains. In this paper we seek to address these limitations by developing new tools for characterizing the limiting distribution. First, we provide necessary conditions for stochastic stability via a coarse, and less computationally intensive, state space analysis. Next, we identify necessary conditions for stochastic stability when smooth convergence requirements are relaxed. Finally, we establish similar tools for stochastic stability analysis in Markov chains over a continuous state space.