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<p>We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the “hitting time of the stationary distribution”. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the <i>Yaglom limit</i>. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.</p>

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Quasi-Stationary Distributions of Non-Absorbing Markov Chains

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  • 出版模式：

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  • 发表日期：

    2025年3月14日

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  • Roberto Fernandez,
  • Francesco Manzo,
  • Matteo Quattropani,
  • Elisabetta Scoppola

摘要

We consider reversible ergodic Markov chains with finite state space, and we introduce a new notion of quasi-stationary distribution that does not require the presence of any absorbing state. In our setting, the hitting time of the absorbing set is replaced by an optimal strong stationary time, representing the “hitting time of the stationary distribution”. On the one hand, we show that our notion of quasi-stationary distribution corresponds to the natural generalization of the Yaglom limit. On the other hand, similarly to the classical quasi-stationary distribution, we show that it can be written in terms of the eigenvectors of the underlying Markov kernel, and it is therefore amenable of a geometric interpretation. Moreover, we recover the usual exponential behavior that characterizes quasi-stationary distributions and metastable systems. We also provide some toy examples, which show that the phenomenology is richer compared to the absorbing case. Finally, we present some counterexamples, showing that the assumption on the reversibility cannot be weakened in general.