Quasi-Stationary Distributions of Non-Absorbing Markov Chains

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Roberto Fernandez, Francesco Manzo, Matteo Quattropani, Elisabetta Scoppola
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引用次数: 0

Abstract

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.

非吸收马尔可夫链的拟平稳分布
考虑具有有限状态空间的可逆遍历马尔可夫链,引入了不需要任何吸收态存在的拟平稳分布的新概念。在我们的设置中,将吸收集的命中时间替换为最优强平稳时间,表示“平稳分布的命中时间”。一方面,我们证明了拟平稳分布的概念对应于Yaglom极限的自然推广。另一方面,与经典的准平稳分布相似,我们证明了它可以用底层马尔可夫核的特征向量来表示,因此它可以进行几何解释。此外,我们恢复了表征准平稳分布和亚稳态系统的通常指数行为。我们还提供了一些简单的例子,这些例子表明现象学比吸收案例更丰富。最后,我们提出了一些反例,表明对可逆性的假设一般不能削弱。
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来源期刊
Journal of Statistical Physics
Journal of Statistical Physics 物理-物理:数学物理
CiteScore
3.10
自引率
12.50%
发文量
152
审稿时长
3-6 weeks
期刊介绍: The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.
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