Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility
Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility
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渐近指数命中时间和亚稳态：一种不可逆的路径方法
DOI：
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发表时间：
2014
期刊：
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通讯作者：
E. Scoppola
中科院分区：
文献类型：
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作者：
R. Fernández;F. Manzo;F. Nardi;E. Scoppola
我们从参考配置 x0 或其吸引盆开始，研究马尔可夫过程对目标集 G 的命中时间，并讨论其与亚稳态的关系。报告了三种类型的结果：（1）基于亚稳态的路径方式方法开发了通用理论，其通用性在于它不假设过程的可逆性，不只关注罕见事件的命中时间，也不假设特定的起始测量。我们仅考虑自然假设，即 G 的平均命中时间渐近地长于参考配置 x0 或 G 的平均重现时间。尽管其数学简单，但该方法在指数校正上产生了精确且明确的界限。 (2) 我们比较并关联文献中提出的不同亚稳态条件。这与无限体积系统的演化特别相关。 (3)我们引入早期渐近指数行为的概念来控制时间尺度渐近小于平均时间尺度。这种控制对于具有无界状态空间的系统特别相关，其中导致退出亚稳态的成核可能发生在体积中的任何位置。我们为这种早期指数性的重现时间提供了自然充分的条件，并表明它导致了概率密度函数的估计。
We study the hitting times of Markov processes to target set G, starting from a reference configuration x0 or its basin of attraction and we discuss its relation to metastability. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which is general in that it does not assume reversibility of the process, does not focus only on hitting times to rare events and does not assume a particular starting measure. We consider only the natural hypothesis that the mean hitting time to G is asymptotically longer than the mean recurrence time to the refernce configuration x0 or G. Despite its mathematical simplicity, the approach yields precise and explicit bounds on the corrections to exponentiality. (2) We compare and relate different metastability conditions proposed in the literature. This is specially relevant for evolutions of infinite-volume systems. (3) We introduce the notion of early asymptotic exponential behavior to control time scales asymptotically smaller than the mean-time scale. This control is particularly relevant for systems with unbounded state space where nucleations leading to exit from metastability can happen anywhere in the volume. We provide natural sufficient conditions on recurrence times for this early exponentiality to hold and show that it leads to estimations of probability density functions.