渐近指数首中时和亚稳定性：不可逆的路径方法
Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility
摘要：
研究了马尔可夫过程从一个参考构形$x_0$或它的吸引区域到目标集$G$的首中时.配置$x_0$可以对应于（Meta）稳定井的底部，而目标$G$可以是井的一组鞍（出口）点，或者一组另外的（Meta）稳定配置。三种类型的结果报告：（1）一个一般性的理论，发展的基础上的路径明智的方法亚稳，它有三个重要属性。首先，它是一般性的，因为它不假设过程的可逆性，不只关注罕见事件的首中时，也不假设特定的起始措施。其次，它只依赖于自然假设，即$G$的平均首中时渐近长于$x_0$或$G$的平均重现时间。第三，尽管其数学上的简单性，该方法产生精确和明确的界限上的修正指数。(2)我们比较和相关的文献中提出的不同的亚稳态条件，以消除潜在的混乱来源。这是特别相关的无限体积系统的演变，其治疗取决于相关参数（温度，字段）是否以及如何调整。(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地小于平均时间尺度.这种控制对于具有无界状态空间的系统特别相关，其中导致从亚稳态退出的成核可以发生在体积中的任何地方。我们提供了自然的充分条件，这个早期的指数举行的递归时间，并表明它导致估计的密度函数。
英文摘要：
We study the hitting times of Markov processes to target set $G$, starting from a reference configuration $x_0$ or its basin of attraction. The configuration $x_0$ can correspond to the bottom of a (meta)stable well, while the target $G$ could be either a set of saddle (exit) points of the well, or a set of further (meta)stable configurations. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which has three important attributes. First, it 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. Second, it relies only on the natural hypothesis that the mean hitting time to $G$ is asymptotically longer than the mean recurrence time to $x_0$ or $G$. Third, 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 so to eliminate potential sources of confusion. This is specially relevant for evolutions of infinite-volume systems, whose treatment depends on whether and how relevant parameters (temperature, fields) are adjusted. (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.
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