具有可数多个亚稳态的相互作用随机系统的亚稳态：超越正常回归
Metastability of interacting stochastic systems with countably many metastable states: beyond positive recurrence
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- Yonsei University（延世大学）
- Inha University（仁荷大学）
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中文总结 AI 辅助
本文将预解式亚稳态理论扩展至可数多亚稳态、非零常返的马尔可夫过程，应用于两类非紧随机系统，验证了相关收敛性与可忽略性，拓宽了理论适用范围。
中文摘要 AI 辅助
亚稳态通常针对具有有限个亚稳态的系统进行建模，且大多处于正常回归（positive recurrence）的设定中。本文将用于研究亚稳态的预解式（resolvent）框架扩展到具有可数多个亚稳态的马尔可夫过程，从而涵盖零常返（null-recurrent）和暂态（transient）动力学。对于局部紧波兰空间（Polish spaces）上的一类过程，我们在温和的边界正则性假设下证明，微观预解式解的渐近平坦性，结合两个紧性条件，等价于投影迹过程依分布收敛到极限马尔可夫链，以及在亚稳态集合外花费的时间可忽略不计。我们将该框架应用于两个非紧随机系统：首先，研究可数无限、一致局部有限图上的凝聚包含过程，在该过程可能为零常返或暂态的设定下，证明凝聚体位置收敛到底层图上的加权随机游走，同时远离完全凝聚构型的时间可忽略；其次，研究具有可数多个稳定平衡点的小噪声一维朗之万扩散（Langevin diffusion），且不假设其遍历性，在与最小能垒相关的Eyring-Kramers时间尺度下，我们建立每个势阱内的局部平衡、势阱索引过程收敛到整数集（$\boldsymbol{\text{Z}}$）上的显式近邻马尔可夫链，以及阱间 excursion 的可忽略性。这些结果共同拓宽了基于预解式的亚稳态理论的适用范围，超越了有限亚稳态空间和正常回归的范畴。
英文摘要
Metastability is typically formulated for systems with finitely many metastable states, most often in positively recurrent settings. In this article, we extend the resolvent framework for metastability to Markov processes with countably many metastable states, thereby encompassing null-recurrent and transient dynamics. For a family of processes on locally compact Polish spaces, we prove, under a mild boundary regularity assumption, that the asymptotic flatness of microscopic resolvent solutions, supplemented by two compactness conditions, is equivalent to convergence in law of the projected trace processes to a limiting Markov chain and to the negligibility of the time spent outside the metastable sets. We apply this framework to two non-compact stochastic systems. First, we study a condensing inclusion process on a countably infinite, uniformly locally finite graph, in a setting where the process may be null recurrent or transient. We prove that the condensate location converges to a weighted random walk on the underlying graph, while the time spent away from the fully condensed configurations is negligible. Second, we study a small-noise one-dimensional Langevin diffusion with countably many stable equilibria, without assuming ergodicity. In the Eyring-Kramers time scale associated with the minimal energy barrier, we establish local equilibration inside each well, convergence of the well-index process to an explicit nearest-neighbor Markov chain on $\mathbb Z$, and negligibility of inter-well excursions. Together, these results broaden the scope of resolvent-based metastability theory beyond finite metastable state spaces and positive recurrence.
