Metastability of interacting stochastic systems with countably many metastable states: beyond positive recurrence

Abstract

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.