Abrupt Convergence and Escape Behavior for Birth and Death Chains

TL;DR

This paper explores the relationship between abrupt convergence (cut-off) and escape behavior in birth-death chains, showing they are connected through time reversal in systems with energy wells.

Contribution

It establishes a link between cut-off phenomena and escape behavior in birth-death chains, especially under energy landscape assumptions, revealing their time-reversal symmetry.

Findings

Demonstrates cut-off and escape phenomena occur simultaneously under certain conditions.

Shows the law of escape trajectories is the time reverse of cut-off paths in reversible systems.

Provides conditions under which energy wells lead to both phenomena.

Abstract

We link two phenomena concerning the asymptotical behavior of stochastic processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape behavior usually associated to exit from metastability. The former is characterized by convergence at asymptotically deterministic times, while the convergence times for the latter are exponentially distributed. We compare and study both phenomena for discrete-time birth-and-death chains on Z with drift towards zero. In particular, this includes energy-driven evolutions with energy functions in the form of a single well. Under suitable drift hypotheses, we show that there is both an abrupt convergence towards zero and escape behavior in the other direction. Furthermore, as the evolutions are reversible, the law of the final escape trajectory coincides with the time reverse of the law of cut-off paths. Thus, for evolutions defined by one-dimensional energy wells with sufficiently steep walls, cut-off and escape behavior are related by time inversion.