Metastability of diffusion processes in narrow tubes

TL;DR

This paper investigates the metastable behavior of diffusion processes in narrow tubes, focusing on entropic barriers, characteristic time scales, and the narrow escape problem, providing detailed asymptotic analysis and estimates.

Contribution

It introduces a detailed analysis of metastability in narrow tubes, identifying multiple time scales and refining understanding of the narrow escape problem with new estimates.

Findings

Identification of a sequence of characteristic time scales for metastability.

Asymptotic characterization of diffusion behavior at different time scales.

New estimates for exit locations and conditional exit time distributions.

Abstract

We study the metastable behavior of diffusion processes in narrow tube domains, where the metastability is induced by entropic barriers. We identify a sequence of characteristic time scales $\{T_\epsilon^i\}_{1 \leq i \leq \abs{V'}}$ and characterize the asymptotic behavior of the diffusion process both at intermediate time scales and at the first critical time scale. Our analysis relies on a refined understanding of the narrow escape problem in domains with bottlenecks, in particular on estimates for the exit place and on the conditional distribution of the exit time given the exit place, results that may be of independent interest.