Breaking the chain

Abstract

We consider the motion of a Brownian particle in $\mathbb{R}$, moving between a particle fixed at the origin and another moving deterministically away at slow speed $\epsilon >0$. The middle particle interacts with its neighbours via a potential of finite range $b>0$, with a unique minimum at $a>0$, where $b<2a$. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is greater than a distance $b$ from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where $\epsilon =\epsilon (\sigma )$ and $\sigma >0$ is the noise intensity.

13 pages, 2 figures. v2: Corrected a mistake in proof of second part of main theorem