Breaking the chain

We consider the motion of a Brownian particle in R , moving between a particle fixed at the origin and another moving deterministically away at slow speed ε > 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 < 2 a . We say that the chain of particles breaks on the left- or right-hand side when the middle particle is at a distance greater than 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 ε = ε ( σ ) and σ > 0 is the noise intensity.