Stochastic backtrackings

A particle moves with constant velocity in the positive direction. It is also subject to randomly occuring instantaneous negative displacements called backtrackings. The magnitudes of these displacements are independent and identically distributed random variables, and it is assumed that the net velocity of the pa?ticle is still positive. The backtrackings are registered but the times of their occurence are not. The resulting collection of overlapping intervals on the real line is studied as a stochastic process evolving in space. A fairly simple Markovian representation is derived and the stationary distribution obtained. Intimate connections with certain linear immigration-birth-death processes are exhibited. Some comments are made concerning estimation and sampling procedures and the identifiability of the displacement distribution from limited information.