Detecting a local perturbation in a continuous scenery

A continuous one-dimensional scenery is a double-infinite sequence of points (thought of as locations of bells) in R. Assume that a scenery X is observed along the path of a Brownian motion in the following way: when the Brownian motion encounters a bell different from the last one visited, we hear a ring. The trajectory of the Brownian motion is unknown, whilst the scenery X is known except in some finite interval. We prove that given only the sequence of times of rings, we can a.s. reconstruct the scenery X entirely. For this we take the scenery X to be a local perturbation of a Poisson scenery X ′. We present an explicit reconstruction algorithm. This problem is the continuous analog of the “detection of a defect in a discrete scenery” as studied by Kesten [13] and Howard [9, 10]. Many of the essential techniques used with discrete sceneries do not work with continuous sceneries.