Detecting a Local Perturbation in a Continuous Scenery

Abstract

A continuous one-dimensional scenery is a double-infinite sequence of points (thought of as locations of <em>bells</em>) 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". Many of the essential techniques used with discrete sceneries do not work with continuous sceneries.