Open Access
2007 Detecting a Local Perturbation in a Continuous Scenery
Heinrich Matzinger, Serguei Popov
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Electron. J. Probab. 12: 637-660 (2007). DOI: 10.1214/EJP.v12-409


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.


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Heinrich Matzinger. Serguei Popov. "Detecting a Local Perturbation in a Continuous Scenery." Electron. J. Probab. 12 637 - 660, 2007.


Accepted: 13 May 2007; Published: 2007
First available in Project Euclid: 1 June 2016

zbMATH: 1127.60094
MathSciNet: MR2318405
Digital Object Identifier: 10.1214/EJP.v12-409

Primary: 60J65
Secondary: 60K37

Keywords: Brownian motion , detecting defects in sceneries seen along random walks , localization test , Poisson process

Vol.12 • 2007
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