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On search for a Brownian target

Published online by Cambridge University Press:  14 July 2016

Thomas L. Corwin
Thomas L. Corwin*
Affiliation:
The Johns Hopkins University
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Abstract

A target is assumed to move according to a Wiener process in ℝ1. The probability of detecting the target is computed in terms of the search effort which accumulates along the target's path. Under certain independence assumptions this probability is given by the expectation of an exponential functional of the process. It is shown in this note that the failure probability in a search for a Wiener target is asymptotically proportional to , where T is the accumulated time spent searching. The asymptotic failure probability is also shown to be independent of the position of the search in ℝ1. In a similar fashion, it is shown that the failure probability in a search for a Wiener target in ℝ2 is independent of the position of the search and asymptotically proportional to (c log T + l)–1, c >0

Keywords

SEARCHMOVING TARGETEXPONENTIAL FORMULA
Type
Short Communications
Information
Journal of Applied Probability , Volume 17 , Issue 1 , March 1980 , pp. 243 - 247
Copyright
Copyright © Applied Probability Trust 1980 

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References

[1] Hellman, O. B. (1970) On the effect of search upon the probability distribution of a target whose motion is a diffusion process. Ann. Math. Statist. 41, 1717–1724.CrossRefGoogle Scholar
[2] Kallianpur, G. and Robbins, H. (1953) Ergodic property of the Brownian motion process. Proc. Nat. Acad. Sci. USA 39, 525–533.CrossRefGoogle ScholarPubMed
[3] Saretsalo, L. (1973) On the optimal search for a target whose motion is a Markov process. J. Appl. Prob. 10, 847–856.CrossRefGoogle Scholar