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Critical growth of a semi-linear process

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Ilya Molchanov ,
Vadim Shcherbakov  and
Sergei Zuyev
Ilya Molchanov*
Affiliation:
University of Berne
Vadim Shcherbakov*
Affiliation:
University of Glasgow
Sergei Zuyev*
Affiliation:
University of Strathclyde
*
∗ Postal address: Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland. Email address: ilya@stat.unibe.ch
∗∗ On leave from the Laboratory of Large Random Systems, Faculty of Mathematics and Mechanics, Moscow State University. Current address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands. Email address: v.shcherbakov@cwi.nl
∗∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK. Email address: sergei@stams.strath.ac.uk
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Abstract

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X t − may be used to describe the soil-drying process between rain showers. This is a backward recurrence time process that corresponds to the renewal process of instances of rain. If a bacterium moves according to another process h, then the fact that h(t) stays above X t − means that the bacterium never hits a dry patch of soil and so survives. We describe a critical behaviour of h that separates the cases when survival is possible with a positive probability from the cases when this probability vanishes. An explicit formula for the survival probability is obtained in case h is linear and rain showers follow a Poisson process.

Keywords

Borel-Cantelli lemmapoint processrenewal processsemi-linear process

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 2 , June 2004 , pp. 355 - 367
Copyright
Copyright © Applied Probability Trust 2004 

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