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Critical growth of a semi-linear process

  • Ilya Molchanov (a1), Vadim Shcherbakov (a2) and Sergei Zuyev (a3)


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.


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Postal address: Department of Mathematical Statistics and Actuarial Science, University of Berne, Sidlerstr. 5, CH-3012 Berne, Switzerland. Email address:
∗∗ 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:
∗∗∗ Postal address: Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G1 1XH, UK. Email address:


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