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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 Xt 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 Xt 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.

Corresponding author
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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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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