Critical growth of a semi-linear process

Reference:

Shcherbakov, V., Molchanov, I. and Zuyev, S., 2004. Critical growth of a semi-linear process. Journal of Applied Probability, 41, 355--367.

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Abstract

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.

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