Critical growth of a semi-linear process

This paper is motivated by the modelling of leaching of bacteria through soil. A semi-linear process X-t(-) maybe 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 It 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.