On the passage properties of the gradual capture of a diffusive particle in the presence of drift

Abstract

We investigate a stochastic process consisting of a two-dimensional particle with anisotropic diffusion, mortality rate and a drift velocity, in the presence of an absorbing boundary. After the particle has encountered the boundary, the process is restarted with updated values of its diffusion constants and drift velocity. We then derive the joint probability that, after $M$ encounters, the particle is absorbed at a point of the boundary at a given time and show that, under certain conditions, the eventual hitting probability corresponds to a Bessel distribution. In the context of the man-mosquitoes problem, the mosquito is able to gradually capture the man, after which the mosquito follows a diffusion process with no drift. Our results are compared thoroughly with Monte Carlo simulations showing excellent agreement.