Gradual diffusive capture: slow death by many mosquito bites

Abstract

We study the dynamics of a single diffusing particle (a ‘man’) with diffusivity DM that is attacked by another diffusing particle (a ‘mosquito’) with fixed diffusivity Dm. Each time the mosquito meets and bites the man, the diffusivity of the man is reduced by a fixed amount, while the diffusivity of the mosquito is unchanged. The mosquito is also displaced by a small distance ±a with respect to the man after each encounter. The man is defined as dead when DM reaches zero. At the moment when the man dies, his probability distribution of displacements x is given by a Cauchy form, which asymptotically decays as x−2, while the distribution of times t when the man dies decays asymptotically as t−3/2, which has the same form as the one-dimensional first-passage probability.